Answers
C H APTER 1
Prerequisite Skills, pages 2-3
1. a) 7 b) -5 c) 1 1 d) 5 e) 8x + 7 f) - 12x + 7
2. a) 1 b) 10 c) 6 d) 0 e) 24x2 - l 8x + 3 f) 1 8x2 - 9x + 1
3. a) m = 3, b = 2 b) m = -1 b = 1. c) m = 5 b
7
'
2
'
2
d) m = -5, b = - 1 1 e) m = -1, b = 1
9. a) vertical stretch and a reflection in the x-axis b) vertical
compression c) horizontal compression d) horizontal stretch
and a reflection in the y-axis e) reflection in the y-axis
10. a) i) f(x) = x + 5
iii) {x E IR}, {y E IR}
b) i) f (x) = -5(x + 1 )2 - 2
=
2
y = 3x + 5 b) y = 4x + 4 c) y = -4x + 3 1
d) y = - 7x + 1 2
5. a) linear b) neither c ) quadratic
6. a) {x E IR}, {y E IR, y :::o: 1 } b) {x E IR, x ­ - 5}, {y E IR, y of- 0)
4. a)
{
c) x E IR, x :s
i},
(
i)
(x ­2 )
8 . a ) x-intercepts -6, l ; vertex
{
{x e IR}, ye IR, y
iii) {x E IR}, {y E IR, y :s -2}
V1: ·S(X+1l'2-2
{y E IR, y :::o: 0 }
7. Answers may vary. Sample answers:
a) y = -3(x + l )(x - 1 ) b) y = - 2x2 - 3x + 3
c) y = 4 x +
ii)
:::o:
2
2 9
-
}
(-Z,
4
-
c) i ) f(x) = - 3 [2(x
)
289 ; opens up;
8
U)
+
4)] + 6
iU){xeOf,{yeOI
1 1 . a) i) vertical stretch by a factor of 2, reflection in the
x-axis, translation 3 units left, translation 1 unit up
ii) y = - 2(x + 3) + 1 b) i) vertical compression by a factor
of ' translation 2 units down ii) y = x2 - 2
12. vertical stretch by a factor of 3, horizontal stretch by a
factor of 2, reflection in the y-axis, translation 1 unit right,
translation 2 units up
t
b) x-intercepts approximately 3 .29, 4. 71; vertex (4, 1 );
opens down; {xe IR}, {y e IR, y :s 1 }
t
1.1 Power Functions, pages 1 1 -1 4
1 . a ) No. b) Yes. c) Yes. d) Yes. e) No. f) No.
c) x-intercepts approximately - 1.47, 7.47; vertex (3, 5);
opens down; {x E IR}, {ye IR, y :s 5 }
2. a) 4, 5 b) 1 , - 1 c) 2, 8 d) 3, -- e) 0, -5 f) 2, 1
1
4
3. a) i) even ii) negative iii) {xe IR}, {y E IR, y :s 0} iv) line
v) quadrant 3 to quadrant 4 b) i) odd ii) positive
iii) {x E IR}, {y E IR} iv) point v) quadrant 3 to quadrant 1
c) i) odd ii) negative iii) {xe IR}, {ye IR} iv) point
v) quadrant 2 to quadrant 4 d) i) even ii) negative
iii) {x E IR}, {y E IR, y :s 0} iv) line v) quadrant 3 to quadrant 4
e) i) odd ii) negative iii) {x E IR}, {y E IR} iv) point
v) quadrant 2 to quadrant 4
4
End Behaviour
y= Sx,y=
y= -x',y= -0.1x"
1 y= x',y= 2x•
y= -x",y= -9x10
Extends from quadrant 3 to quadrant 1
2
2
Extends from quadrant to quadrant 4
Extends from quadrant to quadrant
}
(l, ...L); opens down; {x
x-intercepts 1, ; vertex
3
6 12
y E IR, y ,; 1
12
{
e)
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Advanced Functions
5. a )
e
Function
4x'
Extends from quadrant 3 to quadrant 4
IR } ,
b) {r E IR, 0 :s r :s 10}; {A E IR, 0 :s A :s lOOn)
c) Answers may vary. Sample answer: similarities-vertex
(0, 0), x-intercept, y-intercept, end behaviour; differences­
domain, range, shape
•
Answers
12. a) Answers may vary. Sample answer: similarities­
quadrant 3 to quadrant 1, domain, range, point symmetry,
shape; difference-shifted vertically
6. a)
{r E IR, 0 :5 r :5 1 0} , {C E IR, 0 :5 C :5 207t }
c) Answers may vary. Sample answer: similarities-end
b)
behaviour; differences-domain, range, shape
7. a) power (cubic) b) exponential c) periodic d) power (constant)
e)
none of these f) none of these g) power (quadratic)
b) Answers may vary. Sample answer: similarities-quadrant 2
to quadrant 1 , domain, line symmetry, shape; differences­
range, shifted vertically
v1=x·
ll J
c) c is a vertical shift of x", n E 1'\J
13. Answers may vary. Sample answer: path of a river: y
b) (x E IR}, {y E IR}, quadrant 3 to quadrant 1, point
symmetry about (0, 0); x-intercept 0, y-intercept 0
V=1
=
IR},
x\
. ;I.
(x E IR}, quadrant 2 to quadrant 1; x-intercept 0,
y-intercept 0
10. Answers may vary. Sample answer: similarities-extend
from quadrant 1 to quadrant 3 (positive leading coefficient),
(x E IR}, (y E IR}, point symmetry about (0, 0); differences­
shape, extend from quadrant 2 to quadrant 4 (negative
leading coefficient)
1 1 . Answers may vary. Sample answer: similarities-extend
from quadrant 2 to quadrant 1 (positive leading coefficient),
domain, line symmetry; differences-shape, range, extend
from quadrant 3 to quadrant 4 (negative leading coefficient)
l):=lJ ?tr-
y = (-x)2"+ 1 has the same graph as y = -x2" + 1 , n is a
1
1
non-negative integer, ( -x)2" + 1 = ( - 1 )2"+ 1 (x)2" + = -x2"+
b)
I
V1=W2
\" /
X=1
E
xW
=
· '
X=1
b)
(x E IR}, {y E IR}; cross-section of a valley: y = x2• {x
{y E IR, y :::0: 0}
14. a) y = ( -x)2" is the same graph as y = x2", n is a
non-negative integer, ( - x)2" = ( - 1 )2"(xf" = x2"
iii)
V1:(·X)'·
V=1
N=2
\
'(: ·12B
Answers
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525
c) Answers may vary. Sample answer: y ( -x)2" has the
same graph as y = x2", n is a non-negative integer,
( - xf" = ( - 1 )2"(x)2" = x2"; y = ( -x)2" + 1 has the same
graph as y = -x2" + ', n is a non-negative integer,
( - x)2"+ t = ( - l )2" + ' (x?" + 1 -x2" + 1
15. a) Answers may vary. Sample answer: For the graph of
y = ax", if a > 0, vertical stretch by a factor of a if 0 <a< 1
vertical compression by a factor of a; if 1 <a< 0, vertical
compression by a factor of a and a reflection in the x-axis;
if a< - 1, vertical stretch by a factor of a and a reflection
in the x-axis
16. a) vertical stretch by a factor of 2, translation 3 units
right, translation 1 unit up b) vertical stretch by a factor
of 2, translation 3 units right, translation 1 unit up
c)
=
6.
=3
c ) End
Behaviour
(quadrants)
d) Symmetry
-
2 to 4
point
Sb)
5
4
+
2 to 1
line
+
3 to 1
point
Sd)
3
6
-
3 to4
none
Graph
Sa)
=
v1= · \ l j
b) Sign of
Leading
Coe fficient
a) Least
Degree
Sc)
7. a) i) 3 ii) + iii) 1 b) i) 4 ii) - iii) - 1
8. a) quartic b) fourth, 0.03 c) quadrant 2 to quadrant 1
d) x 2: 0 e) Answers may vary. Sample answer: They
represent when the profit is equal to zero. f) $ 1 039 500
9. a) i) cubic (degree 3) ii) 2
b)
V=1
1 7. a) a is a vertical stretch or compression; h is a shift left
or right; k is a shift up or down
Answers may vary. Sample answers:
b)
b)
Answers may vary.
x 2: 0, V(x) 2: 0
11. a)
18.
19.
1 24
(4, 6), (6, 9)
1 .2 Characteristics of Polynomial Functions, pages 26-29
5 c) 4 d) 5 e) 3 f) 6
1. a) 4 b)
2. a)-d)
Graph
Sign of
Leading
Coe fficient
End
Behaviour
(quadrants)
1a)
+
2 to 1
2
1
1b)
+
3 to 1
2
2
-
3 to 4
1
2
-
2 to 4
2
2
-
2 to 4
point
1
1
-
3 to 4
line
2
3
1c)
1d)
1e)
lf)
I
I
I
Local
M a ximum
Points(#)
Symmetry
Local
Minimum
Points(#)
V(x) = 4x(x - 35)(x - 20); x-intercepts 35, 20, 0 c) 24
12. a) cubic b) third, -4. 2 c) quadrant 2 to 4
d) {d E IR, d 2: 0}, {r E IR, r 2: 0}
13. a) Answers may vary. Sample answer: quadrant 2 to
quadrant 1, {x E IR}, {P(t) E IR, P(t) 2: 1 1 732}, no x-intercepts
b) 144 c) 12 000 d) 69 000 e) 1 3 years
15. Answers may vary. Sample answers:
,.------...
a)
b)
X=1
d) number of maximums and minimums is less than or equal
to the degree of the function plus one; number of local
maximums and local minimums is less than or equal to
the degree of the function minus one
3.
i ) End Behaviour
(quadrants)
ii) Constant Finite
Differences
2 to 1
2nd
2
2 to 4
3rd
- 24
3 to4
4th
-168
3 to 1
5th
72
2 to4
1st
-1
3 to4
6th
-720
a)
b)
c)
d)
e)
f)
iii) Value of Constant
Finite Differences
1
4. a) 2, -4 b) 4, -2 c) 3, -2 d) 4, 1 e) 3, 6 f) 5, 2
5. a) odd b) even c) odd d) even
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Advanced Functions
•
Answers
16. a)
b)
1 to 5
c) i)
V1= '2(X-3HX+2)( X-3)
iii)
V1= ·( + )'2(X-1l'2(X+2)(_
rdiT
X=2
17. a) i) cubic ii) cubic iii) cubic
b)
V1=( + )(X-1lC2X+5)
r
!
I
=
V=B
c) Answers may vary. Sample answer: The number of
x-intercepts equals the number of roots of the equation.
18. a) i) S(r) = 6n,Z(r + 1) ii) V( r ) = 3 n b) Answers may vary.
Sample answer: S(r) cubic, two x-intercepts, {r E }, {S E },
quadrant 3 to quadrant 1; V(r) cubic, one x-intercept,
{r E }, { V E }, quadrant 3 to quadrant 1
19.•)
1
no symmetry about the origin or about the y-axis because these
functions are neither even nor odd.
S. a) i) even ii) line b) i) odd ii) point c) i) neither ii) neither
d) i) neither ii) neither e) i) even ii) line
6. a) y = - 2(x + 3 )(x + 1 )(x ­2) b) y = - 3(x + 2)2(x ­1 )2
c) y = 0.5(x + 2 )3 (x - 1 fdl y = (x + 5 ) 3
7. a) y
-l (x + 2)2(x ­3), neither
4
V1:(X+ l'3
x=-2
V= ·1
4. b) line, even; this function has line symmetry about the y-axis
because it is an even function. a) c) d) neither, neither; there is
+ =t=
b)
y = 2(x + 1 )3 (x ­1 ), neither
V1:2(X+1l'3(X-1))
b)
<)+
1 .3 Equations and Graphs of Polynomial Functions,
pages 39-41
1. a) i) 3, + ii) quadrant 3 to quadrant 1 iii) 4, -3,
-t,
-±,
i
=-<
c) y
=
\
f/.6
- 2 (x
+
1f(x - 3 )2, neither
d) y = - l (x + 3)(x + 2 f(x - 2 )2, neither
2
4, - ii) quadrant 3 to quadrant 4 iii) -2, 2, 1 , - 1
4, - 1 ,
c) i) 5 , + ii) quadrant 3 to quadrant 1 ii i)
b) i)
d) i) 6, - ii) quadrant 3 to quadrant 4 iii) -5, 5
2. a) i) -4,
-±,
1
positive, -4 < x <
-±
ii)
x > 1;
< x < 1 iii) no zeros of order 2 or 3
negative x < -4,
b) i) - 1 , 4 ii) negative for all intervals iii) could have zeros
of order 2 c) i) -3, 1 ii) positive x < -3, x > 1 ; negative - 3
< x < 1 iii) could have zeros of order 3 d) i) -5, 3
ii) positive x < - 5, - 5 < x < 3; negative x > 3 iii) could
have zeros of order 2 e) i) -2, 3 ii) positive -2 < x < 3,
x > 3; negative x < - 2 iii) could have zeros of order 2 and 3
3. a) i) -2, 3,
' all order 1 ii) - 3, ­1 , 1 , 2, 3, all order 1
*
iii) order 2: -4, 1 ; order 1 : -2, l iv) order 3:
; order 2: 5;
2
3
order 1: -6 b) graph in part ii) is even; others are neither
r:T-:-c�-::.,.:-.:-:3)-::(X,...·1,-)'=-=-.�-::"-:.:-:-c)-_,
8. a) point b) line c) pointd) point
9. Answers may vary. Sample answers:
a)
V1=2(
+1l(X-2
\�!
/
xj
c)
b)
v=-16
ThV1= ·( +3)(X+1)(X-2l'2
Answers
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10. a) Answers may vary. b) Answers may vary. Sample
answer: The equation provides information about the
x-intercepts, the degree of the function, the sign of the
leading coefficient, and the order of the zeros.
c)
b)
y = 4[3(x +
y= (3x)'
y= 4(3x)'
( - 2, 1296)
(-2, 5 1 84)
(-1, 1 )
(-1, 81)
(-1, 324)
(-1, 318)
(0, 0)
(0, 0)
(0, 0)
(0, 5178)
(1, 1 )
(1, 81)
(1, 324)
(1, 26 238)
(2, 16)
(2, 1296)
(2, 5 1 84)
(2, 82 938)
( - 2, -6)
-6},
d) {x E IR}, [ y E IR, y
d) The maximum height is approximately 35.3 m above
the platform. The minimum height is approximately 5 . 1 m
below the platform.
-1
1 1 . a) 4, 1, - 1 , 2
b)
( - 2, - 6), X = -2
\
v;, v;
ii)
b) i)
even
ii)
odd
V1=& ·s-7x·3-3X
{\
N
v=-
=1
13. Answers may vary. Sample answers:
a) c = - 5
b) c = 0
r-c--:::-::-:-:-::-�-::-:-:-:,..-;--..,
c) c = 7
d) c == 1 6.95
e) c = 25
3
- 1 (vertical translation 1 unit down), n = 3
b) a = 0.4 (vertical compression by a factor of 0.4), d= - 2
(horizontal translation o f 2 units left), n = 2 c) c = 5
(vertical translation of 5 units up), n = 3 d) a =
f (x) = (3x - 2 )(3x + 2)(x - 5) (x + 5),
g(x) = 2(3x ­2)(3x + 2)(x ­5)(x + 5)
b) y = - 3 .2(3x - 2)(x - 5 ) c) y = 3.2(3x - 2 )(x ­5)
1 5. Answers may vary. Sample answer: shifts from the origin,
and an odd function must go through the origin.
16. a) 0.35 b) 2 . 5 1 7
1 .4 Transformations, pages 49-52
1. a) a = 4 (vertical stretch by a factor of 4), k = 3
} ), d = - 2
(translation 2 units left), c = - 6 (translation 6 units down)
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Advanced Functions
•
=
Answers
i
i
(vertical compression by a factor of ), k = - 1 (reflection
in the y -axis), d = 4 (horizontal translation 4 units right),
c = 1 (vertical translation 1 unit up), n = 3 e) a = 2
(vertical stretch by a factor of 2), k = (horizontal stretch
t
by a factor of 3 ), c = - 5 (vertical translation 5 units down),
n = 4 f) a = 8 (vertical stretch by a factor of 8), k = 2
(horizontal compression by a factor of 1), c = 24 (vertical
2
translation 24 units up), n = 3
5. a) ii b) iv c) i d) iii
6. a) 2 units left, 1 unit down, y = (x + 2) 2 - 1
b) 4 units right, 5 units up, y = (x - 4)3 + 5
1
7. a ) a = -3, k = 2, d= -4, c = 1
b) a : vertical stretch by a factor of 3 and a reflection in the
x-axis; k: horizontal stretch by a factor of 2; d: 4 units left;
c: 1 unit up c) {x E IR}, [ y E IR, y ::s 1 }, ( -4, 1 ), x = -4
d) vertical stretch, horizontal stretch, left, up; horizontal
stretch, vertical stretch, up, left
8. a) vertical compression by a factor of 0.5 and a reflection
in the x-axis, translation 4 units right; f (x) = -0.5(x - 4)3
b) reflection in x-axis, horizontal compression by a factor
of 1., translation 1 unit up; ((x) = - (4x}' + 1 c) vertical
4
stretch by a factor of 2, horizontal stretch by a factor of 3,
translation 5 units right, translation 2 units down;
f (x) = 2 (x - 5J - 2
9. a) f (x) = - 0.5 (x - 4 ) 3
f (x) = - (44 + 1
r
[i
14. a) Answers may vary. Sample answer:
(horizontal compression by a factor of
I
2. a) ii b) iv c) iii d) i
3. a) iii b ) iv c) ii d) i
1
4. a) k = (3 horizontal compression by a factor of-),
c
12. a ) i) 3, 2, - 2, - 3 ii) 0, -
2)]' - 6
y=Jt
(-2, 16)
V1= ·(
f (x) = 2
[±
r
(x - 5J - 2
V1:2((1/3)( ·5))•3-2
=.5
)
)· •1
f
J
v=·ts
/
I
I
I
b) f(x ) =- 0.5 ( x - 4)\ {x E IR}, {y e IR}; f(x ) = -(4x}4 + 1 ,
{ x E IR } , { y e IR , y :5 1 }, vertex (0, 1 ), axis o f symmetry x = 0;
f(x) = 2
]
[f
(x ­5} 3 - 2, {x e IR}, {y e IR}
=
[t
10. a) i) y = -x4 + 2 ii) {x e IR}, {y e IR, y :5 2}, vertex (0, 2),
axis of symmetry x = 0 b ) i) y = -(x- 5 )3 ii) {x e IR}, {y e IR}
c) i) y = (x + 3}4- 5 ii) {x e IR}, {y e IR, y -5}, vertex
( - 3, -5), axis of symmetry x = -3
11. y = x3, y = (x- 4)\ y = (x + 4) 3 , y = x4- 6,
4
y = (x - 4 ) - 6, y = (x + 4}4 ­6, y = -x4 + 6,
y = -( x- 4}4 + 6, y = -(x + 4}4 + 6
3},
12. a) i) f(x) = (x + 2)4 + 3 ii) {x e IR}, {y e IR, y
vertex (- 2, 3 ), axis of symmetry x = -2
bl n f(x )
r
( x + 1 2 iiJ 1x e �RJ, !y e �RJ
c) i) f(x ) = - 3( x + 1}4 - 6 ii) {x e IR}, {y e IR, y :5 -6},
vertex ( -1, -6), axis of symmetry x = - 1
d) i) f(x) = -
[-t f
[% r
- 3 ii) {x E IR}, {y E IR, y :5 - 3},
(x- 1
vertex (1, -3), axis of symmetry x = 1
e) i) f(x )
=
-7
( x + l}
+ 9 ii) {x E IR}, {y
E
IR, y :59},
vertex ( - 1 , 9), axis of symmetry x = - 1
14. a) shift 2 right
V2=<X-2l'3-<X-2l'2
X=U
t.m
)
c) 0, 1 ; 3, 2
15. a) vertical stretch by a factor of 3, reflection in the x-axis,
translation 4 units left
b)
c) 0, -4
16. a) h(x ) =
b) h(x ) = -
3( x + 1 )(x
(x +
2
17. $3 . 1 2
/' "+
+I
18. a) x\ x8 , x 1 6 ) x
"
1
+ 6)(x-
l
)
1) - 5
- 1)
1.5 Slopes of Secants and Average Rate of Change,
pages 62-64
1. e)
2. a) zero b)
3. a) 0 b)
constant and positive c) constant and negative
f d) -t
4. 3.89 %/year
5. a ) 6.45%/year
b) positive 0 :5 x < 8.28; negative 8.28 < x :5 13;
zero: x = 8.28 c) i ) $ 1 3/year ii) $ 8.20/year iii) $4. 80/year
iv) $ 1 7.40/year d) Answers may vary.
7. a) V =
cubic, {x e IR}, { y e IR}; S =
quadratic,
1-n?,
{x E IR}, {y E IR, y
4nr:
0}
b) volume: i) 9529.50 ii) 6387.91; surface area: i) 6 9 1 . 1 5
ii) 565.49 c) 3 1 4 . 1 6 e m d ) 570.07 e) Answers may vary.
8. Answers will vary.
9. a) y = -0.2x(x - 28) b) secant c) 2.4, 1 .6, 0.4, 0, 0, 0,
- 0.4, - 1 .6, -2.4; calculated using two points on the curve
d) steepness of the crossbeams e) have the same steepness in
a different direction
10. a) i) - 20 L/min ii) - 50.66 L/min iii) - 0.54 L/min
b) slows down
c)
[\="
X=30
V= B0.2
ii)
d) i)
iii)
L
L
1 1 . a) - 1470, - 14 1 0, ­1 350, ­1 2 90, - 1 230, - 1 1 70,
- 1 1 10, - 1 050, - 990, - 930 b) greatest: between 0 and 1;
least: between 9 and 10 c) quadratic
Differences
Time ( h )
Amo unt o f Water (l)
First
0
1 8 750
Second
1
1 7 280
- 1 470
2
1 5 870
-1410
60
60
-
3
1 4 520
-1350
4
1 3 230
-1290
60
5
12 000
-1230
60
6
60
vary. c) 1 999-2000: 1 1 . 1 %/year; 2000­200 1 :
8.7%/year; 2001 -2002: 2.8 %/year; 2002­2003: 3.2 %/year
d) greatest: between 1 999 and 2000; least: between 2001
and 2002 e) Answers may vary.
10 830
-1170
7
9 720
-1110
60
8
8 670
-1050
60
9
7 680
-990
60
6. a)
10
6 750
-930
60
b) Answers may
d) The first differences are the same as the average rates of
change for the same interval .
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529
e)
g) 25 h
QuadRe9
'"'=ax 2 +bx+c
a=30
b=-1500
c=18750
12. a) velocity of the ball b) i) 5.3 ii) 7.75 iii) 9.71 iv) 10.151
v)
10.195 vii) 10.200 c) instantaneous rate at 1 s
p2 ­100
2p
e) polynomial function, 3, 5
1 .6 Slopes ofTangents and Instantaneous Rate of
Change, pages 7 1 -73
1 . a) (5, 3) b) (3, 7) - 2 d) instantaneous rate of change
c)
5
2. a) A positive; B 0; C negative
b) A 4 m/s, C -6 m/s c) velocity at 2 s and 7 s
=
l1h
3:5t:53.1
b)
0.1
- 2.89
3 :5 t :5 3.01
- 0.024 49
0.01
-2.449
3 :5t :5 3.001
- 0.002 404 9
0.001
- 2.4049
velocity is decreasing at a rate of approximately 2.4 m/s
ii) between 1 .3 % and 2.2% per year c) Answers may vary.
6. a) 2.21 b) i) between 0.3 and 0.84 ii) between 3.66 and 4.52
iii) between 1 . 8 6 and 2.76 c) Answers may vary.
7. a) - 1 5. 2 m/s b) 2.2 m/s
L -
v
X=i
t.<=2.2X+ .
f\
X=i
End Behaviour
Function
Extends from quadrant
2 to quadrant 4
lih
lit
4 . 25 m/s
S. a) 3.9%/year b) i) between 5.9% and 6.2% per year
c)
3.
3 to quadrant 1
lit
-0.289
2. a) i) odd ii) negative iii) (x E IR}, (y E IR} iv) quadrant 2 to
quadrant 4 v) point b) i) even ii) positive
iii) (x E IR}, (y E IR, y
0} iv) quadrant 2 to quadrant 1 v) line
Extends from quadrant
3. a)
Interval
rates of change between i) 8 and 10 years ii) 8 and 1 2 years
iii) 8 and 13 years d) when h = 0 e) 246
1
13. :±: v'4l
2
14 . lm l = 4VS
Chapter 1 Review, pages 74-77
1 . a) polynomial function, 4, 3 b) polynomial function, 2, - 1
13. h( p ) =
at x
2
12. a) 0.5h + 1 2 h + 246 b) i) 272 ii) 302 iii) 3 1 8.5 c) Average
V=7.6
d) average velocity between 1 s and 3 s (decelerating 15.2 rnls);
velocity at 1 s is 2.2 m/s
8, a) Earth -44. 1 0 m/s; Venus -40.05 m/s
b) Earth - 29.44 m/s; Venus -26.7 m/s
c) a rock falls faster on Earth than on Venus
9. a) $ 1 10.50/MP3 b) $ 1 1 8/MP3 c) Answers may vary.
Sample answer: The cost of producing 200 MP3 players
is more than the average cost of between 100 and 200 MP3
players. d) No.
1 0. a) $327.25/MP3 b) $3 1 1/MP3 c) The average revenue
(between 1 00 and 200) is more than the revenue for 200 MP3
players. d) P(x) = 250x ­0.000 475x3 e) $216.75/MP3
f) $ 1 93/MP3 g) The average profit between 100 and 200 MP3
players is more than the profit for 200 MP3 players.
1 1 . a) i) $ 1 9 440 ii) - $10 800 b) Answers may vary. Sample
answer: The profits are increasing between 2000 and 6000
basketballs and decreasing between 1 6 000 and 20 000
basketballs. c) i) $21 1 50 ii) - $ 1 0 440 d) losing money
when making 18 000 basketballs
e)
Extends from quadrant
2 to quadrant 1
Extends from quadrant
c)
3 to quadrant 4
Reasons
positive leading coefficient
c)y = 4x'
and odd degree
a)y= -x'
b)y=
negative leading
coefficient and odd degree
1x',
positive leading coefficient
d ) y = O.U
and even degree
none
4 . a) iii b) iv ii
S. i) a) fourth b) 1 2 c) none ii) a) fifth b) 120 c) point
iii) a) sixth b) - 720 c) none
6. a) i) 1 ii) 5 iii) 4 iv) 2 v) 3 vi) 3
b) i)
-5 ii)
-t iii) f iv) 9 v) 7 vi) - 3
c)
7. a) i) 3 ii) negative iii) -4 b ) i) 5 ii) positive iii) 2
8. a ) quadratic b) i) 2nd ii) -9.8 quadrant 3 to
d) 0 :=; t :=; 26. 73, 0 :=; h :=; 3500
~
X:10
quadrant 4
\'=3010
e) when the parachutist lands
9. a) i) 3, positive ii) 0, - 2, 4 iii) positive - 2 < x < 0, x > 4;
negative x < - 2, 0 < x < 4
.
...
1
1;
. . ­2 < x < 2
b) 1' ) 4 , negat1ve ''11) - 2 , 2, 3 111) posmve
negative: x < -2,
i
< x < 3, x > 3
c) i) 5, negative ii) -3, - 1 , 3 iii) positive x < -3,
-1 < x < 3; negative - 3 < x < - 1 , x > 3
10. a)
V1=<X+1)(X-3)(X+2)
c)
b)
V1: ·X(X+1)(X+2)'2
X=·.
/ h
V=.
1 1 . a) Answers may vary. Sample answer:
y = (x + 3 ) (x + 1 ) (x ­2)\ y = 2(x + 3)(x + 1 ) (x - 2 )2
b) y =
(x + 3)(x + 1 ) (x - 2 )2
i
530
MHR
•
Adva nced Functions
•
Answers
12. a) point symmetry b) line c) neither
13 . a) y
- (x + 2)(x - 4)2
�� (x + 2f(x - O.S)(x - 4)
=
b)
y= -
14 . a) i) vertical compression by a factor of
t' reflection in the
c) horizontal compression, vertical compression, reflection
in the y-axis, 3 units left, 1 unit down horizontal compression,
vertical compression, reflection in the y-axis, 1 unit down,
3 units left
x-axis, translation 2 units down; y = -lx3 - 2
4
ii) {x E IR), {y E IR} b) i) vertical stretch by a factor of 5,
horizontal stretch by a factor of 2, translation 3 units right,
translation 1 unit up; y = s
{y E IR, y
[t
(x
r
3) + 1
ii)
{x E IR),
1 ) , vertex (3, 1 ), axis of symmetry x = 3
1 5. a) i) y =
-H�·
(x
+
4
f
+
1
ii)
{x E IR), {y E IR , y
1),
vertex ( -4, 1 ) , axis of symmetry x = -4
y = - 5 [4(x + 2)]3 + 7 ii) {x E IR), {y E IR)
16. a), c), d )
1 7. a ) $ 1 4 , $10 b) i ) negative ii) positive iii) zero
c) i) ­$ 1 .6/month ii) $2/month iii) $0/month
18. a) 5.7%/year b) between 6.7% and 6.9% in 2000;
4.6% in 2002 c) Answers may vary.
b) i)
Practice Test, pages 78-79
1. Answers may vary. Sample answers for false:
A. False;
B. False;
C. True
D. False;
y x3 + x2
y = x2
y x4 + x
2. Answers may vary. Sample answers for false:
A. False; degree n means constant nth differences
B. True
C. False; may have line symmetry
D. False; may have higher degree
3. Answers may vary. Sample answers for false:
A. False; y = x2 does not equal y
x2
=
=
B. False; stretches and compressions
C. False; does not matter
D. False; reflection in the y-axis
E. True
4 . a) i b) iii c) ii
S. i) a) third b) - 1 2 c) point ii) a) fifth
iii) a) sixth b) - 720 c) line
( -_t )
first
b) 1 20 c) point
6. a) Answers may vary. Sample answers:
y = 2x(x + l )(x - W, y = -x(x + l)(x - 3)2
y = - 3x(x + l )(x - 3 )2
b)
<)
positive -1 < x < 0; negative x < - 1 , 0 < x < 3, x > 3
7. y = -x(x + l f(x - l ) (x - 2 )3
8. a) a =
t
time at a constant rate b) drop in temperature versus time
at a constant rate c) acceleration d) no change in revenue
over a period of time
1 1 . 4.66 %/year
12. a) 3 125 L b) i) - 245 L/min ii) - 20 L/min c) Answers may
vary. Sample answer: Average rate of change of volume
over time is negative and increasing.
d)L
=10
=
-_t
9. y = - 2(x - W - 5
10. Answers may vary. Sample answer: a) distance versus
(vertical compression by a factor of
(horizontal compression by a factor of
i'
-_t), k = - 2
reflection in the
y-axis), d = -3 (translation 3 units left), c = - 1
(translation 1 unit down)
b) {x E IR), {y E IR, y
- 1 ), vertex ( - 3, - 1 ), axis of
symmetry x = - 3
V:67
e) slopes of secants
13. a) 1 m/s b) 1 . 9 m/s c) speed is increasing
C H APTER 2
Prerequisite Skills, pages 92-93
1 . a) 1 24 R4 b) 1 6 1 R 1 6 c) 147 R9 d) 358 R 1 3
1 69
2. a) -22 b) -6 c) - 5 1 d) - 1 3. 875 e) -27
3. a) x4 + x3 - 7x2 + 3x + 3
b)
2x4 + 4x3 - 1 5x2 + x - 1 9
c ) 3x4 + l lx3 - 7x2 + 25x - 2
d) x2 - 2 e) x2 - 45 f) x2 - 2x - 2
4 . a) (x - 2)(x + 2) b) (5m - 7 ) (Sm + 7)
d)
c) (4y ­3) (4y + 3 ) 3 (2c - 3)(2c + 3)
e) 2(x - 2)(x + 2 )(x2 + 4) f) 3(n2 - 2)(n2 + 2)
S. a) (x + 3 )(x + 2) b ) (x - 4)(x - 5 )
c) (b + 7 ) ( b - 2) d ) (2x + 3 ) (x - 5 )
e) (2x ­3)2 f ) (2a - 1 ) (3a - 2 )
g ) (3m - 4)2 h) (m - 3 ) (3m - 1 )
3
6. a) x = - 3 or x = 5 b) x = - 1 or x = 4
1
3
3
c) x = - - or x = - d) x = - or x = 5
3
2
2
1
5
1
e) x = -2 or x = 1 f) x = 7 or x =
3
7. a) x = - 1 .3 or x = 0 . 1
b) x = 0.7 or x = 2.8
c ) x = - 1 .2 o r x = 0 . 7
d ) x = - 1 .3 or x = 2 . 5
1
8. a) y = - (x + 4)(x - 1 )
3
b) y - 3x(x- 3 ) c) y = -4(x + 3 ) (x - 4)
d) y = 2(x + l )(x - 5 ) e) y = - 3 (2x + 1 ) (2x - 3 )
9 . a ) i ) - 4 and 1 ii) above the x-axis: x < -4 and x > 1 ;
below the x-axis: -4 < x < 1 b) i ) - 1 , 1 , and 2 ii) above
the x-axis: - 1 < x < 1 and x > 2; below the x-axis: x < - 1
and 1 < x < 2 c) i) -2, - 1 , 1 , and 2 ii) above the x-axis:
-2 < x < - 1 and 1 < x < 2; below the x-axis: x < - 2
and - 1 < x < 1 and x > 2
=
A nswers
•
MHR
531
_
2.1 The Remainder Theorem, pages 9 1 -93
x3 + 3xz - 2x + 5 = xz + 2x - 4 + 9 _ b) x* - 1
1. a)
x+1
x+ 1
c) (x + 1 )(x2 + 2x - 4 ) + 9
3x4 - 4x3 - 6x2 + 1 7x - 8 = x3 2x + 3 + _4_
2. a)
3x - 4
3x - 4
_
b) x *
c) 3x4 - 4x3 - 6x2 + 1 7x - 8 = (3x - 4) (x3 - 2x + 3 ) + 4
x3 + 7x2 - 3x + 4 xz + Sx 3. a)
1 3 + _lQ_ ' x* - 2
x+2
x+2
6x3 + x2 - 14x - 6
2 _ * 2:_
_
_
X
2 X2 X 4 +
3X + 2
3
3x + 2'
1
8x
0x3
9x2
7
+
1 1 2x2 _ x 2 + _ _ x* 2:..
<)
Sx - 2
Sx + 2 '
5
d) - 4x4 + 1 1 x - 7 = 4x3 - 12x2 - 36x - 97 '
x-3
x-3
x*3
6x3 + xz + 7x + 3
3
2x2 - x + 3 - _ _ , x* _ 2:_
e)
3
3x + 2
3x + 2
4x2 - 31 = 4xz + 8x + 1 2 + _s_ , x* l
8xJ
+
f)
2
2x-3
2x - 3
3
8x3
+ 6x2 - 6x
2.
l
+
2x2
+
3x
+
x*
g)
4
4 4(4x - 3 ) '
4x - 3
=
b)
_
=
=
_
_
=
=
4. a) 27 b) - 9 c) -2
5. (x + S)(x + 3) (2x + 1 )
6. (3x - 2 ) em by (3x - 2 ) em by ( x + 4) em
7. a) 1 6 b) 3 1 <) 36 d) 2 1 1 e) 4
8. a) 1 6 - 1 3 c) -23
9. a) 9 b) 15 c) 41 d) -4
10. a) k = 3 b) 123
11. a) c 4 28
12. b = 1 1
13. k = 3
14. a) -<) 8
1 5. a) -b) 1 5
16. a )
b)
P(t)
=
0 b) (3x - 2) i s a factor of 6x3 + 23x2 - 6x - 8 .
<) (3x - 2 )(x + 4)(2x + 1 )
17. a) rc ( 9x2 + 24x + 1 6 ); this result represents the area of
the base of the cylindrical container, i.e., the area of a circle.
b) rc (3x + 4)2(x + 3 )
)
Valueofx
Rad ius(cm)
2
10
3
5
1 571
7
5 630
13
6
5
19
8
7
25
16
4
22
6
28
8
Volume(cm3)
Height(cm)
3 186
9 073
9
13 685
11
27 093
19 635
10
18. a) -5t2 + 1St + 1 = (t - b)( -St - Sb + 1 5 ) - 5b2 +
( - 5t2 + 1St + 1 ) - ( - 5b2 + 1 5b + 1 )
t-b
d) Answers may vary. Sample answer: At t b, there is a hole
in the graph; the graph is discontinuous at t = b.
e) 1; at 3 s, the height of the j avelin is 1 m.
19. a) 1l b) At 1.5 s, the shot-put is 2.4 m above the ground.
5
1 1 n = 59
15b + 1 c)
=
2o. m = -5,
21. a
532
=
14
-3, b
MHR
•
5
=
2
-3
Advanced Fu nctions
•
=
2
-3 or k
23. 3
24. A =8
=
4
2.2 The Factor Theorem, pages 1 02-1 03
1. a) x - 4 b) x + 3 <) 3x - 2 d) 4x + 1
2. a) No. Yes. c) Yes.
3. a) (x - 2 )(x + 1 )(x + 4) (x - 3 ) (x + 1 )(x + 6 )
c) (x - 4)(x - 2 ) ( x + 3)
4 . a ) (x - 3 ) (x + 1 )(x + 3 ) b ) (x - 4)(x - 1 ) ( x + 4)
c) (x - 6)(x + 6 ) (2x - 1 ) d ) (x - 7)(x - 2)(x + 2)
b)
b)
(x - S)(x + 5 )(3x + 2 ) f) x(x - 4)(x + 4)(2x + 3)
- 3 )(x + 2) (3x + 4) b) (x - 3 ) (x - 1 )(2x - 1 )
3 ) (2x - 1 ) (3x + 5 ) d) (x - 1 ) ( x + 1 )(4x + 3 )
- 1 )(x + 1 )(x + 2 ) (x - 2)(x + 1 ) (x + 5 )
S)(x - 2)(x + 2 ) d) ( x + 4)(x2 + x - 1 )
e) (x - S)(x - 2)(x +3 ) f) (x - 3 ) (x + 2)(x - 1 )(x- 2 )
g ) (x- 4)(x - 2)(x +1 )(x + 3 )
7. a) (2x - 1 ) (2x + 1 )2 b) (x - 1 ) (x + 2)(2x + 3 )
c) (x - 1 ) (x + 2) (5x - 2) d) ( x - 1 )(x + 1 )(2x - 1 )(3x + 2 )
e ) (x - 2 ) ( x + 2 )(5x2 + x - 2 ) f ) (x - 3 )(x + 4) (3x + 1 )
g) (x - 2)(2x - 1 ) (3x - 1 )
8. (x + 4)(2x - 1 )(3x + 2 )
9. k
-3
10. k
-1
11. a) (x - 1 )(x + 2)(2x + 3 ) ( x + 1 ) (2x - 3 ) (2x + 1 )
c) (x - 1 ) (2x + 5 ) (3x - 2 ) d) (x - 2) (4x2 + 3 )
e) (2x - 1 ) (x2 + x + 1 ) f) (x - 4)(x - 1 ) (x + 2)(x + 3 )
12. a) i) (x - 1 ) (x2 + x + 1 ) ii) (x - 2 )(x2 + 2x + 4)
iii) (x - 3 )(x2 + 3x + 9) iv) (x - 4)(x2 + 4x + 16)
b ) x3 - a3 = (x - a)(x2 + ax + a2 ) < ) (x - 5 ) (x2 + Sx + 25)
d) i) (2x- 1 )(4x2 + 2x + 1) ii) (5x2 - 2) (25x4 + 1 0x2 + 4)
iii) (4x4 - 3 ) ( 1 6x8 + 12x4 + 9)
iv)
x - 4y2
x2 - xy2 + 16l
2
13. a) i) (x + 1 ) (x2 - x + 1 ) ii) (x + 2) (x2 - 2x + 4)
iii) (x + 3 )(x2 - 3x + 9 ) iv) (x + 4)(x2 - 4x + 16)
b ) x3 + a3 = (x + a) (x2 - ax + a2) c) (x + 5) (x2 - Sx + 25 )
d) i) (2x + 1 ) (4x2 - 2x + 1 ) ii) (5x2 + 2)(25x4 - 1 0x2 + 4)
iii) 4x4 + 3 ) ( 1 6x8 - 12x4 + 9)
iv) lx - 4l _±_x2 - xy2 + 1 6y4
5
5
25
14. (x2 + x + 1 ) (x2 - x + 1 )
15. a) ( x - 3 ) (x + 3 ) (2x - 1 ) (2x + 1 )
b) (x - 4) (x + 4) (3x - 2 ) ( 3x + 2 )
17. a) ( x - 2)(x - 1 )(x + 1 )(x + 2)(2x + 3 )
(x- 2)(x + 1 )(x + 2f(2x - 1 ) (2x + 1 )
18. m
0.7, n = - 0 . 9
19. a ) (x + 4)(4x + 3 )(2x - 1 )
_2_ (x - 3 ) (x + 1 ) (3x - 2) (2x + 3 )
10
20. a) i) (x - 1 )(x + 1 ) (x2 + 1 ) ii) (x - 2)(x + 2) (x2 + 4 )
iii) (x - 1 )(x4 + x3 + x2 + x + 1 )
iv) ( x - 2 ) (x4 + 2x3 + 4x2 + 8 x + 1 6 )
b) x" - a " ( x - a) (x" - 1 + ax" - 2 + a2x" - 3 + . . . +
a" - 3x2 + a"- 2x + a"- 1 ), where n is a positive integer.
c) (x - 1 ) (x5 + x4 + x3 + x2 + x + 1 )
d) i) (x - 5 ) (x2 + 25) ii) (x- 3) (x4 + 3x3 + 9x2 + 27x + 8 1 )
21. Yes, but only if n is odd. Let n = 2k + 1 .
Then xz• + 1 + az• + 1 = x2k xz• 1 a + xz• - 2 az xz• - 3a3 +
. .. - ;a2• - 1 + a2•. If n is even, then x" + a" is not factorable.
22. 7x - 5
e)
5. a) (x
c) (x 6. a) (x
c) (x -
b)
=
b)
=
b)
=
22. k
Answers
b)
(t
)(
(
)(
)
f
)
=
b)
=
_
-
_
2.3 Polynomial Equations, pages 110-112
5
1. a) x = 0 or x = -2 or x
b) x = 1 or x = 4 or x = - 3
=
2
c) x = - - or x = - 9 or x = 2
3
2
d) x = 7 or x = -3 or x = - 1
e ) x = 0.25 o r x = 1 .5 o r x = - 8
f) x = 2.5 or x = -2.5 or x 7
g) x = 1 .6 or x = -3 or x = 0.5
2. a) x = - 3 or x = -1 or x = 1
b) x = - 1 or x = 3 or x = 4
c) x = - 2 or x = - 1 or x = 2 or x = 3
d) x = -5 or x = -2 or x = 1
e) x = -3 or x = - 1 or x = 0 or x = 2
3. a) x = 4 b) x = 1 or x = - 1 c) x = 4 or x = - 4
d ) x = - 1 o r x = 1 o r x = 5 o r x = -5
e) x = 1 .5 or x = - 1 .5
f) x = 7 or x = - 7 or x = -3 or x = -4
g) x = - 3 or x = 0.5 or x = 5 or x = -5
4
1
4. a) -5, 0, 9 b) ­9, 0, 9 c) - , 0 , 3
2
d) -2, - 1 , 2 e) - 2, 2 f) - 1 , 0, 1 , 2 g) -5, -2, 2, 5
5. Answers may vary. Sample answers:
a) False. If the graph of a quartic function has four x-intercepts,
then the corresponding quartic equation has four real roots.
b) True. c) False. A polynomial equation of degree 3 has three
or fewer real roots. d) False. If a polynomial equation is not
factorable, the roots can be determined by graphing. e) True.
6. a) x = -2 or x = 3 b) x = 5 or x = -2 or x = 1
c) x = 1 or x = 3 d) x = -2 or x = 3
e) x = -2 or x = 2 or x = 3 f) x = -4 or x = 1 g) x = - 1
7. a) x = -2 or x = - 1 or x = 1 .5 b) x = -0.5 or x = 3
2
2
c) x = -2 or x = -3 or x = 3
d) x = -2 or x = 0.6 or x = 3 e) x = 0 or x = 2
f) x = -2 or x = 0 or x = 0.5 or x = 2
g) x = -3 or x = - 1 or x = 2 or x = 3.6
8 . a) x = - 1 or x = 2 or x = 4 b) x = 3
c) x = 1 d) x = - 1 or x = 2 e) no real roots
9. a) x ='= -2.2 or x ='= 0.5 or x ='= 1 . 7
b ) x ='= -4.5 o r x ='= - 0.6 o r x ='= 0.6
c) x ='= - 1 .2 or x ='= 1 .2 d) x ='= - 1 . 3
e) x ='= - 1 .4 o r x ='= 1 .9 f) x = - 1 o r x ='= 0.4 o r x ='= 1 .4
g) no real roots
10. 2 m by 2 m by 5 m
11. 1 3 m by 3 m by 3 m
12. Answers may vary. Sample answer: Yes, for example,
x3 + 2.
13. Answers may vary. Sample answer: No. If the radical
part of the quadratic formula is negative, then two non-real
roots occur.
14. 7 h
15. 0 m or 8 m or approximately 12.9 m from the end
16. a) {x e IR, 0 < x < 992} b) 22 000
=
18. a) k = 3 b) x ='= - 1 . 3 or x ='= 0.8
19. 24 em by 20 em by 4 em, or 20 em by
20. a) x = 3 or x =
-3
b) x3 - 2x2 - l4x + 40
1 6 em by 6 em
3iVJ or x = - 3 - 3iVJ
2
2
+
21. 9 em by 8 em by 7 em or 12 em by 10 em by 8 em
69
63 = 0
22. X J - 4x2 ­ X +
4
2
23. 1 5°
24. 20
2.4 Families of Polynomial Functions, pages 119-122
1. a) y = k (x + 7)(x + 3), k e IR, k of. 0
b) Answers may vary. Sample answer:
y = 2(x + 7)(x + 3), y = - 3 (x + 7)(x + 3 )
c) y = (x + 7)(x + 3 )
5
2. C (has different zeros)
3. A, Band D (same zeros)
4. A, C, E (zeros are - 3, -2, 1 ); B, D, F (zeros are - 1 , 2, 3 )
5 . a ) y = k(x + 5)(x - 2)(x - 3 )
b ) y = k(x - 1 )(x ­6)(x + 3)
c ) y = k(x + 4)(x + 1 ) (x - 9)
d ) y = kx(x + 7)(x ­2 )(x - 5)
6. a) A: y = (x + 2) (x - 1 )(x - 3);
B: y = - l (x + 2)(x - 1 ) (x - 3);
2
1
C: y = - (x + 2)(x - 2 )(x - 3);
2
D: y = 2(x + 2)(x - 1 )(x - 3)
7 . a ) y = kx(x + 4)(x - 2 ) b ) Answers may vary. Sample
answer: y = x(x + 4)(x - 2), y = - 2x(x + 4)(x - 2)
c) y = lx(x + 4) (x ­2) d) Answers may vary.
4
Sample answers:
8. a) y = k(x + 2)(x + 1 ) (2x - 1 )
b) Answers may vary. Sample answer:
y = - (x + 2)(x + 1 ) (2x ­1 ), y = (x + 2) (x + 1 ) (2x ­1 )
i
c) y = - 3 (x + 2) (x + 1 )(2x - 1 )
d) Answers may vary. Sample answer:
c) x = 3 or x = 8; If the selling price is $3 per bottle or $8 per
bottle, then 17 200 bottles of sunscreen will be sold per month.
17. a) x = 3 b) x ='= - 0.6 or x ='= 0 . 1 or x ='= 3 .9 or x ='= 4.6
Answers
•
MHR
533
9. a) y = k(x + 4)(x + l ) (x - 2)(x - 3)
c) family of functions with the same zeros
b ) Answers
may vary. Sample answer:
y = 2(x + 4 ) (x + l ) (x - 2)(x - 3),
y = - 3(x + 4)(x + l ) (x - 2)(x - 3 )
1
c) y = - (x + 4)(x + 1 ) (x - 2)(x - 3 )
6
d ) Answers may vary. Sample answer:
n= ·<116l< · )( +1l< -2lL
10. a) y = k (2x + 5 ) (x + 1) (2x - 7)(x - 3 )
Answers may vary. Sample answer:
(2x + 5 ) (x + 1) (2x - 7)(x - 3),
y = 2(2x + 5 ) (x + 1) (2x - 7)(x - 3 )
5
c) y = (2x + 5 ) (x + 1 ) (2x - 7)(x - 3 )
U
d) Answers may vary. Sample answer:
b)
y= -
approximately 27.16 em by 15.16 em by 4.42 em or
26 em by 14 em by 5 em
22. a) Answers may vary. Sample answer:
y = k (3x - 2)(x - 5 ) (x + 3 ) (x + 2 )
b ) 4 c) Answers may vary. Sample answer:
y = - (3x - 2)(x - 5 ) (x + 3 ) (x + 2 )
5
d) y = (3x - 2 )(x - 5 ) (x + 3 ) (x + 2 )
5
23. Answers may vary.
24. 24 em
25. g(x1 - 1) = x4 - x2 - 3
e)
t
2.5 Solving Inequalities Using Technology, pages 129-131
1. a) - 7 < X :5: - 1 b) - 2 < X :5: 6 C) X < - 3, X 2:: 4
d) X :5: -1, X 2:: 1
2. a) X < -1, -1 < X < 5, X > 5
b) X < - 7, -7 < X < 0, 0 < X < 2, X > 2
C)
X < -6, - 6 < X < 0, 0 < X < 1, X > 1
2 2
d) X < -4, -4 < X < -2, -2 < X < 5' 5 < X < 4.3, X > 4.3
3.
11. a) y = k(2x3 - 3x2 - 4x - 1)
�� (2x3 - 3x2 - 4x - 1)
12. a) y = k(x4 + 2x3 - 26x2 - 6x + 1 17)
b) y = - i (x4 2x3 - 26x2 - 6x + 117)
b) y =
+
13. a) y = k(x4 - 2x3 - 10x2 + 20x - 8 )
b) y =
4(x4 - 2x3 - 10x2 + 20x - 8 )
14. y = -2(x + 2)(x - 1 ) (x - 3 )
15. y = ( x + 3)2 ( x - 1 ) (2x - 3 )
16. y = - 2x(2x + 7)(x + 2)(x - 1 )
1 7 . Set A : no; Set B: yes
18. a) V = x(48 - 2x)(30 - x) b) 44.31 em by 2 8 . 1 6 em
by 1.84 em or 18.6 em by 1 1 .4 em by 10.8 em
c) volume doubles; volume triples; family of functions
with zeros 24, 30, 0 d) Answers may vary. Sample answer:
19. y = kx(x - 30)(x - 60) (x - 90)(x - 120 ) (x - 150)
20. a ) V = x(36 - 2x) (24 - 2x)
b) i) V = 2x(36 - 2x)(24 - 2x) ii) V = 3x(36 - 2x)(24 - 2x)
534
MHR
•
Advanced Functions
•
Answers
4. a) f(x) > 0 when x < -2 or 1 < x < 6
b) f(x) < 0 when -3.6 < x < 0 or x > 4.7
5. a) i) -6, 3 ii) -6 < X < 3 iii) X < - 6, X > 3
b) i) -2, 5 ii) X < -2, X > 5 iii) -2 < X < 5
C) i) - 4, 3 ii) -4 < X < 3, X > 5 iii) X < -4, 3 < X < 5
d) i) -4, 1 ii) X < -4 iii) -4 < X < 1, X > 1
6. a) - 3 < X < 4 b) - 5 :5: X :5: - 3
C) 1 < X < 2, X > 3 d ) -4 :5: X :5: - 3, X 2:: - 1
e) X < -3, 2 < X < 3 f) X :5: -4, -1 :5: X :5: 4
7. a) x :o; -4 or x 2:: 0.5 b) - 0.5 < x < 3
c) x :5: -4 or - 2 :o; x :o; 1 d) - 5 < x < - 1 or x > 4
e) x < -5 or -2 < x < 7 f) x :5: 7
8. a) - 4.65 < X < 0.65 b) -2.43 < X < 1 .10
c) x :o; -2 .17 or -0.3 1 :5: x :5: 1.48
d) - 2.12 :o; x :5: - 0.43 or x 2:: 0.55
e) x < -1.93 or - 0.48 < x < 1.08
f) -1.34 :5: X :5: 1.25
9. a) approximately x > - 0.67
1
b) x :5: -4 or - 2 :5: x :o; 6 c) x :5: -4 or-- :5: x :5: 3
3
d) x < -2. or -1 < x < 2 e) x < -2 or -.!. < x < 3
2
3
f) x < - 3 or -1 < x < - 1 or x > 4
2
-
1 3 . a ) approximately 7 < n < 11 or 1 9 < n < 20, so
10. approximately 0.50 < t < 6.03, or between about
0.5 s and 6.03 s.
1 1 . a) approximately 2.73 < t < 5 . 5 1 , or between later in
the second week and halfway through the fifth week
b) There are no tent caterpillars left.
12. a) between 0 and approximately 4.47 years
b) after approximately 4.94 years
13.-14. Answers may vary.
15. Answers may vary. Sample answers:
a) (3x + 2) (5x - 4)(2x - 7) > 0,
- 30x3 - 1 09x2 - 2x - 56< 0
b) x3 - 2x2 - l Ox+8 > 0, -x3 + 2x2+ l Ox - 8 < 0
16. approximately -0.66 s x s 2.45
1 7. a) {x E IR, - 1 s x s 0 } ,
E IR, 0 s s
b) {x E IR, x < -1 , x > 1 }, {y E IR, > 0}
19.1 5:34
20. 2V6
{y
y
S
5
2 b) X > -2 C) X
S
2. a) x < -2 or x > 4
•I
-5
-4 -3
-2 -1
3
b) x s -2 or x
0
b) 6
-5
S
...
I
$
-4 -3
S
1
s
I
-1
-2 - 1
9
0
c) -4
II(
X
I
3
2
x
s
4
I
0
3
2
+
5
6
4
4
$
1
•
0
2
I
I
2
I
3
1
7
4
I
+
9
+
2
5
10
I
5
,..
)o
3
2
6. a) approximately x
-5.09 b) x < -4 or - 3
- 1 or l s x s 3 d) true for all intervals
5
1
7. a) - 5 s x s 1 b) - 2 < x < - - or x > 3
2
1
12
c) - 1 < x < -2 or x >
s
s
x
s
1 or x
....
s
-t
2
10
3
5
- 7 or .!. s x s 1
3
2
1
c) x < -2.5 or -- < x < 1 d) -4 s x s -2 or x 7
3
5 b) - 3 < x < 5 c)
5. a) x s 3 or x
s x s
3 s x s - 1 or x 1
d) x < -2 or 1 < x < 3 e) -2
d) - 1 - Vl?
1. a) i) 37 ii) x3 + 9x2 - Sx + 3
5
3
8
2
4. a) -2 < x < - 1 or x > 3 b) x
c) x
Chapter 2 Review, pages 1 40-1 41
2
I
-1
3
2
2
3. a) x < - 3 or x > 2
•I
1
4
-1
-2
1
-3d) X< 1 e) X< -2 f) X
0
y
y ±}
2.6 Solving Factorable Polynomial Inequalities
Algebraical ly, pages 1 38-1 39
1. a) X
between 7 and 1 1 years from today and between 1 9 and
20 years from today b) approximately 12 < n < 1 8.6, so
between 12 and 1 9 years from today c) Not valid beyond
20 years. 20 years from today the population will have
fallen to 5560, and in the next year it would fall below 0,
which is not possible.
14. x4 - 76x2 + 1 1 56 s 0, -x4 + 76x2 - 1 1 56
0
1 5. 3v'U
25
16. = ±x 3
3
- l + Vl?
---:
...:.
:..__
-
2
_
8. 22 em by 24 em by 1 0 em
9.after 10 years (in 2009)
11. 8
12. approximately x < 0.59 or 1 < x < 3.41
}
s
x
s
-2
= x2 + 1 1x + 1 7 + _l_Z_,
x-2
x-2
x*2
+ x - 11
l
2x3
2x2
= 4x2 - 2x + 1 b) i) -12 ii)
__
3x + 1 _g_ '
3x + 1
1
X* 3
27 ii) - 8x4 - 4x + l Ox3 - x2 + 1 5
c) i)
2x - 1
2
27
= -4x3+ 3x2 + x - l +
x * .!.
2
2(2x
- 1 )'
2
77
2. a) k
b) 162
27
3. b = - 34
4. a) (x - 3 ) (x - 2)(x + 1 ) b) (x - 4)(x+2) (3x + 1 )
c) (x - 3 ) (x - l ) (x + 6 )(5x + 2)
5. a) 4(x - 2)(x + 1 )(x+2 ) b) (x - 2) (5x - 3)(5x + 3)
c) x(x - 2)(x + 2)(2x + 5)
6. a) (2x - 1) metres by (x+3 ) metres by (x+ 1) metres
b) 4 m by 2 m by 1 m
7. k = -2
8 . x = - 4 or x = -2 or x = 3
1 + v'105
1 - v'105
or x = ----,9 . a) x = -4 or x = 4 b) x =
4
4
10. a) x = - 1 or x = -0.5 or x = 0.8
b) x = - 0.7 or x = 0.9 or x = 8 . 8
1 1 . V = l(l- 5 )(2/+ 1 ); approximately 8 . 5 5 e m b y 3.55 em
by 1 8 . 1 0 em
12. B (different zeros)
3
13. a) = k(x - 4x2 - x) b) = - 2(x3 - 4x2 - x)
1 4. - 3 (x + 2)2(x- 1 )
15. a) x s -4.2 or x
1 .2 b) ­.!. < x < 3 or x > 4
2
1
c) x < - 1 .7 or 0.4 < x < 3.3 d) x s -4 or - - s x s 3
3
e) x < -2.2 or x > 2.2
16. approximately between 0.8 s and 7.6 s and between
20 s and 23.6 s
3
2
4
1
1 7. a) - 5 < x < 4 b) -2 s x s 3 or x
c) x < -5 or x > 5
3
1
7
1
18. a) x s -3 or x 4 b) x s -4 or 3 s x s 2
=
-
y
--
y
c) approximately x < -2.4 or x > 4.3
Chapter 2 Practice Test, pages 1 42-1 43
1. c
2. c
3. D
x - 4xz + 3x - 7 = xz - 7x +24 - _]J_
4. a) J
x+3
b) x* - 3 c) (x + 3 ) (x2 - 7x + 24) - 79
5. a) k
=
x+3
t b) 1 93
An swers
•
MHR
535
6. a) (x - 4)(x - 2)(x + 1 ) b) (x - 3)(x + 2)(x + 3 )
c) (x - 2)(x + 3)(x + 4) d ) (x ­1 ) (x + 2) (5x + 2 )
6. a) (x + 4)(x + 3) b) (Sx - 2)(x - 3) c) (3x + 8)(2x - 1 )
d) ( x + 1 ) (x + 3 ) (x - 2 ) e) (2x + W(3x - 2 )
f) (3x - 4)(9x2 + 12x + 1 6)
1 e) - 7, 1 f) 6, - 5
3 d) -5, - 6
1. a) 8, -4 b) - 5, - 1 c) 3,
3
2
2
-4 + Vi4
5 + V73
8. a) -2 ± Vl b)
d) no x-intercepts
c) 6
2
-4 ±
v'10 f) 1 ± 2Vl
e)
3
9. a) X > 6 b) X :=; 11. C) X > .!. d) X > - 5 e) X > _ 2_ f) X > - 9
2
2
2
10. a) -2 < X < 2 b) X < -3 Or X > 6 C) -m < X < Vi3
e ) ( x + 2)(x + 3)(x + 4 ) f ) ( x + 1 ) (x + 2)(x + 3 ) (2x + 1 )
7. x = -5 or x = 3 or x = -2
8. a) x = 2 b) x = - 1 1 or x = 1 1 c) x = -5 or x = 5
d) x = - 3 or x = 3 or x = -2 or x = 5
9. a) x = - 2 or x = - 1 b) x = -4 or x = 1 or x = 3
c) x = - 1 .75 or x = 1 .5 or x = 1 .75
2
3
2
x = - - or x = O or x = - or x = 5
3
3
10. Answers may vary.
1
11. a) y = - x(x + 3 )(2x + 3 )(x - 2 )
2
3
b) X < -3, - < X < 0, X > 2
2
12. a) y = k(x4 ­6x3 - 1 7x2 + 1 20x - 50)
d) x < - 5 or x > 2 e) - 7 :=; x :=;
y = - (x4 ­6x3 ­1 7x2 + 120x - 50)
5
13. a) V = x(20 - 2x) ( 1 8 - x) b) approximately 1 6. 7 em
by 16.34 em by 1 . 7 em or 5.8 em by 10.9 em by 7.1 em
c) V = k(20 - 2x) ( 1 8 - x)x
14. a) x :=; -0.9 or 0.4 :=; x :=; 7.4
b) - 2.0 < x < -0.6 or 0.9 < x < 4.7
15. a) approximately x < - 3.6 or - 1 . 1 < x < 1.7
b) - 1 .5 :=; x :=; -1 or approximately x :=; - 1 . 7 or
approximately x 1 . 7
4
5
4
16. a ) - 3 < x < 3 b) x < 0 c) x :=; - 3 or - 2 :=; x :=; 3
1
d) x :=; - 3 or ­1 :=; x :=; - or x
2
2
17. a) V = x(32 - 2x)(40
2x)
b) i) V = 2x(32 - 2x)(40 - 2x)
ii) V = x(32 - 2x)(40 - 2x) c) family of functions
d) approximately 2 < x < 10.9 or x > 23.1
2
c)
Answers may vary. Sample answer: A line or curve that
the graph approaches more and more closely. For f(x) =
the vertical asymptote is x = 0.
b) X = -4, y = 0
2. a) X = 3, y = 0
,
V= -.2
d) X = 5, y = 0
V1=11<2 -10)
l
I
=0
V=·.t
3. a) {x E IR}, {y E !RJ b) {x E IRJ, {y E IR, y 4 J
c) {x E !RJ , {y E !R J d) {x E IR, x * OJ, {y E IR, y of. O J
e) {x E IR, x * 4 J , {y E IR, y * O J f) {x E IR, x of. OJ , {y E IR, y of. O J
4. a) - 1 3 b) .!. c) ­ d) - 1 e) ­1 3 f) £
536
5
b) 0.71 c) 0 . 1 8 d) 0.38 e) 0.53 f) 0.35
MHR
•
-00
0
As x
+
8
0
t(x)
+oo
- 00
-oo
0
0
= 2, y = 0 ii) X = -3, y = 0
-i b) i) X = - 6 ii) y = 0 iii) t
i) X = 1 ii) y = 0 iii) 5 d) i) X = - 7 ii) y = 0 iii) - t
3 . a) i) X = 5 ii) y = 0 iii)
C)
1
""") 12
e) '•) x = ­3 "")
u y=0m
5 • a) Y =
9
-oo
+ oo
Prerequ isite Skills, pages 146-147
3
+oo
-5
1
b) ')• y = x -1 2 "11) y = x + 3
1.
S. a) 0.38
t(x)
- 5+
8
2. a) i) X
(
0
+ oo
C H A PTE R 3
v=.m
0
-00
Asx
2
-oo
+ oo
b)
.!.
+oo
2+
i
j
x :=; - 6 or x
3.1 Reciprocal of a Linear Function, pages 153-155
1. a)
As x
t(x)
b)
c) x = 8, y = 0
- 1 f)
Adva nced Functions
•
Answers
1
b) = 1 c) = 1 d) y = - x +1 4
x - 3 Y x + 3 Y 2x - 1
The slope is negative and
decreasing for x < 3.
The slope is negative and
increasing for x > 3 .
The slope i s negative and
decreasing for x <
-I.
The slope is negative and
.
. for x < - 7 .
mcreasmg
2
The slope is positive and
increasing for x < -4.
The slope is positive and
decreasing for x > -4.
d)
V1=51G-2X)
X=O
7. a) V1=1/(X·1)
b)
c)
:;
The slope is positive and
3.
.
. for x < 2
mcreasmg
C
v=
6666?
ll
X=O t1
V1=11<X+l
-
The slope is positive and
decreasing for x >
(x E IR, x of. 1 }, (y E IR, y of. 0},
X = 1, y = 0
(x E IR, x of. -4},
(y E IR, y * 0}, X = 1 , y = 0
X=O l V=.25
V1=11!2X+1l I
f)
V1: /(5-X)
v=.a
B. a) F =
{
x E IR, X of.
=
-
v
l
=
v=-.2
t },
5
(y E IR, y * 0}, X = 2'
y=0
(x E IR, x of. 5}, (y E IR, y of. 0},
X = 5, y = 0
(
?t--
1
X- 1
1
9' y
4x + 4
3850
10. a) t =
V1=11(2X-5)
b)
{
X E IR, x
{
x E IR, x of. -
of.
i },
i}
,
1
(y E IR, y * 0}, X = -2,
y=0
::__L
c) 300 N d) The force is halved.
14. a)
c)
1 y=0
(y E IR, y * 0}, X = 4'
8' y =
J
l
V1=11GX-5)
X=O
(x E IR, x of. -4},
(y E IR, y * 0}, X = -4, y = 0
x=o ( v=-.2s
V1= '3/(2X-5) )
x=o
h)
c)
b)
n
=
X:O ' V=1
d) V1: '1]+ )
y.r
)
jL
r-V1=-1-/(-X--5)_____---..
When 0 < b < 1 , - < -2. When b < 0, ­ > 0.
b
b
12. a)
f'-
e)
c) 7.7 h, or 7 h 42 min
d) As the speed increases, the rate of change of the time
decreases.
1 1 . a) Answers may vary.
b) The equation of the asymptote is x = - . When b = 1 ,
b 2 < 0.
the asymptote is x = -2. When b > 1 , -2 < b
V1:3t<X-2)+
X=O
t-..
b)
V1=1:/·\____
\___
15
V2:1/(2X-5)
Answers may vary. Sample answer: The reciprocal of the
y-coordinates on either side of the x-intercept of y = 2x ­5
1- .
are the y-coordinates of f(x) = -2x
­5
yz
16. X = _ ' y of. Z, X of. 0, y of. 0, Z of. 0
y z
17. 14
18. E
Answers
•
MHR
537
Extension, page 1 56
2. a) no; does not divide pixels evenly
b) yes; divides pixels evenly
3. Answers may vary. Sample answer: Xmax
=
--- b) y =
1
4 . a)
y
=
d) y
=
-
(x - 1 )2
1
(x + 4)2
5. a) i) (x E IR, x of- -3, x of- 3 ) ii) x
iii) y-intercept
47
-
3.2 Reciprocal of a Quadratic Function, pages 1 64-1 67
1. a)
As x _,
fix)_,
1+
0
0
+ oo
f!x) 4
5
Change in
- 00
Slope
+ oo
5•
..)
11
0
As x4
x=O
-
O<x<3
-
x>3
+
0
-
-
-
-
-
+
+
+
+
vi) (y E IR, y * 0) b) i) (x E IR, x * - 3, x * 5 )
0
+ oo
-3 < x < O
+
Sign of Slope
-00
- 00
x < -3
Sign of f(x)
-00
-4
- 4·
x = -3, x
5, y
=
=
1
0 111
...) y-mtercept
.
- 1.5
f{x) _,
-00
-6
- 00
-6 ·
-oo
0
+ oo
0
v)
2. a) x = 4; (x E IR, x * 4)
b) x = 2, x = - 7; (x E IR, x * 2, x of- -7) c) (x E IR)
d) x = - 5, x
5; (x E IR, x * 5, x of- - 5 )
x < -3
Interval
+
Sign off(x)
3, x = 1 ; (x E IR, x of- 3, x of- 1 )
- 4' X = - 3 · (X E iR1' X -:f- -4' X -:f- - 3 )
h) ( x E IR)
x E IR, x * -2, x *
g) x = -2 , x =
;{
Interval
x<1
x> 1
+
Sign of f{x)
Slope
1 <x<5
-
x>5
+
0
-
-
-
+
-
{
vTo9
-
vi) (y E IR, y of- 0) c) i) x E IR, x *
ii) x
-
+
Change in Slope
x=1
-
+
Change i n
+
+
Sign of Slope
Sign of Slope
}
-3 < x < 1
-
+
=
=
3. a)
0
v)
I nterval
As x _,
=
=
- 00
- 00
f) X
- 3, y
+ oo
1
e) x
=
3, x
+ oo
3+
c)
=
-oo
3
b)
1
1
c) y ­
- - -(x + 2 ) (x - 4)
x2 - 9
=
-5
::+::
,y
2
=
-5
::+::
+
v'109}
2
1
.
.
21
0 111..) y-mtercept
-
b)
x < -2
I nterval
+
Sign of f{x)
+
Sign of Slope
+
Change in
Slope
-2 < x < 1
-
x= 1
+
-
-
1 <x<4
-
0
+
-
x < -3
-
Interval
Sign off{x)
-
Sign of Slope
-
Change i n
SloP.e
d)
Interval
Sign of f(x)
Sign of Slope
Change in Slope
538
MHR
•
+
-
Interval
-3 < x < O
x=O
O<x<3
+
+
+
-
0
-
+
-
+
+
+
l
x < -4
-
x > -4
-
-
+
+
Advanced Fu nctions
x> 3
x<
+
-5 -
-5
J
- vlo9
2
v1o9
2
< x < - 2.5
X = - 2.5
- 2.5 < x <
x>
-
•
v)
-
-
c)
x>4
-5
-5
+
v1o9
2
vi) (y E IR, y * 0)
Answers
v1o9
2
+
I
I
Sign of
Sign of
Change i n
f(x)
Slope
Slope
-
-
-
+
-
+
+
0
+
+
+
+
-
+
-
{
d) i) x
E
IR, x
iii) y-intercept
=f.
2, x
--t
=f.
- t} ii) x = 2, x = - t, y = 0
increasing for x < - 3 and
-3 < x <
decreasing for
--t,
_ .l < x < 2 and x > 2
2
1 <x<2
3 an d
.
for 2
mcreasmg
x > l, decreasing for
2
1
1
1
x < - - and - - < x < ­
2
2
2
v)
x<
Interval
_
+
Sign of f{x)
..!_
3
_
..!_ < x <
3
-
+
+
Sign of Slope
Change in
v)
E
Y1=11<X>•2l
Interval
Y=.5
Sign of f{x)
-
Change in Slope
{
vi) y
E
IR, 0 < y <
t}
i<x<2
-
+
-
x=O
x>O
+
0
+
-
i;
X=O
n
X=O
g)
increasing for x < -5 and
- 5 < x < -4, decreasing for
-4 < x < - 3 and x > - 3
1f\_
Y=B
Y1= '1I<X>+3l
X:O
+
no x- or y-intercepts; for x < 0, the function is positive
and increasing (positive slope); for x > 0, the function is
positive and decreasing (negative slope) b) {x E IR, x =f. 1 ) ;
{y E IR, y > 0 } ; asymptotes x = 1 , y = 0 ; y-intercept 1 ; for
x < 1 , the function is positive and increasing (positive slope);
for x > 1, the function is positive and decreasing (negative
slope) c) {x E IR, x =f. - 2}; {y E IR, y > 0}; asymptotes
x = -2, y = 0; y-intercept
for x < -2, the function
is positive and increasing (positive slope); for x > -2, the
function is positive and decreasing (negative slope)
increasing for x < - 1 and
- 1 < x < 0, decreasing for
0 < x < 1 and x > 1
b)
Y1=B/(Xl+1)
-
6. Answers may vary. a) 0 b) 0.009 c) 0.01 1 d) 1 .250 e) 0
7. a) {x E IR, x =f. 0); {y E IR, y > 0}; asymptotes x = 0, y = 0;
X=O
Y1=1/j+BX•1c
e)
t
+
+
x>2
+
x<O
+
Sign of Slope
6
0
-
IR, y =f. 0)
e) i) {x E IR} ii) y = 0 iii) y-intercept
iV)
-
+
Slope
vii) {x
5
x=-
6
<) I)
Y: .3333333
•
b) Answers may vary.
1,
.
. for x < 3
mcreasmg
decreasing for x >
ii)
t
V1= '5/(Kl+5K+Bl
K= '2.\I�
5 Y: ·2.8571 3
10. Answers may vary. Sample answers: a) f(x) and g(x) will
have the same shape reflected in the x-axis.
2- is a vertical stretch of k(x) = --1- by
b) h(x) = -x2 - 9
x1 - 9
a factor of 2.
c) m(x) and n(x) will have the same shape but different
asymptotes.
1
1 1 . Answers may vary. Sample answers: a) y = -:;--.;::._--:2
+
x
x-6
b) y = 1 1 c) y = - 1 ) 2
x +2
(x + 3
136 · 9
12. a) I =
d
Y:.06666667
increasing for x > 6,
decreasing for x < 6
increasing for x > 0,
decreasing for x < 0
' :=t-
9. a),
increasing for x < 0,
decreasing for x > 0
c) I = 1 368.9; rate of change
is approximately -2737.8.
Answers
•
MHR
539
b)
13. a)
;
V1= X/(X•B
-
=_j
16 a)
•
C)
b)
V1=1/(2Xl
X=O
V=.5
x>O
-
+
+
+
V1=CM+1l/( -Xl
)!�
x < -1
-
Interval
V=.2:S
Sign of f(x)
d)
)
)
+
+
Sign of Slope
Change in Slope
r�-1
-
-
V1=1/( X>l
X=1
J
V1=11(
-8 < x < O
+
+
+
Sign of Slope
V=
M=1
x < -8
Sign of f{x)
Change in Slope
c)
Interval
l<
X=1
1 286
V=
x=-H
b) i) approximately 8 1 7.4 N ii) approximately 3 1 0.5 N
c) h
5 1 554.5 km
14. a)
-=:::::
d)
X=1
-1 < x < 4
+
+
+
x>4
-
+
-
V1=(X+2)/( M-5l
V=1
17. a) symmetric about the origin
b) symmetric about the y-axis
18. Explanations may vary.
a)
�,f�''----
Sign of Slope
19. Answers may vary. Sample answer: y =
(x - a ) (x
1
20. a < - 2 7 or a > l
21 . 7
22. c
ax +
:,
3.3 Rational Functions of the Form f{x)
ex +
pages 1 74-1 76
1. a) x = 7, (x E IR, x =!= 7) b) x = - 5, (x E IR, x =!=
=
=
e) x =
2. a) y
E IR, x =!= - 8) d) x =
E
=
IR,
1 , (y E IR, y
=!=
=!=
l ) b) y = 3, (y E IR, y
=!=
- l)
E IR, y =!=
=!=
y = 2, (y E IR, y
e)
V1:( '2X-3)/(X+5)
=!=
5)
2)
.---
+
Sign of Slope
-
-
-
4. a) i) m3_5
Change n
i Slope
-
-
+
X>
540
MHR
•
Advanced Fu nctions
•
Answers
+
-
-
-
+
-
-
+
+
+
+
-
+
+
t
Sign of Slope
Sign of f{x)
4
+
-
-
V1=0X+1l/(2X+1l
O<x<5
-
x>5
x>
X > - 1 .5
Change in Slope
Sign of f{x)
x<O
4
- 5 < x < - 1 .5
Sign of f(x)
f}
-
x < -5
Sign of Slope
Interval
I nterval
\
-2 < x <
V=1.5
Interval
-5)
3)
=!=
b)
x=-3
IR, x =!=
x = 5, (x E IR,
c) y = 1 , (y E IR , y =!= l) d ) y =
e) y = - 1 , (y E IR, y
E
-
Change in Slope
-\
(x
t' {x
t}
x
-* , {x x -*} f)
f, {Y
t}
f)
- 8,
+
Sign of f(x)
1
c) x
x < -2
Interval
Change in Slope
x < - .!..
2
+
+
+
1
1
-< x < -3
2
-
+
+
x > _ .!.,
3
+
+
-
= -24, m20 = -0.02 ii) m2_5 = -24, m_20 = -0.02
b) The function is decreasing for x < 3 and increasing for
3.
') y = 21 ") y = 25 "') y = 2
b) Answers may vary. Sample answer: The horizontal
asymptote is equal to the coefficient of x in the numerator
divided by the coefficient of x in the denominator.
c) =
6. a) y = 1, x = 9,
b) y = 3, X = -2,
{x E IR, x * 9),
(x E IR, x -2),
{y E IR, y * 1}
(y E IR, y 3)
v
5 . a)
I
- II
- - Ill
E.
c
Y
-:F
*
·
12. a)
X=1
vl
1
increases, the rate of change of the initial velocity decreases.
1 5.
:,.-
1, x 1; (x E IR, x 0, x
0; 0 < x < 1, f(x)
1};
16. a)
t
\
V1:X/(Xl-1
V1:( X-3)/(2X+1l
c)
=
\"
V1=(X+5)/(Xl-X-12l
C)
=
333.9 min
7- b) Answers may vary.
2 + -2x - 1
c) after approximately
:=t
V1=2+(7/(2X-1ll
=
-1 -4, 3 2 0
2. x = 10 b) x = 4 or x = -2 c) x = 11 d) x = 5 or x = - 1
3
3
e) X = -34 f) X = 2
3. a) x = 0 or x
6.71 b) x -0.27 or x - 18.73
c) x 4.34 or x 2.47 or x 0.19 d) x
1.28 or x -1.28
4. a) x < 3 or x > 7
1 1 1 1 1 1 1 1 1 1 1 1 1+ 1 1 1+ 1 1 1
-10 -8 -6 -4 -2 0
2 4 6 8 10
b) -1 < X < 0
"" I I I I I I I I I $ $ 1 I I I I I I I I I •
-10 -8
-4 -2 0 2 4 6 8 10
c) x < -4 or - 1 < x :5 1
• 1 1 1 1 1 1+ 1 1+ 1 + 1 1 1 1 1 1 1 1 1 •
-10 -8 -6 -4 -2 0
2
4
8 10
a)
V1= 0X/(200000+X)
b) The amount of pollutant levels off at
=
3.4 Solve Rational Equations and Inequalities,
pages 1 83- 1 85
b)
c) l d)
1. a)
-
=
C)
Common features:
Answers may vary.
-
V=2
=
1 1 . a)
.66o66667
x,
-2 x 2
X ; 3, X = 2
i) y X - 1, X 2 ii) y
iii) discontinuous at x = -3 (linear)
2x - 3 b) y = x - 4
7. a) y
x+ 1
x-3
8, y = X + 4
x-2
9• y = 5x - 3
2x + 1
10. a)
L
-,
------
17. Answers may vary. Sample answer: When the degree of
the polynomial in the numerator is greater than the degree
of the polynomial in the denominator, you can expect to
get an oblique asymptote.
18. A
0
0
19. a ) quotient remamd er
b)
x
V1:( -Xl/(5+Xl
::\
:\ :j
b) ,_
~
-1, x - 5,
(x E IR, x * -5),
(y E IR, y * -1}
=
E IR, :5 0,
x 1,
asymptotes y =
=
>
* 1 } , (y
y
y-intercept for
is negative and
decreasing and the slope is negative and decreasing; for >
i s positive and decreasing and the slope i s negative and
increasing. Comparison: Answers may vary.
y>
f(x)
*
e) y
X•1»
14. Answers may vary. Sample answer: As the mass of the club
V1= X/(X+
1
c) y = 2 x = -'
2'
{x E IR, x -:F -i},
(y E IR, y 2)
t
n=2
30 giL.
-
=
=
=
=
=
=
=
I(
)o
-6
6
Answers
•
MHR
541
d) -5 < x
:s:
2 or x > 4
• I I I I I + I I I I I I + I + I I I I I I ,.
-10 -8
-6
-4 -2
0
2
4
6
e) x < -4 or - 1 < x < 4 or x > 5
8
f) 0
-6
-4 -2
< x < 3 or x > 6
0
2
4
6
8
10
• 1 1 1 1 1 1 1 1 1 + I I I I + + I I I I I ,.
- 1 0 -8
-6
-4 -2
0
2
4
6
8
10
5. a) x < - 7 or - 2 < x < 1 or x > 5 b) - 3 < x < _!_
2
c) - 6 < x < -5 or - 1 :s: x :s: 4 d) x < _!_ or l :s: x :s: 2
2
3
or x > 5
2x
-3
= 0
6. Answers may vary. Sample answer:
(x - 3 )(x + 5 )
7 . x < -4 o r ­1 < x < 0 o r x > 2; points o f intersection
( -4, 0) and (0, 0)
8. - 5 < x < 0 or 3 < x < 7
1 b) x = - 3 ± V14
9. a) x = c) x = 2 or x = 6
3
5
+ 3v'2
-3 f) no solution
{x E IR, x =F 1 } e) x =
2
10. a) x < 0 or x > 9
b) x < 0 or x > 3
d)
'1'1= - 71:0:+3:
\'l
x=·1o
.
10
• I I I I I I + I I + I I I I + + I I I I I ,.
-10 -8
b)
.
..
·.
-=
T
l.
. . . . . ... ..
.
.
:
.
.·
.
·
. .· = . .:
--
.
.
:
.
_
V .2
X 1
_
_
_
3.5 Making Connections With Rational Functions and
Equations, pages 1 89-1 91
b) The light intensity is less.
c) When d is close to 0, the
light intensity is very great.
1. a)
2. a) V =
5000
p
c) The volume is halved.
b)
l
d) x < 1 or x > 3 1
..
or x > 5
- Vi
b) x = y = Vi or x = y = T
2
19. a) X = 2 b) X > 2
20. a)
,... ,. .:rc_ d)
:::rl:
(
X=3
V=3.5
-±)
(
discontinuous at 4,
x=2
( -±)
discontinuous at - 3,
(
discontinuous at 0, - 1
12
5. a)
)
/v=.
•
Advanced Fu nctions
•
Answers
)
( -f)
discontinuous at - 1 ,
f)
( )
discontinuous at 1, _!_
4
and 2,
(
t)
b) Answers may vary.
Sample answer: Average
profit is modelled by
P(x)
----_x- = slope of secant.
c) The average profit is the greatest when x = 200.
d) 9 . 1 8 X 10- 4
MHR
.
.•
12. x < - 7 or - 3 < x < -2 or x > 5 versus - 7 < x < - 3
or - 2 < x < 5
13. x < - 5 or .]___ :s: x < 4 versus - 1 < x :s: .]___ or x > 3
13
13
14. a ) __!__ + __!__ = _!_ b ) 4 0 c) b = l
x
2a
3
3
2b
15. a) i) 3600 lux ii) 2.25 lux b) i) 1 4 1 .4 m ii) 0 < d :s: 2Vs
16. 2 < I < l
2
18. a) l = 3 1 .26 em, w = 0.74 em
542
.
27
3- + 8
2- b)
21 . a) +
5(x + 2 )
x- 1
x + 3 5 (x - 3 )
1- + _
5
3_
c) X + 2
X ­3
(X - 3 ) 2
discontinuous at 0,
11. x < -
. .
. . .
. .
:
3. a) X = 1 b) ­1 < X < _!_
3
4. a) V1=XI<X>·2Xl
......---:
c) O < x < 2
25
'12 1/<X> l
. . . -=== ===-
12. a)
b) The systolic pressure decreases and gets closer to
25.
0.58 s and then
increases gradually, getting closer to 0.
d) rate of change of P(t) at t 5 is -1.48; for R (t) it is 0.84
13. 1
c) The rate of change decreases until t ==
7. Answers may vary. Sample answers:
a)
c
=10 -�v= oo
--,
,-----
c
14. a)
=1o -�1·= 5.5 556 :
The cost is j ust slightly greater per person than the original
model. The cost decreases at a greater rate at first.
b)
The cost is much greater per person. The gap between the
graphs decreases as the nurnber of passengers increases.
The cost decreases at a slower rate.
c)
V1=1oooo?<1o•Nl
n=tsooo?<12• l
=2��v=.o
0.0418
1.414
gradually decreases to C as time increases close to 0.
15. increasing for 0 < R < 0.40
��
b) The curve increases to reach a maximum concentration
mg/cm3 when t ==
min and then
of C =
16. False because the function is discontinuous at point
1 1. A
18. x = 3 +
=
1t 2kn, k 0, ± 1, ± 2, ± 3, . . .
(i.e., . . . , - 531t ' l' 73n ' . . · )
L
-7, 0 5, y = 0
2, 0
1
2
2. a) y = -- b) y
x-1
x +-4
3. a) V1=5/(M -3l
\
'\ 5
M=1
v=-2.
5
{x E IR, x * 3}, {y E IR, y * 0}, -3, x = 3, y = 0
=
=
-.
1 __i_
l d) y = x - 1
x _
{x E IR, x *
C)
10. a)
c::/
{
X=10
Y:3.1B1B1B2
b) approximately 8.39 h c) approximately 5.85 h
100 000
1 1 . a)
b) V1=100000/(1TM>l
h
=
"'
(2, 4 ) .
Chapter 3 Review, pages 1 92-1 93
b) X =
y = C) X =
y
1. a) X =
The cost per person is greater. As the number of passengers
increases, the cost per person decreases and the graphs get
closer. The cost decreases at a slightly slower rate.
b) a slanting asymptote c) x -
,.-�-�=-0-.1:_X_I(-M>-�-N+-2l-----,
=
l
x=2
Y=7957.7 72
V1=11<2X-3l
4), {y E IR, y * 0), 4,1 x 4, y = 0
=
D
3 }' {y E IR, y * 0), -3,1 X - 2'3 y - 0
X E IR, X * 2
d)
Y:·.571 2B6
{x E IR, x * - }, {y E IR, y * 0}, -2, x = - , y = 0
X=2
•
Answers
•
MHR
543
E
X = 3, X = -4, {X IJ\11 , X * -4, X * 3)
X = -3, {x IJ\11, X * -3)
c) x = -6, x = -2, {x e 1R, x * -6, x * -2)
S. a) i) X = -5, X = -1, y = 0 ii) l_
5
V1=1/(:l•iX+5)
4. a)
b)
E
decreasing and the slope is negative and decreasing; for x > 2,
f(x) is positive and decreasing and the slope is negative
and increasing
V1=XI(X-2l
t
···)
L
increasing for x < -5 and -5 < x < -3, decreasing for
-3 < x < -1 and x > -1
{X X * -5, X * -1}, {y y > 0, y -t}
b) i) X = 8, X = -3, y = 0 ii) - 1
24
iv)
Y)
E
IJ\11 ,
E IJ\11 ,
::5
l
X=i
v=-1
b) x = -3, y = -1, {x e IR, x * -1}, {y e IR, y * -3}
y-intercept 0; for x < -1, f(x) is negative and decreasing
and the slope is negative and decreasing; for -1 < x < 0,
f(x) is positive and decreasing and the slope is negative and
increasing; for x > 0, f(x) is negative and decreasing and
the slope is negative and increasing
V1= ·3U(X+1ll
increasing for x < -3 and -3 < x < 2.5, decreasing
for 2.5 < x < 8 and x > 8
v) {x e iR, x * 8, x * -3}, {y e !R, y > O, y s - 1 1 }
c) i) x = 3, y = 0 ii) -t
iv)
iii)
:tV
increasing for x > 3, decreasing for x < 3
IJ\11, X * 3), {y IJ\11, y < 0}
d) i) y = 0 ii) iii) V1= ·U(X>+5)
iv)
Y)
E
{X E
-.
e
Interval
-
x< 2
2
<x<
x =
-l
4
3
-4
_i < x < l
4
x > l
Sign of Slope
Change in Slope
+
+
+
-
0
-
-
-
-
+
1
· Y = - (x + 4)(x - 5)
8. a) y = 1 b) y = -2 c) y = 1
9. a) x = 2, y = 1 , {x e IR, x * 2},{y e IJ\11 , y * 1), y-intercept 0;
for x < 0, f(x) is positive and decreasing and the slope is
negative and decreasing; for 0 < x < 2, f(x) is negative and
7
544
MHR
•
Advanced Functions
•
E IJ\11 ,
E
X=O
V=
iv) increasing for x > 0, decreasing for x < 0
v) {x e IJ\11} , {y IJ\11, - s y < 0}
6
'---
:----\ V=-1.5
c)x = -4, y = 1, {x e IJ\11, x * -4), {y e IJ\11, y * 1}, y-intercept
- ±, x-intercept 2; for x < -4, f(x) is positive and increasing
and the slope is positive and increasing; for -4 < x < 2,
f(x) is negative and increasing and the slope is positive and
decreasing; for x > 2, f(x) is positive and increasing and the
slope is positive and decreasing
V1:(X-2l/(X+ l
__/
x=-1 .t-1
d) X = t' y = 3, {x IJ\11, X * ±} , {y
y * 3);
.
1
1 1,x
"'( ) 1s positive
y-mtercept - 2, x-mtercept
.
-3; for x < - 3,
and decreasing and the slope is negative and decreasing;
for -% < x < t' f(x) is negative and decreasing and the
slope is negative and decreasing; for x > t' f(x) is positive
and decreasing and the slope is negative and increasing
V1:(iX+2li(2X-1l
'
Answers
:=t=
.
. .
-1
f(x) = 4x
3x + 2
15 b) x = - 9 or x 3
11. a) x =
T
12. a) x = 0 or x 0.86 b) x ='= 40.88 or x 0.12 c) x ='= 1.64
13. a) x < -5 or x > - Z
2
b) -3 < x < -2 or x 1
c) x < -4 or -2 < x < 5 or5 x > 6
d) - 7 < x < -5 or x > - 3
14. a ) -4 < x < -1 or 2 < x < 3
b) x < -8 or x > -21 and x * 3
10.
='=
=
='=
15. a)
( t)
b) Profit increases as sales
mcrease.
c) The rate of change of the
profit at 100 t is 1 . 875
and approximately 0.208
at 500 t, so the rate of
change is decreasing.
b) discontinuous at
16. a) discontinuous at 0,
l
(
7,
V1:Xt(X>+SX
ii)
13. Answers may vary. Sample answer: x
0, y = 0, slopes
increasing and decreasing faster as n increases.
n even:
• For x < 0, f(x) is positive and the slope is positive and
increasing.
• For x > 0, f(x) is positive and the slope is negative and
increasing.
n odd:
• For x < 0, f(x) is negative and the slope is negative and
decreasing.
• For x > 0, f(x) is positive and the slope is negative and
increasing.
=
Chapter 1 to 3 Review, pages 1 96-1 97
1 . a)
Chapter 3 Practice Test, pages 1 94-1 95
1. c
2. B
3. A
4. Answers may vary. Sample answers:
2_ b) y =
a) y = _
x+2
6
(x +4)(x ­3 )
5 . a ) i ) {x E } , {y E , - 2 :5 y < 0} ii) y-intercept - 2
iii) y = 0 iv) decreasing for x < 0, increasing for x > 0
b)
V1= · /(X>+ )
v
X=1
v= -unn3
6. Yes; -1- will always have an asymptote at y = 0.
f(x)
14
7. a) X =
-
8. a) X
-2, X > - 8
5
b) X = 3
7
3
or 2
5
-x + 2
9. a) Answers may vary. Sample answer: y =
2( x + 1 )
- 2x + 4
b) Yes. Sample answer: y =
4(x + 1 )
401 800 000
<
1 0. a) g =
b)
b) X < - 1
dz
\_
1 1 . a)
v=1oo s
- 1,
1 , and 2; y-intercept 2
"T
x-intercepts approximately ­2.88 and 3 . 6 3 ; y-intercept
- 16
2. a) The graph extends from quadrant 2 to quadrant 1 ,
thus, a s x - oo , y oo , and as x oo , y oo .
The graph is not symmetric.
b) The graph extends from quadrant 3 to quadrant 1 ,
thus, a s x - oo , y - oo , and a s x oo , y oo .
The graph has point symmetry about (0.55, 0).
3. a) i) - 6 1 ii) - 3 7 b) ­49 approximates the instantaneous
rate of change at x = 2.
4. a)
c) d ='= 8 1 83.3 km
V1= HBOOOOOIX>
x=2oo
< X <
x-intercepts
X= 1 ..:.:..._
'-'-'-'
::...
''-'
,._
V1=100X/(X+30)
)(
X=5 ��-V=1 . B571
.
b) {t E , t :2: 0}, {P E , 0 :5 P < 1 00}
c) The percent lost can get close to 100%, but not equal
to 100%.
12. a)
X=
��V=12.S
�-
b) The power output increases
from 0 Q to 2 Q. The power
decreases from 2 Q to 20 n.
c) The power is constant at
R = 2 (not changing).
b) 4 c) y = - 2(x - 1 )2(x + 3 )2
d) Answers may vary. Sample answer: Reflects and stretches
the graph. Also, since the function has even degree, a negative
leading coefficient means the graph extends from quadrant 3
to quadrant 4 and has at least one maximum point.
6. a) - 4 b) 1 2 c) local minimum
An swers
•
MHR
545
7. a) maximum approximately ( 1 2.25, 9.64), minima
approximately (3. 14, ­14. 1 6), (26.61, - 70.80)
b) between (3.14, - 14.16) and ( 12.25, 9.64) approximately
2.6 1 ; between ( 1 2.25, 9.64), and (26.6 1 , - 70.80),
approximately -5.60 c) x = 32
8 . Answers may vary. Sample answers:
y = 2x(x + 7)(x ­3 f; y = - lx(x + 7)(x - 3 )2
3
9. y = k(x ­2)2(x + 5). Answers may vary. Sample answers:
y = 2(x - 2)2(x + 5), y-intercept 1 0; y = - 3(x - 2)2(x + 5),
y-intercept - 60
10. a) 4x3 - 5x2 + 6x + 2 = (2x + 1) 2x2 - Zx +
- .l.l,
2
4
4
_l
X -:f.
2
b) 3x4 - 5x2 ­28 = (x - 2 )(3x3 + 6x2 + 7x + 14), x -:f. 2
9
1 1 . a) 3 8 b)
12.)
(
_
;
12. a) No. b) Yes.
13. k = _ Q
2
14. a) (x - 3 ) (x2 + 3x + 9) b) (x - 2)(2x2 + 8x + 3 )
1 5 . a ) x = -4 o r x = 1 o r x = 5
-4 - v'31
-4 + v'31
b) x = - 3 or x =
or x =
5
5
16. a) x ::; 1 or x 2:: 6 b) x < - 3 or -2 < x < 2
17. from 0 min to 10 min
18. A
19. a) f(x)
0 b) f(x) 0 c) f(x) oo d) f(x)
- oo
20. {x e IR, x -:f. - 1 ), {y e IR, y -:f. 1)
21. a) i ) {x e IR, x -:f. 2), {y e IR, y -:f. 3 ); x-intercept .l
1 asymptotes x = 2 , y = 3 ; neganve s ope
.
y-mtercept
- 4;
x < 2, x > 2; decreasing x < 2; increasing x > 2
=--
-
ji)
Y1:(6X+1)/(2X- )
Prerequisite Skills, pages 200-201
1 . a) cos e = ± ' tan e = l b) sin e = _ Q ' tan e =
5
4
13
_
c) sin e =
V2 '
V2 '
V3 ' cos e = .l ' tan e = V3
2
2
9. a) .----------:
-----,
X
ii)
b) x = -1-, y = -1-c) csc45° = V2, sec45° = V2, cot45° = 1
V2
V2
1 cos 1 35° = ---,
1 tan 135° = - 1 ,
10. sin 1 35° = --,
V2
V2
esc 135° = V2, sec 1 35° = -V2, cot 1 35° = - 1 ,
sin 225° = - -1-,cos 225° = - -1-, tan225° = 1 ,
V2
3x + 6
x-1
23. a) X = - 2.2 b) X ='= 2. 1 5
24. a) x < - 2.75 or x > -2 b) ­1
25. a)
V2
csc 225° = -V2, sec 225° = -V2, cot 225° = 1 ,
sin 3 1 5° = - -1-, cos 3 1 5° = -1- , tan 3 1 5° = - 1 ,
V2
22. f(x) =
::5
x
<
1 or x > 2
V2
csc 3 1 5° = -V2, sec 3 15° = V2, cot 3 1 5° = - 1
1 1 . a) 5 b) 1 3 c) 1 7 d) 1 0
12. a ) a2 + 2ab + b2 b) c2 - d2 c) 6x2 - x y - 2y2
d) sin2 x + 2 sin x cos y + cos2 y
4.1 Radian Measure, pages 202-21 0
b) {x E IR, X
{
2::
0), P(x)
E
IR,
- ,; P(x) < 5}
c) The profit is always less than $5000.
26. Since t represents time, t 2:: 0; t -:f. 10 because the
denominator cannot be zero.
MHR
•
Q
,
b) i) {x e IR, x -:f. - 3, x -:f. 3), {y e IR, y -:f. 0); no x-intercept,
.
1 asymptotes x = - 3 , x = 3 , y = 0 ; positive
y-mtercept - -;
..
9
slope x < ­3, - 3 < x < 0; negative slope 0 < x < 3, x > 3;
decreasing 0 < x < 3, -3 < x < 0; increasing x < -3, x > 3
546
_
5
c) sinx = _]_ cos x = - 24 d) cos x = -..!.l. ' tanx = .!...
15
17
25
25 '
2 . a) 0.25 8 8 b) 0.5592 c) 3 .7321 d ) 0.9848 e) -0.9205
f) 2.7475 g) -0. 8480 h) 0.978 1
3. a) 4 1 ° b) 65° c) 83° d) 1 17°
4. a) .l b) 3 c) i d) _1_
3 V3
2
.) 3
5
13
12
5 . a ) secx = ) ' cotx = 4 b ) csc e = - 5 , cot e = - 5
8
17
25
25
c) csc x = ?' secx = d) sec e = -8, cot e = 1.5
24
6. a) 1.7434 b) - 1 .2361 c) 2.1445 d) 1 . 1 792 e) - 1 .2690
f) 1 .0724 g) - 1 .5890 h) 1 .0038
7. a) 53° b) 54° c) 1 8° d) 1 39°
V3
1
1
. e = l'
8. a) Sin
COS e = l' tan e = YJ
b) sin e = -1- cos e = -1- tan e = 1
-
.
\____
C H A PTER 4
Advanced Fu nctions
•
Answers
c) 2 n d) 5 n
6
3
2. a)
c)
d)
18 24 36
12
5
3
3. a) b) n c) n d) n
4
4
2
4. a) b)
c)
d)
12 20 60
8
1 . a)
3
b)
2
b)
'
'
21t
b)
c) 71t d) 71t e) 5 1t f) 51t
9
18
4
6
3
12
6. a) 0.40 b ) 0.89 c) 1 .43 d) 2.23 e) 4 . 1 9 f ) 5.76
7. a) 36° b) 20° c) 75° d) 50° e) 1 35° f) 270°
8. a) 1 34.1 o b) 1 79.9° c) 301 .9° d) 431 .4° e) 3 9.0° f) 98.5°
9 . 1 1 8.75 em
10. a) 720°Is b) 4n rad/s
21t 2 1t
1 1.
5' 5 ' 5
12. Answers may vary depending on speed.
1 3. a) 0.000 2 9 1 b) 1 862 m c) Answers may vary.
14. b) 8 km
1 S. 0.009 053 rad, 0.5°
16. 0.5 rad, 28.6°
17. 400Jt rad/s, approximately 1 256.6 rad/s
8 1t m b) 83.8 m
18. a)
1 9 . a) It must follow the rotation of Earth. b) 24 h
c) 0.000 023n rad/s d) It is the same.
20. a) 1 00 grad b) 3 1t
4
cos 5 1t = tan 5 1t = 1 ,
c) sin 51t = 4
4
4
esc 5 Jt = -V2, sec 5 1t = -V2, cot 51t = 1
4
4
4
d) sin 1t = 0, cos 1t = - 1 , tan 1t = 0, csc 1t = undefined,
see n = - 1 , cot 1t = undefined
9. a) 20(v'2 ­1) m b) 20(VJ - V2) m c) 8.3 m horizontally,
6.4 m vertically
10. a) (20v'2 - 30) m
b) The kite moves farther from Sarah, since the horizontal
distance at is now greater than at ­
3
4
c) (30VJ - 20v'2) m; the altitude increases since the
vertical distance of the kite at has increased.
22. Using v =
b) 21t c) 9:00 d) 1 1 :00 e) 5 1t
4
3
1S. b) 0.500n radians d) The values are approximately the same.
16. a) 0 b) 1
1 7. a) 1 b) 0
18. a) 0 b) undefined
1
21. a) sin( 150 grads) = -1- , cos( 150 grads) = - . r:; '
v2
V2
tan( 150 grads) = - 1 , csc(150 grads) = V2,
sec( 150 grads) = -V2, cot( 150 grads) = - 1
b) Answers may vary.
22. d) 0.00 :S: X :S: 0.30 e) 0.000 :S: X :S: 0.100
23. Answers may vary.
24. A
S. a)
,y/G X MEacrh
11
, where G = 6.673 X 10­
R
N·m2/kg2, ME,.rh = 5.98 X 1 024 kg, and R is the radius of
orbit for the satellite, the orbital speed of a geostationary
satellite is approximately 3073 m/s.
23. Using the modern definition of a nautical mile, 1 852 m,
the knots are approximately 1 5 .4 m apart.
24. the same
2S. a) A = r28 b) 45.24 cm2
B 2,
26. a) A 1 ,
c 2, 7n , D 2, 3 n
4
2
3
n
bl il
iil (5, 2.2 1 ) iii! 5,
2
(
i
1),
(Y2, )
(
),
( ) ( )
( )
4.2 Trigonometric Ratios and Special Angles, pages 2 1 1 -2 1 9
1 . a) i ) 0.4226 ii) 0.3090 iii) -2. 1445 iv) 0.2588 b) i) 0.4223
ii) 0.3087 iii) -2 . 1 452 iv) 0.2586 c) The degree measures are
approximately the same as the radian measures.
2. a) i) 0.9356 ii) - 0. 8 1 87 iii) - 0.09 1 8 iv) 0.0076
b) i) 0.9336 ii) - 0.8 1 92 iii) - 0.0875 iv) 0.0000
c) The degree measures are approximately the same as the
radian measures.
3. a) 0.7071 b) 0.9010 c) - 0.5774 d) 0.4142
4. a) 3.8637 b) 1 .6243 c) -0.6745 d) -2.6695
S. a) 1 .2 1 23 b) ­3.7599 c) 14.5955 d) 1 .0582
6. a) - 2.0000 b) 1 .5270 c) -0.3249 d) -2.7475
V3
7. a) sin 2 1t =
, cos 2 1t = _ .!., tan 21t = - VJ
3
2
3
2
3
VJ
5
5
!
tan 1t = - -1b) sin 1t = . . cos 5Jt = 2'
6
6
2 '
6
V3
c) sin 3 1t = - 1 ' cos 3 1t = 0 tan 3 1t = undefined
'
2
2
2
7
1
7
d) sin 1t = - - cos 1t = -1- tan 71t = - 1
4
4
4
V2 '
V2 '
V3
8. a) sin 71t = _ .!. cos 71t = tan 71t = -1-'
6
6
2'
6
2 '
V3
7
7
7
1t
1t
1t
= '3
esc
= - 2 sec
= _ ..1.._ ' cot
'
...j3
6
V -'
6
6
V3
4
4
4
.!.
b) sin 1t = cos 1t = _ ' tan 1t = '3
3
Y -' ,
2 '
3
2
3
esc 41t = - ..l_ ' sec 41t - 2 cot 41t = -1'
3
3
3
V3
V3
·
·
1
d) 1 .7 m horizontally, 23.7 m vertically
1 1 . a)
12. a)
v;
1
b)
2
b)
2
13. a) 30v'2 m b) 15v'6 m
14. a)
2S. B
%
( %)
4.3 Equivalent Trigonometric Expressions, pages 220-227
1. sin
2. cos
3. - sin
4. -esc
s.
6.
5 1t
14
= cos
= sin
%
(
-
­
)
(I + %)
= sec ( + )
= cos
18
7 . 2 1t
9
8. 3 1t
7
9. a) 0.6549 b) 0.6549
10. a) 0.8391 b) - 0.8391
1 1 . 0.12
1 2 . 0.93
13. 2.32
14. 2.91
1 S. sin ( 1t - x) = sinx, cos ( 1t - x) = -cos x,
tan (Jt - x) = - tan x, csc (Jt - x) = cscx,
sec ( 1t - x) = - secx, cot ( n - x) = -cot x
=
Answers
•
MHR
547
16. sin ( n + x) = - sinx, cos ( n + x) = -cos x,
tan ( n + x) = tanx, esc ( n + x) = -cscx,
sec ( n + x) = -secx, cot ( n + x) = cotx
3 1t - x = -cosx, cos 2
3 1t - x = -smx,
17. sm 2
(
.
(
(
)
)
)
(
(
)
)
.
tan 31t - x = cotx, esc 3 1t - x = - secx,
2
2
sec 31t - x = -cscx, cot 3 1t - x = tan x
2
2
18. sin 3 1t + x = -cosx, cos 3 1t + x = sinx,
2
2
tan 31t + x = -cotx, esc 3 1t + x = - secx,
2
2
sec 3 1t + x = cscx, cor 31t + x = - ran x
2
2
19. sin (2 n - x) = - sinx, cos (2 n - x) = cos x,
tan (2n - x) = -ran x, esc (2n - x) = -csc x,
sec (2n - x) = secx, cot ( 2n - x) = -cotx
20. Answers may vary.
2 1 1t
21. Answers may vary. Sample answer: -cos
26
z
z
z
v
1[
v
v
cot
8
=
22. a) r = g tan - ­8 =
b) 255 m
g
g tan 8
2
24. Answers may vary. Sample answer: 1t
16
25. Answers may vary. Sample answer:
12
26. Answers may vary.
28. a) iJ 1 ,
- 1 , - 2 n iil 5,
-5, 5 n
6
3
3
b) i)
ii) ( -4, 0)
,
(
(
(
)
)
)
( ), {
(f )
+ 2 nk, k
(
E
Z ii)
-
(
(
( )
-
c) i )
(
)
)
)
)
-
--
) ( - ), {
+ 2 nk, k
E
)
1. a) sin
(
(
4
+
12
)
)
2
(
4
-
.
(
4
(
V3
548
+
)
V3
V3
MHR
•
)
'
V3
V3 .
V3
(
-
(
)
·1
12 ' 2
-) · V3
) V3
) V3
V3 V3
V3
V3
V3
Adva nced Functions
sin( 8 + 8)
sin 8 cos 8 + cos 8 sin 8
= 2 sin 8 cos 8
13. cos 2x = cos (x + x)
= cos x cos x - smx smx
= cos2 x - sin2 x
527
b) ­33 6 c) 2.86
15. a)
625
625
16. For question 1 2:
Plot1
Plot
+
+
+
+
Plot3
'Y 1 Bs i n (2 X )
,y a2s i n< X > co s < X
)
'Y 3 =
,y =
'-Y s =
'-Y 6 =
For question 1 3 :
Plot1
Plot2
Plot3
' Y 1 Bcos <2X )
'Y acos ( X ) -s i n (
X)
,y 3 =
,y =
'Y s =
'-Y 6 =
For question 14a):
Plot1
Plot
Plot3
' Y 1 Bcos < 2 X )
' Y 2 B l -2si n ( X ) 2
'Y3=
,y =
'- 'r' s =
'-Y 6 =
''r' 7 =
Plot1
· 1 d) cos
12 ' 2
4
12 ' 2
7n -­
3n + ­
n · - b) sm 1t · - 2. a) sm 5
15 ' 2
5
15 '
2
c) cos 2 1t + 5 1t · O d) cos 1 01t - 5 1t '· 9
18
9
18
2
+ 1
-1 +
+ 1
1
3. a) V3 l b)
c)
d)
V
V
2Vl
2V
2 l
2 l
+ 1
+ 1 b)
4. a)
2 Vl
2 Vl
-1 -1 +
5. a)
b)
V
2 l
2 Vl
1 1 b)
6. a)
l
2 Vl
2V
­1
1 +
7. a) -V3 l
b)
2V
2 Vl
8. a) cos x = ± b) sin y = ll
5
13
6
9. a) 3 b) _ 33 c) _ ..!f d) 56
65
65
65 65
10. a) cos x = _ _g b) sin y = ±
13
5
1 1 . a) _ 33 b) 6 3 c) _ 56 d) _ ..!f
65 65
65
65
c) cos
=
=
For question 14b):
Z
4.4 Compound Angle Formulas, pages 228-235
V3 b) sin (
'·
12. sin 28
•
Answers
Plot
Plot3
'- Y 1 Bco s ( 2 X )
''r' B2cos ( X )
,y 3 =
,.,.. =
'Y s =
'-Y 6 =
'Y 7 =
-1
1 7. a ) h 1 = 1 2 sin x
18. a) 66.5°; Answers may vary. The Sun is not seen at all
at this latitude.
b) - 2 3 .5°; Answers may vary. The negative sign represents
a latitude in the southern hemisphere. The Sun appears
directly overhead at noon.
sinx cos y + cos x sin y
20. a) tan (x + y ) =
.
cos x cos y - sm x smy
.
c) Both sides of the equation equal -
v;.
v;.
21. b) Both sides of the equation equal
22. a) tan 2x
b)
Plot1
=
Plot2
2 tan
1 - tan-x
Plot3
,y 1 atan < 2 X )
W 2 B2tan ( X ) / ( l -t
an ( X ) )
'-Y 3 =
,.,.. =
'Y s =
'-Y 6 =
c) Both sides of the formula equal approximately 1 . 70 3 6.
b) sin x ­siny
25. 0. 71 rad
=
V3.
( ) ( )
23. a) Both sides of the formula equal
x+y
x-y
2 sin -- cos -2
2
14. a) sin
27. a)
8
8-
!!'._
6
sin 8
b)
0.01
0.05
0.1 0
0.1 5
0.25
0.35
0.01 000
0.04998
0.09983
0.1 4944
0.24740
0.34285
0.01 000
0.04998
0.09983
0.1 4944
0.24740
0.34290
O.Ql
0.05
0. 1 0
0.1 5
0.25
0.35
0.99995
0.99875
0.99500
0.98875
0.96875
0.93875
0.99995
0.99875
0.99500
0.98877
0.96891
0.93937
8
1 -
f.
2
cos 8
28. a) sin e
=
e
+!
b) cos e = 1 c) tan e = e
4.5 Prove Trigonometric Identities, pages 236-241
Plot1 Plot2 Plot3
9. b)
'Y 1 EI 1 -2cos ( X ) 2
'Y 2 Eis i n (X ) cos ( X )
< tan < X ) - 1 /tan ( X )
)
'Y3 =
,y =
,y =
1 4 . Answers may vary.
1 7. a) Yes, the graphs appear to be the same. identity
18. a) Answers may vary. Graphs are different.
b)
b) While the left side results in both positive and negative
values, the right side is restricted to positive values only.
20. a) Yes, the graphs appear to be the same. b) identity
22. tan 2x =
2 tan x
1 - tan2x
Chapter 4 Review, pages 244-245
1 . a) 0.58 b) 2.41 c) 4.40 d) 6.06
2. a) 7 1 .0° b) 1 6 1 .6• c) 273 .9• d) 395.9•
5
3. a) 1t
c)
d)
b)
b)
b)
b)
Revolutions
per Minute
Degrees per
Second
Radians per
Second
1 6 rpm
33
96"/s
1 7t
87t
18
(
rpm
rad/s
45 rpm
78 rpm
270"/s
468"/s
31t
rad/s
2
397t
15 rad/s
)
)
==
0.4567
= cot
- 41t = tan 41t
18
9
9
2
b) 5.6713; tan l 3 n = tan 3 7t = cot
9
2
18
18
13. 1 77t
22
-
(
V3
3
- __
'
4
2
b) sin ( 51 7t2 - 4 ) '- 12
)
(
V3
c) cos 5 7t + '· _ 1_ d) cos 57t - '2
2
12 4
12 4
24
1.
15. a) cos x =
b) sin y =
c) ±
5
25 5
336
527
16. a)
625 625
1
- .__ V3
=..
17. ___::...__:_
V2
2 .
b)
19.
b)
Plot1 Plot 2 Plot3
,y1 El l /cos ( X )
'Y 2 EI ( 2 ( cos ( X ) s i n
< 2 X ) -s i n ( X ) cos ( 2
X ) ) / s i n <2X)
,y 3 =
,y =
'Y s =
21. a ) No; the graphs are not the same for all values.
b) Rewrite 3x as 2x + x. Then, use the addition formula for
cosine to expand cos (2x + x). Next, apply the appropriate
double angle formulas and simplify.
22. Not an identity. Let x = 0; L.S. =f. R.S.
Chapter 4 Practice Test, pages 246-247
1. B
2. c
3. c
4. D
5. B
6. c
7. A
8. a) 1 3°/day, 0.23 rad/day 88 4 1 2 km/day
b)
V6
11
11
11
esc 47t == 1 .0993, sec 47t == 2.4072, cot 47t
11
11
11
V2. + 1
8. a) 2 b)
V2.
V3
15
9.
m
4
3
10. n
14
12. a) 5.6713; cot
)
)
10. a) J.Q_ m b) 32.7 m
4
4
4
7. sin 7t = 0.9096, cos 7t = 0.4154, tan 7t = 2. 1 897,
11.
+
(
)
= sin
- 71t = cos 71t
18
18
2
9
0.3420 · sin 81t = sin
+ 77t = cos 7n
'
9
2
18
18
- V3 ­1
12. a)
2V2.
36
24
. y = 12
13. a) cos x = sm
c) 25
325
U
14. Yes; the engine's maximum velocity (293.2 rad/s) is
slower than the maximum velocity of the propeller (300 rad/s).
15. (5000 + 2500 V3) km
18. Answers may vary. Sample answer: Let x = 0 and y =
2.4( V3 ­ 1)
m
20.
V3
21. a)
A , 0.5 , B 5 n , 0.5 ; cos x = cos (2n - x)
3
c) cos x = - sin 3 7t - x
2
d) No. An identity must be proven algebraically.
1 1 . a) 0.3420; sin
200"/s
1S rad/s
5 7t
12
V3
9.
2
12 9 15 20
4 . a l 72• so• c) 1 05• dl 1 10·
27t rad/s
5. a) 72•Is
5
6. a),
(
(
(
b)
)
b)
-
b)
(} ) (
( )
)
C H A PT E R 5
Prerequisite Skills, pages 250-25 1
1 . a) 0.5 878 0.9659 c) - 5.6713 d) -0.4142
2. a) 5.9108 32.4765 c) 0.3773 d) - 1 .4479
1
V2.
b)
b)
3 . a) - - b)
V3
c)
2
-
1
2
1 .h
f) v 3
2
- 1 d) - e) -
Answers
•
MHR
549
4.
5.
a) -V3 b) -V3
2
2
2
c) - 1 d) 1 e) 0 f) .vh
Y1=sin(X)
7. The graphs of sine and cosine are periodic because they
repeat a partern of y-values at regular intervals of their domain.
8. amplitude 3 , period 1 800, phase shift 3 0° to the right,
vertical translation 1 unit downward maximum 2,
minimum -4 c) 3 9.r, 1 1 0. 3 °, 2 1 9. 7° d) - 3 .6
9. amplitude 2, period 3 60°, phase shift 90° to the left,
vertical translation 1 unit upward maximum 3 ,
minimum - 1 c) 2 1 0°, 33 00, 570° d) 1
10. 3 1 . 3 ° 1 4 1 . 3 ° c) 74. 3 ° d) 27.9°
11. 0.2 2 . 3 c) 0.9 d) 0.2
X = 1, X = - 2 y = 0
a)
a)
a)a)
12.) a)
b)
b)
b)
b)
J
I\
1/f\
a)
a)
b)
b)
5.1 Graphs of Sine, Cosine, and Tangent Functions,
pages 252-260
( ; ) (I'
(-I, ) ( )
1. a) maxima - 3 , 5 ,
)
b)
2.a)
C)
6. a)
( ), (I' )
(-I, ) ( )
-'�_/
b)
( .±;-)
8.
(
b)
y = sin4x
V1=cos(X-51T/6)
b) y = cos ±3 x c) y = sin l3 x d) y = cos 2x
b)
V1=cos( /3X)
Y1=sin< X)
/'"'-...
r"'\ /:'.,
'..../ '.J
X:O
c)
d)
V1=sinUI3X)
9.
b)
a)
V=1
Y1=cos(2X)
Y:O
a)
b)
5 2 units upward
c) Window variables: x E [0, 4n], Xscl
10.
~
b) y = 5 cos x c) y = -4 sin x d) y = - 2 cos x
550
Advanced Functions
Answers
Y=7
V=1
X:O
y = 3 sin 2x
Window variables: x E [0, 2n] , Xscl
Y1=cos(X)-5
•
(
)
cos x ­ 5 1t c) y = sin x + 3 1t
4
6
d)
Y1=sin<X+ 1T/ )
a)
a)
=
* +-
X=O
3. a) y = 3 sin x
•
+ }-) b) y
Y1=sin<X+1TI3f
X=6.2B31B53
MHR
(
sin x
1 ----
d)
Y1=sin(X)-2
C)
5 ;
3 , 321t , 3
maxima ( -2n, - 4), (0, -4), (2n, -4);
minima ( -n, - 6), (n, - 6)
-1 ;
c) maxima - 3 1t , - 1
2
minima
- 3 , 31t , - 3
2
d) maxima (0, 2), (2n, 2); minima ( 1t , 0)
minima
=
d) y = cos x -
7.
v=-.5
3 ; the function is linear, so the rate of change is the
slope of the line. same
14. 1 4.4 km/h
1 5.
1 7 m/s 15 m/s c) the speed at 0.5 s
13.
y
b)
Y1=1/(X>+X-2)
X:O
5. a)
I' y
E
[ -4, 4]
I' y
E
[ -4, 8 ]
)
1 1 . a)
b)
rad to the left c) y = sin
d) Window variables: x
E
h
x+
)]
[- 6 , S67t ] , Xscl 4 ' y
E
c)
d)
[ -4, 4]
22. a) 3
b)
r-:1.0 71976
!
b) 880 1t
4 0
13. a) 120 b) 1 c) y = 1 20 sin 1 207tx
60
d) Window variables: x E
Xscl -1-, y E [ - 1 50, 1 50],
120
0
3
Yscl 50
12. a)
r·vv
r-:.01666667
Window variables: x
E
[- 37t , 3 7t], Xscl
Y=B.136E " 1 0
[o, .l..] ,
23. a)
b) i)
V=1.50BE '7
14. a) Odd. The graph of y = sin ( -x) is equivalent to the
graph of y = - sinx.
b) Even. The graph of y = cos ( -x) is equivalent to the
graph of y = cosx.
c) Odd. The graph of y = tan ( -x) is equivalent to the
graph of y = -tan x.
1 5. Answers may vary.
16. g) For positive x A , the amplitude gets larger as xA gets
larger. For negative xA , the amplitude gets larger as xA gets
larger, but the graph of y = sin x is reflected in the x-axis.
h) The amplitude range changes.
17. a) a = l b) c = 2. c) The period is 60 s. d) k =
2
2
30
2
1 8. a) d = 0.6sin 7t t
3
b) Window variables: x E [0, 6], y E [ ­1 , 1 ]
( )
b)
d) 3 1t
c)
6 4
24. a) i) r2 = 4 ii) r
3
x2
+
4
'y
E
[- 2,
2]
.
5
8
8 iii) r = 4 sin(} + 1
3 cos + 4 sm
l = 3 6 ii) x2 + l = 3x iii) x2 + y2 = 2x + 2y
=
5.2 Graphs of Reciprocal Trigonometric Functions,
pages 261 -269
1. X ='= 0.20, X ='= 2.94
2. X ='= 1 .05, X ='= 5 .24
3. X ='= 2.90, X ='= 6 .04
4. a) The cosecant function is the reciprocal of the sine
..Jz
(..Jz)
function and sin_ , is the opposite operation of sine.
b)
esc
='=
1 . 5 3 9 3 , sin-'
=
%
S. a) The secant function is the reciprocal function of the
cosine function and cos-' is the opposite operation of cosine.
V3
b) sec
= 1 .5425 cos­
' V3 =
'
6
2
2
6. a) The cotangent function is the reciprocal of the tangent
function and tan­' is the opposite operation of tangent.
1
b) cot 1 ='= 0.642 1, tan­( 1 ) =
7. a) secx = csc x +
or sec = csc x - 3 7t
2
b) Answers may vary. Yes; the phase shift can be increased
or decreased by one period, 27t.
8. a) Window variables: x E [- 21t, 2 1t ], Xscl
y E [ -4, 4]
( )
(
c) The waves will be closer together. The equation becomes
d = 0.6 sin 1tt.
20. b) V1=obs(s;n(r.))
b)
x
='=
0.944 or x
)
='=
(
'
)
- 0.944
1 (2) or approximately
9. b) The range is 0 :s x :s tan­
0 :s x :S 1 . 1 07. c) Assuming the lifeguard swims a portion
c) Yes; it passes the vertical line test.
d) Even; it is symmetric about the y-axis.
471t , 4?7t Xscl
21 . Window variables: x E
y E [ -4, 4]
24 24
2
[-
],
'
of the distance, w :s d :s VSw.
d) Answers may vary. e) The total distance will be shorter.
10. b) 1 . 1 5 m
c)
r.:.7B5 9B16
V=2
r.=1
Answers
•
MHR
551
d) As x approaches 0,
d
approaches infinity. This means that
the angle of elevation on the summer solstice approaches the
horizon and so the length of the awning approaches infinity.
As x approaches
approaches 0. This means that the
angle of elevation on the summer solstice approaches an
overhead location and the length of the awning approaches 0.
11. a) x = 0.70 b) No; x = 0 .4 0 . c) No; x =
12. a) Answers may vary. Sample answer: csc2 x ­1 = cot2 x.
19. a)
I' d
1.28.
=
OO=-O-'-V'i-2
d = 500 secx b) d _lV3
-1
13. a)
C)
=
V1:5QO/co5(X)
X=.785 9816
V=707.10678
15. a) Window variables: x
V
b) i)
CT(X)
v
e
[-
rt ,
4
],
rt Xscl
4
3rt
2
I' y e [ -4, 4]
(J
n
v
V1= t,;n(X)
u
n
X=1.570796
V=
(\
iv)
V1=1/5;n(X- )
u
n
X= .581 893
00
v
11. b) No; the equation is only true for x
18 l
·s
552
e
[- I, I]·
2 cos O
•
Advanced Functions
•
Answers
(r, 6)
(2 , 0)
6
V3
(V3. )
rt4
rt3
rt
3
34rt
Srt
rt
6
( · %)
2
V2
( )
(o. I)
1,
1
0
23rt )
( - __?__ 3rt4 )
(
-1
2
-1,
VI '
V2
(- V3, s;)
- V3
( - 2,
-2
3 rt
2
MHR
I
2
2rt
b) i)
=
0
2
00572
r
rt)
c) amplitude 3, period 2
Window variables: x E [ - 2rr, 2n], Xscl
ii)
d) amplitude
i'
period 81t
Window variables: x E [- 2n, 2n], Xscl
4rr
3
3rr
2
e) amplitude 1 .5, period 1 0
Window variables: x E [ -2n, 2n], Xscl
I' y E [ -4, 4 ]
amplitude 0.75, period 2.5
Window variables: x E [ -2rr, 2n], Xsci
I' y E [ -2, 2]
f)
3rr
2
I' y E [- 2, 2]
51!
3
iii)
4rr
3
I' y E [ -4, 4]
2. a) y = 3 sin 4x b ) y =
3. a) 4 b)
5.3 Sinusoidal Functions of the Form
= a sin [k(x - d)] + c and f(x) = a cos [k(x - d)] + c,
pages 270-279
f(x)
2rr
3
Window variables: x E [- 2n, 2n], Xscl
i
cos 2nx
21t c) rad to the left d) 2 units downward
3
n , Xscl
e) Window variables: x E
y E [ ­8, 4]
51!
3
[o, ]
'
1. a) amplitude 5, period
2'
y E [ - 6, 6]
4. a) 3 b) 2 c) 2 rad to the left d) 1 unit downward
e)
amplitude 3, period 3 1t
2
Window variables: x E [ - 2n, 2n], Xscl
Window variables: x E [0, 4], y E [ - 6, 4]
b)
2'
y E [ -4, 4]
S. a) amplitude 3, period 2n, phase shift
vertical translation 1 unit downward
Window variables: x E [0, 4n] , Xscl
4
rad to the left,
I' y E [ - 6 , 4]
Answers
•
MHR
553
b) amplitude 2, period 4n, phase shift 5TI rad to the right,
vertical translation 4 units upward 6
y E [- 2, 8]
Window variables: x E [0, 8n], Xscl
I'
c)
AA
9. a) amplitude 2, period 12, phase shift 4 rad to the right,
c) amplitude 2, period 1, phase shift 3 rad to the left, vertical
translation 2 units downward
Window variables: x E [0, 2], Xscl 0.5, y E [ - 6, 4)
c)
lO. a) y
6. a) amplitude 3, period 2n, phase shift
vertical translation 6 units upward
Window variables: x E [0, 4n], Xscl
CJV
X=6.2B31B 3
4
I' y
rad to the right,
E
[ - 2, 1 2)
[ ( %)] [ ( )]
[ ( )]
11. a) y = 4cos u x +
r\T\: t,
I'
phase shift 2 rad to the right,
vertical translation 7 units upward
Window variables: x E
[ }],
o,
Xscl
i, y
E
[- 2, 1 6]
4 sin 4 x + 3 TI + 3
4
b) Window variables: x E [ -n, n], Xscl
12. a) y
=
3 cos (0.4nt) + 4.5
I' y
]
(x + 2 ) + 2
[%
E
]
(x - 2 ) - 1.5
[ -2, 8]
[ - 2)] -
3 cos 2TI (x
2
3
b) Window variables: x E [ -6, 6], y E [ - 8 , 4]
13.
a) y
=
[I
c) Yes. y
]
vertical translation 1 unit upward
b) y = 3 sin (x - 1 ) + 1
MHR
•
Advanced Functions
[ ( %)]
14. Answers may vary.
a) y = 1 .5 sin 2 x +
:-:=<•
8. a) amplitude 3, period 4, phase shift 1 rad to the right,
554
[
+ l b) y = 2.5 cos
•
+
1 .5
b) Window variables: x E [ - 2n, 2n] , X sci
�c;v ·-::v
=
2 sin
V= ·7.
c) amplitude 7, period
7. a) h
=
V=B.1213203
vertical translation 5 units downward 3
y E [ - 1 2, 2)
Window variables: x E [0, 1 6n], Xscl
b)
1 b) y
3 sin 2 x -
=
b) amplitude 5, period 8n, phase shift 4TI rad to the left,
X=2 .1327 1
]
[
vertical translation 1 unit upward
b) y = 2 cos (x - 4) + 1
Answers
=
[ ( : )]
1 .5 sin 2 x ­ n
+ 1 .5
I' y E [ - 4, 4)
2 (0, 1), ( rr , -±)
b) i) Window variables: 8
15. a) b)
16. Answers may vary.
17. Answers may vary.
18. a) x =
b) = sin
y E (-7, 7]
E [0, 2rr], 8step 17t2 , x E [-7, 7],
4cosrrt y 4 rrt
E [0, 4], y E [ -6, 6]
c) Window variables: x
ii) Window variables:
y E (-3, 1]
8
E [0, 2rr], 8step 1 0 , x E [-2, 2],
r1=1-sin(l))
-i
19. a) v =
sin x
b) Window variables: x
E [0, 12], y E [ - 2, 2]
iii)
Window variables: 8
iv)
Window variables: 8
y E (-6, 2]
.
E [0, 2rr], 8step 100 x E [-4, 4],
20. a) h = sin
+
b) Window variables: x
25 (70rrt) 50
E '[o, lj,
35 Xscl _l,
70 y E [0, 90]
rvv
X=.0 8 71
.V= O
y E (-8, 4]
-�
c) Only the period changes: h =
25 (80rrt) 50.
sin
+
a csc [k(x d)] + c; a: multiply y-value by a;
k : changes the period to 21t ; d: phase shifts work the same
k
as for sinusoidal functions; c: vertical translations work
the same as for sinusoidal functions.
23. a), b)
s to the right
24. a) Window variables: x [ Xscl .
22.
y
=
-
2.5
c) a =
25. a)
E [- 2rr, 2rr], 8step 1 0 , x E [-8, 8],
E 2rr, 2rr],
$: ·6. 8 18
X= ·6.2BHB
V= ·1.1 8E ·9
v) Window variables:
8
E [0, 2rr], 8step 1 0 , x E [-2, 2],
iv) Window variables:
8
E [0, 2rr], 8step 100 x E [-2, 10],
y E (-2, 2]
2 y E [-7, 7]
2.7 and b 4.2, to two decimal places.
y E (-6, 6]
.
=
Extension, page 280
Part 1
1. a)
$=1
X=.909 97
y
0
0
0.51t
1
1t
0
l.Stt
\':1. 161 68
Smaller increments of 8step make the graph smoother
(more circular).
X
21t
4.
-1
0
The k-value has not been factored out of the bracket.
6. Answers may vary.
Answers
•
MHR
555
20. a) No two integers have a product of - 3 and a sum of 1 .
Part 2
1. a )
Sunrise in Fort Erie, ON
Time
Time (decimals)
Jan 1
7:47
7.78
Feb 1
7:31
7.52
Mar l
6:52
6.87
Apr 1
5:58
5.97
-
May l
Jun 1
5:1 0
5. 1 7
-
4:40
4.67
Aug 1
4:40
4.67
5:06
5.1 0
1---
Jul l
c)
.
b)
Date
+ v'i3
-1 c) 0.45, 2.69, 4.02, 5.41
6
21 . b ) Technology allows you to check all the zeros on the
graph within the domain.
-
22. a ) 1 1 . 93 s, 1 8.07 s
b) Window variables: x E [0, 60], Xscl 5, y E [0, 30], Yscl 5
-
Sep 1
5:40
5.67
Oct l
6:1 2
6.20
Nov 1
6:49
6.82
Dec 1
7:26
7.43
-
23. 0.08
24. 0.53, 1 .04
26. - 2n, -n, 0, n, 2n
27. a) 0.004 s b) No.
28. - 2n, 0, 2n
29. Answers may vary.
.
5.5 Making Connections and Instantaneous
Rate of Change, pages 290-299
1. a)
Window variables: x E [0, 1 2 ] , y E [0, 1 0]
= 1 .553 739 6 sin (0.497 842 4x + 1 .304 949 9)
+ 6.235 505 6
b) amplitude 1 .6, maximum 7.78, minimum 4.67, period 4rc,
phase shift 2.6, vertical translation up 1 . 3
2. a ) y
c)
p
=1
b) 0, rc, 2rc
.
3rc
. .
rc
c) maximum
2
' mm1mum
T
2. a)
Instantaneous
Angle x
0
v=7.7 766 7 .
4. Answers may vary.
5.4 Solve Trigonometric Equations, pages 282-289
1. a) 0.25, 2.89 b) 2.42, 3 . 8 6 c) 1 .37, 4.5 1 d) 1 .32, 4.97
e) 2 . 1 6 , 5.30 f) 3 .55, 5.87
5 rc d) 3rc 7rc
3. a) 4rc 5rc b) 5rc c)
3 ' 3
3' 3
4' 4
4 ' 4
S. a) 0.93, 2.2 1 , 4.07, 5.36 b) 0. 84, 2.30, 3.98, 5 .44
c) 0.88, 2.27, 4.02, 5.41 d) 0.89, 2.26, 4.03, 5.40
e) 0.74, 2.40, 3.88, 5.55
2rc 4rc 5n
5rc 7rc 1 1rc
5rc 7rc 1 1rc )
7 . a)
b)
c
6' 6 ' 6 ' 6
6' 6 ' 6 ' 6
3' 3 ' 3 ' 3
2rc 4n 5n
d)
3' 3 ' 3 ' 3
9. 3rc
2
5rc 3n
10.
6' 6 ' 2
11. 1t
12. 1 . 1 1 , 1 .89, 4.25, 5 .03
13. a) 0.46, 1 . 1 1 b) 0.32, 1 .25 c) 0.42, 1 . 1 5
Iff-
6 6
27t 3
47t
1 .9 1 , 4.37, 3'
18.
19. 1 .25, 2.68, 4.39, 5.82
MHR
•
Advanced Functions
Answers
1t
3
1t
r--
-
2
3
f-f-
1--
•
1t
6
1t
4
21t
14. n o solution
15. 0 . 1 7, 1 .40, 3.3 1 , 4.54
16. 0 . 84, 5.44
17. ' 5rc
556
f{x)
=
1
cosx
Rate of Change
0
0.87
-0.50
0.71
-0.71
0.50
-0.87
0
-1
-0.50
-0.87
37t
4
- 0.71
-0.71
5rr
6
- 0.87
-0.50
1t
-1
71t
6
5rr
4
47t
3
37t
2
-I-I
-r--
0
- 0.87
0.50
- 0.71
0.71
-0.50
0.87
-
-
0
1
5rr
3
0.50
0.87
77t
4
0.71
0.71
1 1 1t
6
0.87
0.50
27t
1
0
•. 16. a) ,---,----y:cr--,
c) Yes.
3. a) i) - 0 . 1 74 ii) -0 . 1 92 iii) - 0 . 1 96 iv) -0. 1 96
b) The instantaneous rate of change of h at t = 20 s is
about - 0.196 m/s.
c) The instantaneous rate of change represents the vertical
speed of the car at t = 20 s.
d) No. The graph of the sine function changes its slope
continually and would not likely yield the same value at
a different value of t.
1 1- •l
Daylight in Sarnia, ON
4. a)
Month
Duration (decimal)
1
9.08
2
9.95
3
1 1 .20
4
1 2.73
5
1 4.1 0
6
1 5. 1 3
7
1 5.32
8
1 4.52
9
1 3. 1 8
10
1 1 .75
11
1 0.30
12
b) T = 3 . 1 2 sin
[
]
X:3.1 1S 27
E
b) none; no maximum; no minimum
f{x)
=
tan x
Rate of Change
0
0
1t
6
1
0.58
1 .33
1t
4
1
2
1 .73
4
1t
3
1t
2
undefined
undefined
3
- 1 .73
4
31t
4
-1
2
5rr
6
-0.58
1 .33
21t
[0, 14], y E [0, 1 6]
G)
Instantaneous
Angle x
9.25
The equation fits the data well.
d) T == 3 . 1 1 sin [0.5 1 (m - 3.63)] + 1 2 . 1 4
The values for a , k, c , and d compare well with those
in the model.
e) phase shift
f) T == 3 . 1 1 cos [0.5 1 (m - 6.71)] + 1 2 . 1 4
S. 1 .5 h/month
6. h = 3 cos (rct) + 4
7. a) ( 1 .5, 4) b) 9.4 m/s c) speed of the spring
8. Answers may vary.
9. Answers may vary.
10. a) a = 4, k = Src b) d = 4 sin 5nt
11. a) y = -x2 + 8 b) y = 2 sin x + 4
12. a) -4 b) 3 . 14 c) different
d) Answers may vary. Sample answer: The cars may fall
off the track.
13. Answers may vary.
14. Answers may vary.
[-1, 1]· c) 0 d) 1
V: .102E '10
c)
(m - 5 ) + 1 2 .2
c) Window variables: x
h
b) No. Restrict the range to the interval
1t
0
1
7rr
6
0.58
1 .33
5rr
4
1
2
41t
3
1 .73
4
31t
2
5rr
3
undefined
undefined
- 1 .73
4
71t
4
-1
2
1 1 1t
6
-0.58
1 .33
21t
1
0
d)
e) Answers may vary.
Answers
•
MHR
557
4. 0.25, 2.89
5. a) The secant function is a reciprocal of the cosine
function and cos­'is the opposite operation of cosine.
1
=
= 1 .32, cos­
b) sec
b)
)
6. b)
1t 3 1t no max1mum;
.
no mm1mum
2' 2;
.
I
.
( )
(7z)
%
Instantaneous
Angle x
flx)
cscx
undefined
0
1[
6
1[
4
1[
3
1[
Rate of Change
undefined
2
-3.46
1 .41
- 1 .41
1.15
-0.67
1
0
2n
3
1.15
0.67
3n
1 .41
1 .41
5n
6
2
3.46
2
4
undefined
1[
7n
6
3.46
5n
4
- 1 .41
1 .41
4n
3
-1.1 5
0.67
3n
2
-1
0
5n
3
- 1 .1 5
- 0.67
7n
- 1 .41
- 1 .41
11n
6
-2
-3.46
undefined
2n
v
·.
c) As x approaches 0, s approaches infinity. This means
that the angle of elevation of the Sun approaches the
horizon and so the length of the shadow approaches infinity.
As x approaches s approaches 0. This means that the
2'
angle of elevation of the Sun approaches an overhead
location and the length of the shadow approaches 0.
7. a) amplitude 3, vertical translation 1 unit downward,
phase shift 1 rad to the right, period 2
b) y = 3 cos[n(x - 1 )] - 1
c)
undefined
-2
4
d)
=
8. a) 3 b) 2 c) 4 rad to the left d) 1 unit downward
e) Window variables: x E [ -n, n], Xscl y E [ ­5, 4]
I'
9. a) 1 .32, 4.97 b) 0.64, 2.50 c) 0.46, 3.61 d) no solution
10. a) ' 5 1t ' 3 7t
6 6 2
5 1t 77t l 17t
6' 6 ' 6 ' 6
12. a) y = 0.9sin t b) 0.37 s, 1 .63 s
undefi ned
11 .
(I )
13. 1 s, 3 s; maxima 0 s, 4 s
14. a) Window variables: x E [0, 12], y E [0, 300], Yscl 20
.
(�
e) Answers may vary.
19. 1
20. a) .l b) LR =
2
6
21. r = a sin 0 + b cos 0 is a circle
z
b2 with center
= a
+
x
(Y - 1 r
( ff
­
. v;t+b2 .
radms
2
:
(f, 1)
and
b)
y = 26 sin
[%(x - 1 )] + 235
c) The equation fits the data reasonably well.
Chapter 5 Review, pages 300-301
1. a) 2 b) 4
2. y
=
3. a) y
cos
[t (
x+
)]
5 sin 601tt b) No. A phase shift can generate
another possible equation.
558
=
MHR
•
Advanced Functions
•
Answers
d) Using sinusoidal regression, an equation that better fits
the data is y = 22.68 sin [0.83(x - 0.96)] + 230. 6 1 .
e) Answers may vary.
b) y
Chapter 5 Practice Test, pages 302-303
1. B
2. c
3. c
4. A
S. D
6. c
7. A
8. a) The cosecant function is a reciprocal of the sine
function and sin ­'is the opposite operation of sine.
b) csc
9. b)
( f)
==
( f)
1 .3 1 , sin ­'
=
}
=
The model appears to fit the data well.
16. a), b)
Phases of the Moon 2007
2 cos
[I
]
(x ­1 ) + 1
b) Window variables: x E [ -n, n], Xscl
[(
)]
3 sin 2 x + 51t + 1
6
b) Window variables: x E [ -n, n], Xscl
12. a) y
=
a
Phase
(percent illumination)
-
3
1 00
11
50
19
0
25
50
33
1 00
41
50
48
0
55
50
62
1 00
50
71
78
[15
0
--
84
50
92
1 00
1 00
50
1 07
0
1 14
50
]
c) y = 50 sin 1t (x + 4.5) + 50
I' y
I' y
d) Window variables: x
Yscl 20
E
E
E
[0, 1 50], Xscl 10, y E [ - 20, 1 20],
[ -4, 4]
[ -4, 6]
13. 0.80, 2.35, 3.88, 5 .49
14. ' ' 5 1t
6 2 6
15. a) Window variables: x E [0, 14], y E [ - 1 0, 30], Ysci 5
a
Date (days from
beginning of year )
r-t--
the angle of inclination of the wire approaches horizontal
and so the length of the wire approaches infinity. As x
approaches , I approaches 4. This means that the angle
of inclinatio of the wire approaches vertical and the
length of the wire approaches 4 m.
10. a) 2 b) c) 2 7t rad to the left d) 3 units downward
3
2
1t y E [ - 8 , 4]
e) Window variables: x E [ -n, n], Xscl l'
=
]
(x - 4) + 8.55
<) lA
c) As x approaches 0, I approaches infinity. This means that
11. a) y
[
1 3.4 sin
e) y == 49.75 sin [0.2 1 (x + 4)] + 52.74; the values for a, k,
c, and d compare well with those in the model.
17. a) Answers may vary. Sample Answer: Using your
model, first find the average rate of change of the percent
of illumination, and then estimate the instantaneous rate
of change.
b) The instantaneous rate of change on January 25 is about
1 0.4%/day.
c) The instantaneous rate of change represents the percent
change in illumination of the Moon per day.
Chapters 4 and 5 Review, pages 304-305
1.
a) 5 7t b) 1 05°
9
2. 3 radians
3. V3 + VI
V6
4. 25( v'3=1) m
s.
6.
0. 1 045
8
Answers
•
MHR
559
7. _ n_
65
. 2
9 . a) sm x + sin2x = tan2 x
cot2x
12. a) 71t b) to the left
12 4
13. a) 6 em b) -1- s c) 200rt d) y = 6 cos (200rtx) + 6
100
e) Window variables: x E [0, 0.02], Xscl 0.01, y E [ -4, 15]
20. a) C(t)
=
10 sin
[t2 (t ­12)] + 20
b) Window variables: x E [0, 72], Xscl 3, y E [0, 40], Yscl 2
�i\7\f\
X=6 -�-V=10
-�
21. a) t = 12 b) 2.6 ppm/h
C H A PTER 6
v
Prerequisite Skills, pages 308-309
to the right d) 2 units upward
2 4
1t , 1t Xscl
e) Window variables: x E
y E [ -4, 6]
4
4
14. a) 3 b)
[-
c)
],
1. a )
V1:3(2'X)
I'
b) i) {x E IR} ii) {y E IR, y > 0} iii) y = 0
2. a) 300 b) 2400 c) i) approximately 2. 74 days
15. b) 0 :S d :S 250Y:i3 c) The total time will be a minimum
when the contestant stays on the pavement.
16. a ) to the right
1t , 1t Xscl
b) Window variables: x E
4
[-
],
I' y
E
+
ii) approximately 38 bacteria
2b 2
1
'
3. a) x7 b) m3 c) k6 d) - 8x '- e) - a f l -:; g) u
V =HX)
S. a )
[ - 3, 5]
X=1
b) {x E
17. a )
h(t)
=
x-
4. a) 4 b) 1 0 c) 27 d) 82
( ;)
60 sin 2 t
!R,
x
-
2::
V=1
0}, { y E IR, y 2:: 0}
X=1 / V=1
b) Window variables: x E [0, 1 0] , y E [ - 80, 80], Yscl 1 0
e) Yes. f) {x E IR, x 2:: 0}, (y E IR, y 2:: 0}
6. a)
( )
2 rt mak mg
2 rt to -,
"
the
c) The value of k would ch ange from 2 1tt .
equation h(t) = 60 sin 3
3
5
18. a) 1 .37, 4.91 b) 0.34, 2.80, 3 .39, 6.03
19. a) Window variables: x E [0, 20], y E [ -2, 14]
D2SL
V=6.55B5256
X=
b) 7 1 34 cones c) 6300 cones
d) 8.7 years e) Answers may vary. Sample answer:
approximately Oct 1 99 1 and June 2000
f) Answers may vary.
560
MHR
•
Advanced Functions
•
Answers
b) (x E IR, } , (y E IR, y 2:: 4 }
c)
d)
e) No. f) (x E IR, x 2:: 4} , (y E IR }
7. Yes. Each curve is a reflection of the other in the line y = x.
8. a) translation of 3 units to the right and 1 unit up.
b) reflection in the x-axis, vertical stretch of factor 2
9 . a) vertical stretch of factor 3 and then translated
down 4 units
b) period doubled, reflection in the y-axis
10.
Vl=< -2)>-7
7. a) iii) b) i) c) ii) d) iv)
t( )x
(ix f
2
8. a) is inverse of 7d): y =
b) is inverse of 7b): y = SX
12.
V1:3(2)'
\
c) is inverse of 7c): y
=
d) is inverse of 7a): y
=
9. a)
.'-.....
=1
V=1.
6.1 The Exponential Function and Its Inverse,
pages 3 1 8-322
1. C, D
2. C: y = 3'; D: y =
l. •) k/
(tr
=2
b) i) m12
b)
=
. .,
' · •I +
X:1
V=.
v1=3'
=1
""""'\
x=·1
b) Y
=
3x
h(x) = 3'
IR
XE
IR
IR
(0, 0)
(0, 0)
none
y-intercept
(0, 0)
(0, 0)
(0, 1 )
x<O
x<O
never
x>O
x>O
XE
IR
IR
XE
IR
function is
negative
XE
increasing
d) increasing
XE
ye
function is
- 0.5
iR
g(x) = x'
range
positive
-
IR
x-intercept
function is
b) i) m_ 3 _2 = -4 ii) m_2_1 = 2 iii) m_ 10 = - 1 iv) m01 =
c) m_ 3 ='= -5.6, m_2 ='= -2.8, m_ 1 ='= - 1 .4, m0 ='= -0. 7,
m1 ='= -0.35
6. a)
XE
V=2.2
0.75 ii) m23 ='= 1 . 1 3 iii) m34 ='= 1 .69 iv) m45 ='= 2.53
c) m, ='= 0.6, m 2 ='= 0.9, m3 ='= 3 .4, m4 ='= 5.1, m 5 ='= 7.6
d) increasing
c), d)
llxl = 3x
Function
domain
equation of
IR
none
asymptote
ye
XE
y e [R. y > O
none
y= O
c) All three functions have the same domain and are increasing.
d) h(x) = 3' is different than the other two functions
for range, x-intercept, y-intercept, y = 0 asymptote,
and positive/negative intervals.
e) For f(x) = 3x, the instantaneous rate of change is
constant. For g(x) = x3 , the instantaneous rate of change
is decreasing and then increasing. For h(x) = 3", the
instantaneous rate of change is increasing.
10. a) i) 10 ii) 20 iii) 40 iv) 80 b) Yes.
j
b) y =
(tf
c) m12
1 1 . a)
=
20 d) i) m1
='=
1 3.9 ii) m2 ='= 27.8 e) Answers may vary.
V=3
Answers
•
MHR
561
b)
V 1 : ( ·2J•H
)(
12. a) {x E IR), {y E IR) b) 0 c) 0
d) For x < 0, the function is negative. For x > 0, the
function is positive. e) The function is increasing for
all intervals. f) None.
13. a) {x E IR), {y E IR) b) 0 c) 0
d) For x < 0, the function is negative. For x > 0, the
function is positive. e) The function is increasing for
all intervals. f) None.
14. a)
c) No. d) i) not real (square root of a negative value) ii) not real
e) Answers may vary.
22. a) m 1 == 0.56 km/s, m34 == 1 . 10 km/s b) m3 == 0.9, m4 == 1.3
23. a) For2 0 < b < 1 , f and r- l have equal X- and y-coordinates
at the point where they intersect the line y x.
b) Yes; when b > 1 , the graphs do not intersect the line y = x.
24. c) i) b > 0 ii) b < 0 d) i) b > 0 ii) b < 0 e) Yes.
=
6.2 Logarithms, pages 328-330
1. a) log4 64 = 3 b) log 1 2 8 = 7 c) log5
-2
V:.S
( )
1
-2
25
d) log± 0.25 e) Iogb y = x f) log 1 0 y = x g) log3 1 y = - 3
27
h) log v = n
3
d) - 3 e) 3 f) 10 g) 6 h) 4
2. a) 6 b) 3 c)
3. a) 3 b) - 1 c) 0 d) - 3 e) -4 f) 6 g) -2 h) 4
4. a) 72 = 49 b) 2 5 = 32 c) 1 04 = 10 000 d) b'" = z e) 23 = 8
f) 54 = 625 g) 1 0- 2 = __
l h) 72Y X
1 00
2
=
( )
=
15. a) {x E IR), {y E IR, y > 0) b) None. c) 1
d) The function is positive for all intervals.
e) The function is decreasing for all intervals.
f) y = 0
16. a) {x E IR, x > 0), {y E IR) b) 1 c) None
d) For 0 < x < 1, the function is positive. For x > 1, the
function is negative.
e) The function is decreasing for x > 0 f) x = 0
17. a) same domain b) the rest of the key features differ
18. a) same range b) the rest of the key features differ
19. a)
6. a) 2.6 b) 3 . 7 c) 6.2 d) - 1 .5
8. a) 2.63 b) -4.43 c) 0.95 d) -0.70 e) 1.23 f) 2.00 g) 2.26 h) 3.00
9 . a) y = l Ox
b)
b), c)
n=&·H
f/
H=1
21. a)
Uor
X
1 . b) Answers may vary.
Sample answer: log 1 1 1 1 = 1 c) logx x = 1
12. a) Answers may vary. Sample answer: The logarithmic
function has decreasing slope and the exponential function
has increasing slope. b) Answers may vary.
13. a) approximately 6.3 days b) approximately 1 2.6 days
14. a) approximately 66.5 m b) No. d == 1 6.2 m.
c) Answers may vary. Sample answer: Drive slower.
15. a) at least 5.73 m b) 7.16 m
17. Answers may vary. Sample answer: a) 3 b) y is an integer
but x is a power of 10.
y
0
1
1
-2
2
4
-8
3
4
562
MHR
16
•
1
' =0
10. a) 1 b) 1 c) 1 d) 1
11. a) logx x = 1 for x >
d) y = 6
2o. y =
L
-
--
Advanced Fu nctions
•
Answers
b)
b)
X
y
1 0°
0
1 01
1
1 0'
2
1 01
3
1 04
4
1 05
5
1 06
6
b)
{y E IR, 2 :S y :S 6}
10. a)
The graph is linear. The semi-log grid has turned each
power into the exponent value.
c) To plot a greater range of values.
19. The graph is a line that is increasing, with positive,
increasing slope. c) Answers may vary.
20. D
21. Vs
b) b)
)
b)
.. F
6.3 Transformations of Logarithmic Functions,
pages 338-340
1. a) iv i i c) i d) iii
2. a) translate right 2 translate left 5 and down 4
c) translate up 1 d) translate left 4 and down 6
c)
b)
8. For y = - log(x - 4): a) {x e IR, x > 4} { ye IR} c) x = 4
For y = log( -x) - 3 : a) {x e IR, x < 4} b) {y e IR} c) x = 0
9. a) V1= ·21o3(X)
18. a) Answers may vary. Sample answer:
= 6.3 c) V. = 1 020 d) The domain is the input
volt;ge, V; > 0; the range is the output voltage, V0, which
can be any real number.
11. Answers may vary. Sample answer: The domain and the
range are the same.
12. Answers may vary. Sample answer: No; the domains are
different, but the ranges are the same.
13. Answers may vary. Sample answer: The domains are
different, but the ranges are the same .
b) V = 6, V
14. a)
:
f L..
V1=1o3<X-3>•
x=
b)
b)
v= .30103
4. a) y = 2 logx
y
V1=21o30(X+ ))-2
b)
d)
_
_
V 1= 1o 3 ( - X)
.. ..
...
V1= ·21o3((1/2)X+3)-3
_
=
log (2x) c) y = log
( ±x) d) y = ± tog x
V= ·2.1092
5. a) vertical compression by a factor of 1.
2
horizontal compression by a factor of 1. and a reflection
5
. c) horizontal stretch by a factor
. the y-ax1s
m
of 2 and a
reflection in the y-axis d) vertical stretch by a factor of 5
and a reflection in the x-axis
6. a)
V1= 1o3(X)
~
X=10
c)
V=
V1=<113>1o3(X)
F
X=1000
7. a)
V=1
V1= ·1o3(X- )
~
X=1
v=·1
b)
d)
b)
V1=1o3((1/3)X)
~
X=30
V=1
V1=1o3( X>
d) i) {x E IR} ii) {y E IR, y
>
0} iii) y = 0 e) y = 1 02x + 8
18 •)
F
X=2
V=2
V1:1o3( ·X)-3
'"'"
X = ·1
V:·3
Answers
•
MHR
563
c) Answers may vary.
20. X = 64
d)
6.4 Power Law of Logarithms, pages 347-348
1 . a) 1 2 b) 3 c) - 8
2. a)
f b) f c) 8 d) 6
1
-2
d)
3. a) t == 1 .66 b) t == 3 .43 c) t == 9.01 t == 27.62
4. a) i) $500 ii) $572 iii) $655.40 b) i) t == 1 0. 2 years
ii) t == 1 6 .2 years
5. a) 2. 854 b) 1 .672 c) 0.356 d) -0.558 e) -4.907 f) - 7.228
6. a) log5 8 b) log9 1 7 c) logt
log1x 1 1 (x + 1 )
±d)
_
7. 10 000 times
8. approximately 1 5 dB
9. 4
10. a) approximately 1 99.53 times as intense
b) approximately 15.85 times as intense
1 1 . approximately 794 328 235 times as intense
12. a) approximately 4 1 .69 times brighter
b) approximately - 1 1 . 6 1
1 4 . Answers may vary. Sample answer: The absolute
magnitude takes distance into consideration.
15. a) i) approximately 158.5 times ii) 1 00 000 times
b) closest star is Biffidus-V, next is Cheryl-XI, farthest
away is Roccolus-III
16.-18. Answers may vary.
19. B
20. VIs
21 . B
22. D
Chapter 6 Review, pages 356-357
1 . a ) v1=J·x
J
i) (x E IR}, (y E IR, y > 0} ii) no x-intercept iii) 1
iv) positive for all intervals v) increasing for all intervals
vi) y = 0
b), C)
d)
9. a) X ='= 4 . 1 92 b) X ='= 3.333 C) X ='= 1 .623 X ='= 8.790
10. a) $400; this is the amount when t = 0.
b) approximately 8 years
11. log (mx) = m log x only when m = 1 and/or when x = 1 .
12. log x" = (log x)" only when n = 1 and/or when x = 1 .
13. a) 1 5 b) 1 5 c) Answers may vary.
14. Answers may vary.
15. a) approximately 6.6 h b) (d E IR, 0 < d :s 1 000},
(t E IR t 2: 0}
fog2 9 log2 25
18. a) -b) -log2 3 log2 1 0
1
19. log2 (2 0)64 = 640
20. a) A = P(1 .008 75 ) 41 b) i) approximately 1 9.9 years
ii) approximately 3 1 .5 years
21. _1_
V3
22. 384
6.5 Making Connections: Logarithmic Scales in the
Physical Sciences, pages 353-355
1 . a) 2 b) approximately 4.5 c) 9 d) approximately 9.8
2. a) [H+ ] = 10- 1 1 b) [W] = 0.001 c) [H+ ] == 3 . 2 x 1 0-9
d) a)
0.000 039 8
3. Answers may vary. Sample answer: The pH scale varies
over several powers of 10. b) Answers may vary.
Sample answer: [H+ ] is very small.
4. a) 5 b) approximately 1 0.6
5. a) Answers may vary. Sample answer: 0 < x < 0.003,
Xscl 1 X 1 0-<, 0 < y < 5, Ysci 1 b) approximately 2. 7
6. a) 1 00 times b) 100 000 times
[W] ==
564
MHR
•
Advanced Functions
•
Answers
V1=J'X
J
------
c) (x E IR, x > 0}; (y E IR}; x-intercepr 1 ; no y-intercept;
r-l is positive for X > 1 and negative for 0 < X < 1 ;
r ' i s increasing for all intervals; vertical asymptote: X = 0
2. a )
:¥
E IR}, (y E IR, y > 0} ii) no x-intercept iii) 1
iv) positive for all intervals v) decreasing for all intervals
vi) y = 0
i) (x
b), Q
c) (x
E
IR, x > 0}; (y E IR}; x-intercept 1 ; no y-inrercept;
r l is positive for 0 < X < 1 and negative for X > 1 ;
r ' i s decreasing for all intervals; vertical asymptote X
3. a) log4 64 = 3 b) log3 28 = x c) log y = 3 d) log 5 1 2
4. a) 27 = 1 2 8 b) bx = n c) 35
5. approximately 5.6
6. a) 4 b) 4 c) -2 -6
7. logx X = 1 , X > 0, X oF 1
d)
= 2 43
d)
6
b 19 =
4
2
=
=
0
9
b) i) {x E IR, x > 5} ii) {y E IR} iii) x = 5
9. a) V1=31o'( ·xl
c) i) (x
d)
E
IR, x < -3} ii) (y E IR} iii) x = - 3
-
b) i) {x E IR, x < 0} ii) {y E IR} iii) x = 0
1 1 . Answers may vary. Sample answer:
a) y = - log x + 4 - 3
b)
(±
)
c) i) {x E IR, x > - 8 } ii) {y E IR} iii) x = - 8
12. a ) 1 5 b) - 6 c) 3 d) 8
13. a) X ='= 2.579 b) X ='= - 1 .5 1 5 C) X ='= 1 .661 d) X ='= 0.322
14. approximately 1 .3 mm
15. b)
16. There is less hydronium ion concentration in Chemical B.
17. No; a great earthquake is 10 000 times as intense as a
light earthquake.
18. a) 3 ( n == 3 . 1 )
b) approximately
,, c
1
( T == 2.683 X 1 0-6)
372 7 1 7
c) approximately 138 9 3 4 times as much
X=.125 -�V=3.1Q721 :::::;
b) { T E IR, 0 < T :s; 1 } , ( n E IR, n
20. Answers may vary.
2!
1}
Chapter 6 Practice Test, pages 358-359
1. D
2. A
3. B
4.
c
5. a) 1 . 839 b) 2 . 1 63
6. a) Compre;s vertically by a factor of 1 , translate 3 left,
translate 6 up, and reflect in the y-axis. 2
8. approximately 1 1 years
9. a) pH == 1 1 .5 b) approximately 0.000 32 <
V<
c) vinegar acidic, ammonia alkaline
10. a)
0.01
b) approximately 54.3 kPa c) approximately 1 .4 km
1 1 . a) approximately 1 1 0 dB b) approximately 1 995 times
C H A PT E R 7
Prerequisite Skills, pages 362-363
1. product law a) x9 b) 6a5b4 c) x4y5 d) 6rs2
e) 5 q2 r f) v3w4 g) 4ab4
l
q
2. quotient law a) k4 b) - 6 n c) 6x3y d) 2ab2 e) - f) v2w2 g) b4
2r
3. power law a ) w8 b) 4u2v6 c) a6b3 d) x6/ e) 8w6 f) a2b8 g) rs6
2a2 product and quotient laws b) - 108k9m '\ product
4. a) b'
and power laws c) 32y5, product, quotient, and power laws
d) a4b, product, quotient, and power laws
-7
1
5. a) x = -6 or x = 4 b) x = 2.5 or x = 2 c) x = 2 or x = 2
=
d) q = 5 or q = -4 e) b = 1 or b -1 f) y = -1 or y = -1
4
2
3
3
g ) x = 2 or x = - 1 h) r = -3 or r = 1 i) q = -4 or q = - 1
2
- 1 ± VS
1
q
= ± v'41 c) x =
+ v'i4 b)
6. a) y = -3 4
2
1
-1
5
d) a =
v'21 e) x =
V29 f) r = ± Vi3
2
2
2
3
3 ± V57
g) m = V3 h) a =
6
2
7. a) 2V2 b) 4VS c) 3Vl d) 9Vl e) 5 V3
f) 2( V3 + 1 ) g) 1 ± V6
8. a) 6 b) c) 6 d) 4 e) 10 f) 4 g) 4 h) -2
±
±
±
i
9. a) 2.096 b) 2.322 c) 3.907 d) 2.044 e) 1 .585 f) 1 . 893
g) 2 . 1 6 1
h) 4.248
10. a) 0.86 b) 1 .29 c) 0.36 d) - 0.17 e) 0.63 f) 0.53 g) 3.17 h) 1 .30
7.1 Equivalent Forms of Exponential Equations,
pages 368-369
log l 4
1. a) 212 b) 29 c) 2 -6 d) 2 Toi2
2. a) 36
I
log i
b) r4 c) 3 logJ d) 3 J;;g3
Answers
•
MHR
565
3. Answers may vary. Sample
answers: a) 44 b) 1 62
16
33
2
4 a) 4 3 b) 43 c) 4T d) 2T
5 a) x = 3 b) x = -2 c) w = 3 d) m
:
=
]_
4
1 1 d) k = 9
6. a) x = - 3 b) x = 6 c) y = 4
7. a) x = 5 b) x = 5 c) Answers may vary.
8. aH) Answers may vary.
9. Answers may vary.
10. a) x = -4 b) k = - 2
11 1 . a) x = -
b)
I
12. 1 0 = b log b ' b > 0
15. a) i) x > 2 ii) x > - 1 1 b) Answers may vary. Sample
2
answer for inequality i}: Graph each side of the inequality
as a separate function. Find their point of intersection.
Test a point to the right of the point of intersection to
ensure that the inequality is true.
,------=-.,...-...
c) Answers may vary.
16. a) i) 2-' > x3 for 0
< x < 1 .37; x 3 > 2x for x > 1 . 3 7
(correct to 2 decimal places). ii) 1 . 1 x > x 1 0 for 0 < x < 1 . 0 1 .
b) for x > 1 .52
7.2 Techniques for Solving Exponential Equations,
pages 375-377
1. a) iii) b) i) c) ii) d) iv)
2. a) 1 0.24 b) 1 1 .53 c) 5S.71 d) 1 S.91 e) -2.SO f) - 0.55
h)
r:=
X=12 --V=.S
log 2
g) - 1 5.63
13. a) approximately 66.4 h b) Answers may vary.
14. approximately 6.44 days
15. a) 6 years b) approximately 10 years
16. a) i) 1 2 s ii) approximately 40 s
-3
3 . a) approximately 35.75 m b) approximately 10.3 min c) No.
log 3
2 log 5
b) x =
log S - log 4
log 3 - log 2
log S + log 3
log ? + 2 1og4
d) X = ____::___
C) X =
_:
.:::...__
log 4 ­2 1og 7
log 3 - log S
5. a) 2. 710 b) 14.425 c) - 3 .240 d) - l . SS3
6. a) a = 1 , b = 1 , c = - 6 b) x = 1 c) 2x = -3
log( 1 + V6)
7. a) a = 1 ' b = -2 c = - 5 b) x = ____:-'-::
--'---'
log S
c) sx = 1 - V6
8. a) approximately 2 1 . 3 min b) approximately 92.06 min
9. a) approximately 1 9.7 min
b)
4. a) x =
_
_
=
c) Answers may vary.
(-f) fo
b) {t E IH!, t ?. 0}, {y E IH!, 0 < y
18. a) y =
19. Answers may vary.
20. D
21. c
22. 1 1
1}
7.3 Product and Quotient Laws of Logarithms,
pages 384-386
1. a) log 54 b) log S c) log3 2 1 d) log5 2
2. a) 1 . 732 b) 0.903 c) 2. 771 d) 0.43 1
3. a) log (2xyz), x > 0, y > 0, z > 0
( )
(m;�3)
b) log, 3ab , a > 0, b > 0, c >0
- 2c
c) log
, m > 0, n > 0, y > 0
d) log(u2vy!W}, u > 0, v > 0, w > 0
4. a) 2 b) 2 c) 2 d) 3
5. a) 3 b) 4 c) 3 d) - 2
6. a) 2 b) approximately 2.301
7. a) log7 c + log7d b) log3 m - log3 n c) log u + 3 log v
d) loga + log b - 2 log c e) log2 2 + log2 5 + 1 + log 5
-f
log5 (25 X 2) = 2 + log5 2
8. Answers may vary.
9. a) logx, x > 0 b) log m, m > 0 c)
f)
d)
f
t logw, w > 0
2
t log k, k > 0
10. a) log (x + 2), x > 2 b) log (x + 4), x > - 3
c) log
( x ; 2 ) , x > 3 d) log (: ) , x > 3
1 1 . a) V
0
= ( 2)
log
V
VI
b) i) V = 1 ii) V = 2 iii) V = 0
0
0
0
12. a) Translate y = logx up ( 1 + log n) units.
b) Answers may vary.
13. no, n > 0, positive integers only
14. a)
,--..,
V 2= 31 .3(X l
V1=21o3(X)
fr
V=.95 2 2H
c) i) The graph is decreasing faster ( shorter time). c) ii) The
l/
graph is decreasing at a slower rate (longer time to decay).
d) i) steeper slope (negative), more would decay in the same
amount of time ii) slope is not as steep, less to decay in
same amount of time
10. No.
11. a) X = 1 Or X ='= 1 .26 b) X ='= 0.16 C) X = - 1 d) X = 1
e) no solution f) x ='= 1 .29
566
MHR
•
Advanced Functions
•
Answers
X=3
t/
V=2.3B56063
14. d) p (x) = q(x); power law
V=2.3B56063
15. a) q(x) = logx6
b)
7.5 Making Connections: Mathematical Modelling With
Exponential and Logarithmic Equations, pages 404-407
'11=61o (X)
X=2
1 . in June of 2037
2. in approximately the year 1 774
3. a) for P = 1 006( 1 .016)' i) P = 4920 ii) t = 1 8 8 .4 years; for
P = 1 000 X 2 4;.5 i) P = 4921 ii) t = 1 8 8 years b) quite close
V=1.B061B
c) Answers may vary. Sample answer: The graph has values
for x < 0 and x > 0 because x6 is an even power; the graph
has a discontinuity at x = 0.
16. Answers may vary.
18. a) approximately 6.7 years b) i) B ii) B c) Answers may vary.
22. V2.
23. - 1
24. B
25. .£ k
c
k+n
_
_
7.4 Techniques for Solving Logarithmic Equations,
pages 391 -392
1. a) x = 12 b) x = 75 c) p = 38 d) w = 1 7
e) k = 1 0 8 f) n = 50
9 f) n = 2
2. a) x 5 b) x = 2 1 c) k = 27 d) x = 25 e) t = =
3 . a ) x = 2 + Vi4 b) x = 4 c) v = 5 3 3
d) y = 1
8
33
e) k = 3 f) p = 1
5 . a ) x = - 2 o r x = 5 b) x = - 50 o r x = 2
6. a) x = _S_ b) k = .!.
511
2
b) X = 2 . 1 6
7. a) X = 9.05
4. Answers may vary.
5. Answers may vary. Sample answer: Rural Ontario
Investment Group takes a little less time to make $80
000 (7.35 years), so it could be considered. Muskoka
Guaranteed Certificate takes the same amount of time,
so it does not need to be considered.
6. a) Answers may vary. Sample answer: increasing in a
curved pattern
I
-
a
b) i) y = 4.21x ­3 .07 ii) y = 0.86x2 - 0.93x + 1 .2
iii) y = 1 .3 7( 1 .65)'
c) Answers may vary. Sample answer: exponential
d) Answers may vary. Sample answers:
i) approximately 207.8 m2
ii) t = 12.44 min e) Answers may vary. Sample answer: circle
7 . a) A = 1000( 1 .02 )4' b) approximately $ 1 3 72.79
c) approximately 8 .75 years
8. a) A = 1 000( 1 .02 )4' - 50 if t < 4.
b) shifts the graph down by 50
9. Answers may vary.
I
b) by 10 000 000 (or 107 )
II
12.-14. Answers may vary.
15. 48 km/h
16. A =
- 1
17. z is increased by 12.5%.
10. a) 1000 =
8 . a ) approximately 1 1 9.54 d B b) 1
9. a) No. b) Yes.
X
10- 1 0 W/m2 c)
1 W/m2
10. a) 2 b) 1
16
1 1 . a) w = 5 :±: SVS
b)
..2
Chapter 7 Review, pages 408-409
1 . a) 43 b) 41 c) 4- 2 d)
1og20
log 0 .8
5
42
2. a) slOgS bl sloSs
3
3. a) X = - l b) X = - 1 8
X=16.1B03
V=.66666667
12. x = - 1 ; graph each side of the equation as a function
and find the point of intersection.
4. a) approximately 5 min b) approximately 21.6 min
--
14. 44
1 5. ± 3 VS or approximately 6.7 or - 6.7
16. D
-=-----,
--
2 log 3
k
2 log 2 + log 3
b) = ---'-log 2 ­log 3
log 3 - log S
6. a) x = -4.301, k = -6.129
log 3
log S
7. a) X =
b) X = 2 Of X =
log 2
log 4
8. a) approximately 2.5 years b) approximately 8.3 years
9. a) 3 b) 2 c) 2 d) approximately 2.05
5. a) x =
10. a) log7 2 b) log
( 3:.-:)
, a > 0, b > 0, c > 0
I
(+)
1 1 . a) 2 log a + Iog b + log c b) log k -
(
)
log m
12. a) log -2- , m > 3 b) log x
m-3
13. a) X = 45 b) X = 5
5 , x > 4, x > - 5
x-4
Answers
•
MHR
567
3. A: y = X + 2; B: y
4. a) {x E IR}, {y E IR}
14. X = 2
"· :=¥
16. a) approximately 1 1 .6 h b) approximately 66%
1 7. a) A = 500( 1 .03W' b) approximately $691 .79
c) approximately 10.67 years
b) {x E IR}, (y E IR, y
=
2x; C: y =
2:::
0}
2:::
0}
±x2 + ±x
d) i) same shape translated up by 5 ii) above the original
function and increasing at a faster rate
Chapter 7 Practice Test, pages 41 0-41 1
1. c
2. A
3. D
4. c
s. 2
c) (x E IR}, (y E IR}
16
3
S. a) log (x + 1 ) = S logx - 2 b) x == 3 . 37
9. a) approximately 35 min b) approximately 232.5 min
(or 3 h 52 min 30 s)
10. approximately 9.9 min
11. a) x = 3
12. a) Yes. b) Yes. c) Answers may vary.
6. a) X = 8 b) X = - C) X = 1 5 d) X = 2.8
d) (x E IR}, (y E IR, y
13. a)
.
S. a) odd b) even c) odd d) even
. .. . . . . . . . . . .
7. a) (x E IR, x
b) y == 0.17x2 - 5.34x + 97.93 c) y == 95.67(0.96)"
d) Answers may vary. Sample answer: Exponential, because
if predicting values for x > 15 .74, the quadratic gives
increasing values, which is not realistic.
e) i) approximately 49.6°C ii) approximately 2 1 . 1 min
f) Answers may vary. Sample answer: The temperature of
the room was constant.
l
b) {x E IR, x
V1=1/( - )
CHAPTER S
0}
0}, ( y E IR, y
·
4}, (y E IR, y
0}
Prerequisite Skills, pages 41 4-41 5
1. b) Pattern A: linear, Pattern B: exponential, Pattern C:
quadratic
2. a)
Pattern
t 1:L1,L2
Pattern B
A
1
x 2, x - 2
x+2
b) v(x) = x - 3, x 2
9. a) i) {x E IR, x 2, x -2}, y E IR, x
{
ii)
./
0, y
V:3
=1
Pattern C
J
iii) asymptotes: x
b) Yes.
568
S. a) u (x ) = -- ,
MHR
•
Advanced Fu nctions
•
Answers
=
( i)
- 2, y = 0; hole: 2,
4}
1
b) i) (x E IR, x
ii)
;6
-2}, (y E IR, y
;6
-5}
8. a) i)
1lL
iii) hole: ( -2, - 5 )
1 1 . a ) i v b ) iii c) i i d) i
12. Answers may vary. Sample answers:
i) 0 s :s t :s 8 s, - 5 em :s d :s 5 em
ii) 0 s :s t :s 3 s, 0 m :s d :s 5 m
iii) 0 s :s t :s 10 s, - 2 m :s d :s 2 m
iv) 0 s :s t :s 10 s, 0 m :s d :s 3 m
X - 3
13. a) y = X + 2 b) y =
C) y = ±VX+S, X
=
-5
4
1 - 1 , X ,., 0
d) y X
14. The inverses of parts a), b), and d) are functions, since
they pass the vertical line test.
8.1 Sums and Differences of Functions, pages 4 1 6-428
1. a) i) blue ii) red iii) yellow
b) i) y = 3x ii) y = x2 - 1 iii) y = 2x
2. a) y = 3x + x2 ­1 + 2x b) i) 71 ii) 1 1 7
3 . a) i) y = 6x + 7 ii) y = 4x ­7 iii) y = - 4x + 7
b) i) y = - 3x + 14 ii) y = -x - 4 iii) y = x + 4
c) i) y = x2 + 5 ii) y = x2 + 3 iii) y
-x2 - 3
=
=
d) i) y = - 3x2 + 7x - 7 ii) y = -3x2 + x + 7
iii) y = 3x2 - x - 7
x + 5, j ( - 1 ) = 4
4. a) h(x) = 7x + 1 , h(2) = 1 5 b) j(x)
c) k (x) = -x - 5, k(O) = - 5
S . a) h(x)
- 4x2 + 2 x + 2, h( - 3 ) = -40
b) j(x) - 4x2 - 2x + 8, j (O) = 8
c) k (x) = 4x2 + 2x - 8, k(3) = 34
6. a)
y
b)
y
==
b) i) 3 units up ii) 3 units down
iii) reflection in the x-axis and 3 units up
c) i) (x E IR}, (y E IR, y > 3 }
ii) (x e !R} , (y e !R, y > - 3 }
iii) ( x E IR}, ( y E IR , y < 3 }
9 . Window variables: x E [ -Jt, 3Jt], Xscl
•l
b)
10. a) i) C
b), <)
X=IO
=
I' y E [ -2 , 2 ]
120 + h ii) R = 2.5h
V=200
The break-even point is the point at which the revenue and
cost are equal. When the vendor has sold 80 hotdogs, the
cost and the revenue are both equal to $200.
d) P(h) l .Sh - 120
=
e) C(h): (h e Z, 0 :s h :s 250}, ( C e IR, 1 20 :s C :s 3 70}
R(h): ( h E z, 0 :S h :S 250}, (R E IR, 0 :S R :S 625}
P(h): {h E Z, 0 :S h :S 250}, {P E IR, - 120 :s P :S 255}
f) $255
11. a) i) If C = 1 00 + h, then the vendor only needs to sell
about 67 hotdogs to break even.
ii) If C 1 20 + 0.9h, then the potential daily profit
becomes $280.
b) Answers may vary. Sample answer: Choose to reduce
the variable cost.
=
Answers
•
MHR
569
Window variables: x E [- 424?rt , 424?rt l , Xscl 2 ,
y E [- 15, 30], Yscl 5
12. a)
::s
c) {t E IR),
13. a)
18. a)
Y, =
{y E IR, 5 y 25} d) i) 5 ii) 25 iii) 15
same b) Window variables: x E [-4, 4], y E [-4, 8]
::s
d) f(x) =
2x e) Answers may vary. The rate of change of
the exponential function is continuously increasing at a
greater rate than the other component functions.
14. a) Yes. b) No. c) The commutative property holds true
for the sum of two functions, but not the difference of
two functions.
15. a)
c) same
d)
c)
total operating costs; Y3 = 20 000 + 15x
X=tooo
v=
ooo
revenue increasing at a constant rate; Y4 = 20x
Y=1
Break-even point: (4000, 80 000), cost equals revenue
profit; Y5 = 5x - 20 000
f)
e) {x
::s
::s
E IR}, {y E IR, -2 y 2 }
shifted to the right and amplitude multiplied by Vl
ii) horizontal line iii) same as when c = 0
g) i) {x E IR}, {y E IR, - Vl y Vl}
ii) {x E IR), {y E IR, y = 0 ) iii) {x E IR), {y E IR, - 2 y 2 )
h) Answers may vary.
16. a) y = -x - 2
b)
c) y = 3
f) i)
::s
d)
f)
$15/game; cost increases per game at a constant rate;
15x
¥ =
2
d)
Y2=sii'JG'(-C)
X=1. 70796
b)
$20 000; not affected by the number of games;
20 000
::s
e)
::s
::s
y = 3, same
Subtracting the functions is the same as adding the opposite.
v
X=BOOO
g) i)
\'= 0000
loss ii) break-even iii) profit
h) i) move to the left ii) move to the right
20. a) Window variables: x E [- 7t,
4 47t ], Xscl , y E [-4, 4]
VITT7
X=i
d)
I I
infinite number e) g(x) intersects h(x) at the point of
inflection due to the variable vertical translation.
21 . a) Window variables: x E [- rt,
4rt ] , Xscl , y E [-4, 4]
[1JLl_
T7r1C7
d)
b)
n=tortG )-:�
X=1
(
J
Y=.
(7
077
infinite number
intersects h(x) at the point of inflection due to the
variable vertical translation.
e) g(x)
570
MHR
•
Advanced Fu nctions
•
Answers
22. Yes
23. The sum of two even functions is even.
24. B
25. D
26. 4
27. negative reciprocals: A = - C; reciprocals: A = C
6. a) Both functions are exponential, with fish increasing and
food decreasing.
Window variables: x E [0, 20), y E [0, 1 500), Yscl 1 00
8.2 Products a nd Quotients of Functions, pages 429-438
1. a) even: A and B; odd: A and C or B and C
b) Two combinations multiply to form an odd function.
2. odd
4. a) y = x3 - 2x2 - 4x + 8, {x E IR), {y E IR)
1 , x 2, x
¥- -2, {x E IR, x ¥- 2, x ¥- - 2),
¥x+2
y E IR, y ¥y ¥- 0
b) y =
{
--
i'
( i) , asymptotes: x = -2 and y = 0
hole: 2,
c) y
}
=
b) P(t): {t E IR), {P E IR, p 2: 0)
F(t): {t E IR), {F E IR, F 2: 0)
c) (9.1 1 , 467.88); In 9. 1 1 years, the number of fish and the
amount of fish food both equal 467. 88.
d) Answers may vary. Sample answer: The amount of
fish food minus the number of fish. When the function is
positive, there is a surplus of food. When the function is
negative, there is not enough food.
VH/
e) 9. 1 1 , same
f) Answers may vary. Sample answer: P(t) should start to
decrease since the amount of food is decreasing.
7. a) Answers may vary. Sample answer:
Ratio of food to fish. If the function is greater than one,
there is more food. The graph is decreasing.
x + 2, x ¥- 2, {x E IR, x ¥- 2), {y E IR, y ¥- 4), hole: (2, 4)
b) approximately (9. 1 1 , 1 ); After 9. 1 1 years, the amount
of food is equal to the number of fish.
c) plenty of food, enough food, not enough food
8. a) semi-circle, even
b) odd
5. a) y = 0.9SX cosx, {x E IR), {y E IR)
r-:-.....,..-c-::---:-----..
b) y
=
��;;, {x E IR), {y E IR)
{
}
(2n + 1 )1t
c) y - 0.9SX
, n E ?l.
cos x ' cos x ¥- 0, x E IR, x ­
2
(2n + 1 )1t
{y E IR, y ¥- 0), asymptotes: x =
, n E 71., and y = 0
2
_
7(0.9S• )(<OS( )) l
d) {x E IR, - 5 ,;:; x ,;:; 5), {y E IR, - 4.76 ,;:; y ,;:; 4.76)
{x E IR, -5 ,;:; x ,;:; 5), {y E IR)
9 . a) odd;
Answers
•
MHR
571
b) odd; (x E IR, - 5 :5 x :5 5, x ¥- -n, 0, n), ( y E IR }
d) Answers may vary. Sample answer: The zeros of this graph
are the points of intersection of the graphs of Pves and PAsk'
10. a ) P(t) is exponential and increasing, while F(t) is linear
and increasing.
Window variables: x E [0, 30], Xscl 2, y E [0, 15]
:5
14. a) y = - 0.036f sin t + 0.252t sin t ­0.04f + 0.28t
5
X:2.1
b) y = 8 + 0.04t - 6 ( 1 .02 )'; Answers may vary. Sample
answers: In 2008, there is a surplus, since the function is
positive. After 20 1 9, there will be a food shortage.
c) (0, 2); In 2000, the maximum is 2, which is the amount of
V : '.7315H9
V:.7313 672B
b) There is approximately a 73% chance of Carlos and
Keiko agreeing to date 2 . 1 days after the dance.
c) after 7 days d) Answers may vary.
15. Yes.
16. No.
17. (x E IR}, (y E IR, y :5 4.04}, x-intercept 0, y-intercept 0;
x < 0, function is negative and increasing; 0 < x < 4.33,
function is positive and decreasing; x > 4.33, function
is positive and decreasing; as x ---7 oo, y ---7 0, and as
X ---7 -oo , y ---7 - oo
the surplus of food.
1 1 . a) decreasing
b) 2000; yes
.
R
R
19. a) i) even ii) even iii) even b) i) even ii) odd iii) [f(x)]" is even
if n is even and odd if n is odd when f(x) is odd.
10. a) 2.45 em
b) original function
c) When -- > 1, there IS a surplus of food. When -- < 1,
there is a shortage of food. For Terra, there is a surplus of
food before 2019 and a shortage of food after 20 1 9.
11 . Answers may vary.
13. a) pM (t) is periodic, with domain (t E IR, t ;;:: 0} and range
{pAsk E IR, 0.05 :5 pAs :5 0.95 }.
k
Pves (t) is quadratic, increasing and then decreasing, with domain
{t E IR, 0 :5 t :5 7} and range (Pves E IR, 0 :5 Pves :5 0.98}.
Window variables: x E [0, 7], y E [0, 2], Yscl 0.5
i) air resistance reduced
Fm
X:6.2831853
21. 10
60
7
5
13
22.
I'
57t ,
b) at the maximum values of p Ask (t), which occur at
2
91t , . . . days after the dance (approximately 1 .6, 7.9, 14.
1,
2
. . . days after the dance)
c) at the maximum value of Pves(t ), which occurs 3.5 days
after the dance
572
MHR
•
Advanced Fu nctions
•
Answers
'2
24. 35 em
V:B.BQ78BB5
ii) pendulum lengthened
8.3 Composite Functions, pages 429-449
=
x2 - l Ox + 25 c) y = x
d) y = x4 + 1 2x3 + 60x2 + 144x + 144 e) y x
2. a) {x E IR}, {y e IR, y s; 2 }
1. a) y = -x2 - 6x - 7 b) y
b) { x E IR } , {y E IR , y
2!
=
0}
b) Answers may vary. Sample answer: The V party will win
if the election is held before 80 days. At 80 days, both the
Red party and the V party have 50% of the vote, while the
Blue party has 0%. c) Answers may vary. Sample answer:
If there were a fourth party, it would have 0% of the vote
(R (t) + B(t) + V(t) = 1 00 for 0 s; t s; 80).
8 . No
I
9 . a) y = :tx3 b) y
x c) y = x d) same
e) 3, 5, - 1 , f(( 1 (x)) = x for all values of x.
10. Yes
=
c) { x E IR } , { y E IR }
d ) { x E IR}, (y E IR, y
b) 20% c) increasing by 0.375 %/day d) Answers may vary.
Sample answer: If the election is held before 22.9 days, the
Blue party will win. If the election is held after 22.9 days,
then the Red party will win. 22.9 days is approximately
when the two functions intersect.
7. a) V(t) = 40 + O. l25t; popularity of the V party;
increasing at a constant rate
2!
9}
=1. 707963
c) Yes d) {x
e) {x E IR}, {y e IR}
3
5. a) decreasing at a constant rate
b) 40% c) decreasing by 0.5 %/day
d) Answers may vary. Sample answer: No. Since the Blue
party starts off with 40% and the Red party with 20%, there
is a good possibility that there are more than two parties.
Knowing the percent of undecided voters would help.
6. a) R(t) = 20 + 0.375t; increasing at a constant rate
E
IR}, {y E IR, 0
s;
y
s;
1}
12. a) y = sin3 x; periodic; {x e IR}, {y E IR, - 1
=1. 707963
4. a) _ 1_ b) -5
V=1
s;
y
s;
1}
V=1
b) Answers may vary. Sample answer: The functions are
both periodic and have a maximum value of 1. The functions
differ in their minimum values (0 versus - 1 ) and period
(7t versus 2n ). One function is even, while the other is odd.
I
13. a) C(P(t)) 1 8 .634 X 2"52 + 58.55
b)
=
E
= 0
c) 60 years
=
V=9 .B37 B
14. ( f ogr l (x) (g­1 r- l )(x)
15. a) parabolic, decreasing
0
V1= ·o.H + )>+BO
n=20•0.37
=20
V=27.
Answers
•
MHR
573
b)
(p E IR, 5 ::; p ::; 23.28} c) No. d) R(p) = [ -0.1(p + Sf + 80]p
V2:( '0.1(X+5l>+BOlX
2. a) i) There will be no minimum or maximum number of
homes, since the cost is higher than the revenue.
be no profit, but a minimum loss of
ii) There will
/\
$153 846.
X=D.OB -V=61B.B3259 ,
$13.08, $618.83
W(N(t)) = 3\/,..10""'0:-+----=-25::-t
b) {t E IR, t 0}, {W E IR,
30}
15. e)
16. a)
2
X=25
18. a)
b), c)
w
3. a) i)
b)
V=B0.777 72
{x E IR, x > 0}
{x E IR, 2n1t < x < (2n + 1)1t, n E &:'}, {y E IR, y < 0}
(4n - 5)7t < x < (4n - 3)7t , n E ,
d) ) {X E IR ,
&:'}
2
2
{y E IR, y < 0}
, e
V1=1o3<<o•< >
X:.7B5 9B16
c)
X < 0, 0 < X < 1 ii) X > 1
+ *-
x < 0, 0 < x < 1, same
.. .)
b)
(x) > 1, u (x) > v(x) for approximately
y = uv(x)
-5 < x < 3.
(x)
ii) When y = u
v(x) > 1, u (x) < v(x) for approximately
X < -5, X > 3.
5. a) Window variables: x E [ -15, 10), y E [ -150, 150], Yscl 50
c) i) When
V: ·.150515
{x E IR, x 0}, {y E IR, y > -9}
{x E IR, X """ -3, X """ 3}, {y E IR, y ,; - , y > o}
20. d(t) = V4Tt
b - dx
21 . r'(x) =
ex 1
+ Vs
23. 0, 1,
2
24. .1_
25
25. 10
,=
19. a)
b)
a
=
or
oo )
b) i) approximately
2
or
ii) approximately ( - oo ,
7. a) Subtract
from
b)
c) Yes. When the graph is below the x-axis,
When the graph is above the x-axis,
(-9.77, -1.23) (2,
-9.77) (-1.23, )
g(x) f(x). y -x2 + Sx - 4
f(x) > g(x).
g(x) > f(x).
8. (0, oo)
-0.77) or (2 , 4)
9. approximately (
-oo,
10. a)
8.4 I nequalities of Combined Functions, pages 450-460
1 . a) i) The minimum number of homes will decrease and
the maximum number of homes will increase. From the
intersection points, the approximate number of homes
becomes
ii) The maximum potential profit will increase. From
the maximum of the difference function, the maximum
million.
potential profit becomes approximately
1. b)
44 ::; n ::; 442.
$3.0
b)
c)
5 ,; p < 15
{p E IR, 5 ::; p < 15}, {N E IR, 0 < N ::; 120}
1 1 . a)
D
=5
V:600
5 ::; p < 15; yes
N(p) maximum is at (4, 121), while the R(p)
maximum is at approximately (9.16, 864.47).
The price affects where the maximum is. d) $9.16
b)
c) The
574
MHR
•
Advanced Fu nctions
•
Answers
•I
b)
profit function
3. Window variables: x E [0, 50], Xscl 5, y E [ - 100, 1 40] ,
~
X=10.69
bl
2. Window variables: x E [0, 50], Xscl 5, y E [ - 1 00, 140],
Yscl 20
12. a)
Yscl 20
bl
•I
V:28).82779 '
c) 6.30 < p < 14.24; profit d) No. e) $ 1 0.69, $283 .83
13 . 0 :s t < 0.22, 0.74 < t < 1 .27, and 1 .79 < t :s 2
14. Answers may vary. Sample answer: f(x) = 5 and
g(x) = - 1
15. a) Window variables: x E [0, 1 6] , Xscl 2, y E [ - 1 0, 30],
Yscl 2
4. a)
b)
V6=V1+V2+V3+'1 +'15
~
Answers may vary.
ii)
S. a) i)
b)
d)
2; where revenue equals cost c) 0.853 < n < 7. 8 1 3; profit
b)
H=
V:).S
:t=
=
=
V= ·.16827 1
X=.01
Answers may vary.
6. a)
e) i) approximately 4333 ii) approximately $0.84/unit
iii) approximately $3633 f) Answers may vary.
16. a) C(n) = 280 + 8n b) R(n) (45 - n)n c) R(n) > C(n)
d) 11 :s n :s 26 e) 1 8 birdhouses at a profit of $61
18. Answers may vary. Sample answer:
f(x) = sinx and g(x) = 1
19. Answers may vary. Sample answers:
a) f(x) = 2x and g(x) = 0.5x2 b) f(x) = 4x and g(x) = x2
20. a) Answers may vary. Sample answer:
f(x) = - (x - 3 )(x + 3) and g(x) x + 3 b) Yes.
21. Answers may vary. Sample answer:
f(x) = (x + 3 ) (x - 4) + 1 and g(x) = 1
22. 8
23. 3 or - 3
24. 1 6
V7=V1+V3+V
V7:s;o(52 1TX)+s;o(6001TX)_
b) Answers
may vary. Sample answer: This is not a pattern
like the others.
7.
a)
8. a)
8.5 Making Connections: Modelling With Combined
Functions, pages 461 -471
1. Window variables: x E [0, 50], Xscl 5, y E [ - 1 00, 140],
Yscl 20
•I
b)
b) Answers may vary. Sample answer: 0 < x < 1 00, skier
going down the hill; 100 < x < 1 60, skier in line;
1 60 < x < 360, skier on lift c) Answers may vary.
Answers
•
MHR
575
9.
a) iJJJW
X:7BO
V=100
<) VWJi
b)
14. Answers may vary. Sample answer:
y
X=1110
V=100
120(0.5°·1x) + 2 sin x
V3=Y1+Y2\
Chapter 8 Review, pages 472-473
1. b) i)
X=720 V:100 ·
10. a) 1, L 1.· 2· · · · · · · · ; ·· · · · · · ·
x=1
=·Y1+Y2
iii)
L.___
X=10 V=1B __
b) Answers may vary. Sample answer: Using regression on
the graphing calculator, a curve of best fit that is quadratic
has a greater value.
c) N(t) ==
+
+
d) N(t) ==
+
+
yes, it gives a
better approximation for each value.
1 1 . a) increasing, producing more high-calibre players
2.
R2
-0.003x2 0.621x 11.003
-0.001x2 0.505x 12.216;
b)
=
15. Answers may vary.
almost equal
c) Answers may vary. Sample answer: It has stayed relatively
constant over the years. d) confirm e) how many more
draftees than retirees there are
a)
L
a) =
ii)
·"),
y
X=1
y
V:'\·2.
x2 + 4x - 5; (x E IR}, (y E IR, y -9}
2=
b) y =
x2 + 4x - 5 + 2x; (x E IR}, {y E IR, y -8.76}
c) y =
x -2 - 2x; {x E IR}, {y E IR, y -2.9}
2=
:::;
= 6x
4. a) W
c) E
=
b) T
= 9x
15x
f) decreasing and
12.
then increasing; surplus, not enough, surplus
number of extra players per team over time
X=
V=.01 BBB1
d)
b) Years when there are no extra players; answers may vary.
c) Answers may vary. Sample answers: P(t) increases;
y = P(t) increases so that it no longer crosses the t-axis;
N(t)
surplus of players to draft
13. Answers may vary.
576
MHR
•
Advanced Fu nctions
•
Answers
a)
$780
l ine symmetry about the y-axis; both
even functions
S.
u (x) and v(x) are
6. a) Window variables:
+
c)
b)
{x
E
x E [- 47241t, 4724n ] , Xscl 2
'
y E[
-4, 4]
Sample answer: The business owner should reduce costs.
588 Hz
13. a)
b)
IR, x ¥= (2n -2 1)n , n E 7L}, [y E IR) d) y = tan x
V1:s;o<5BB1rX)+s;o(6 B"X)_
14. Answers may vary.
Chapter 8 Practice Test, page 474-475
1. A
2. D
3. D
4. A
[x E IR, x ¥= nn, n E 7L), [y E IR)
= 4x2 - 14x
12. a) i) all values except at point A, where C = R
ii) no values b) not profitable c) Answers may vary.
c) Answers may vary. Sample answer: reflection in the y-axis
and shifted right
units
8. a) y
+
5. a) y =
x2 - 5x + 13; [x E IR), [y E IR, y 6.75)
2:
10; [x E IR), [y E IR, y -2.25)
2:
V=6.75
X=2.5
qx2 - 7x /5; {x
+
b) y =
V1=X>-
b) y =
2x2 + 6x - 5; [x E IR), [y E IR, y 2: -9.5)
v
=
X: "1.5
c) y
c) y
V: - .5
4x - 15; [x E IR), [y E IR)
d) y =
E
IR), [y E IR, y -7.25)
2:
5
= x2 + 2x + 1; [x IR), [y IR,
E
E
y 2:
0)
x; [x E IR), [y E IR)
:X
d) y =
X:5
x; x IR, y E IR
E
V:S
9. a)
=
b) Answers may vary. Sample answer:
f(x) = 2x - 3, then r1(x) X +2 3 and f(f-'(x))= X.
Given f(x) x2, then r-'(x) = ±vx and fW'(x)) = X.
10. a) i) approximately x > 9.31 ii) x < 9.31
Given
=
7. Answers may vary.
1 , x ¥= -5, x ¥= -4; [x E IR, x ¥= -5, x ¥= - 4),
x
+
4
IR, y ¥= 0, y ¥= - 1)
8. a) y =
[y E
b) i), ii)
hole:
--
=
(-5, -1), asymptotes: x -4 and y = 0
X= .H
11. Answers may vary.
Answers
•
MHR
577
b) y
=
x + 4, x - 5; {x E IR, x "" -5}, {y E IR, y "" - 1}
Chapters 6 to 8 Review, pages 476-477
1. a) I
X
L2
(-5, -1)
9. a) V(C) = _52_ b) V(SA ) = SA (Sii:
Y 4rr
6
10. odd
11. a) damped harmonic oscillation; {t E IR, t 0},
{x E IR, -9.23 :s x :s 10 }
hole:
3
10 (2t) 0.95' 10
•
x ) 0;
b) i) cos
ii)
c)
em d) at ( t = when the
pendulum bob crosses the rest position the first time
e) at crests and troughs; when the pendulum bob changes
direction f)
s
12. a)
12.7
V1=.s<•· ·•·< - >>
/
b) Answers may vary. Sample answer: y ==
c)
Window variables:
y
2.855x2 - 7.217
(-4.73, 56.67), (-1.94, 3.56), (1.94, 3.56), (4.73, 56.67)
x E [-5, 5], E [-20, 100] , Yscl 10
13. a) Window variables:
x E [0, 40], Xscl 5, y E [-2, 10]
b) oc, c) oc, yes
14. a) Answers may vary. Sample answer:
31 45 31
and g(x) = 0.5(x - 1)(x + 1)
f(x) = (x - 1)(x + 1)
15. a)
b) Answers may vary. Sample answer: The skier encounters
10
s, where the bumps start on the graph.
moguls at
c) {t
:s t :s
:s
:s
E IR, 0
578
MHR
64.3}, {h E IR, 0 h 80}
•
Advanced Functions
•
A nswers
y
2
1
16
-1
1
4
0
1
1
2
2
-
1
4
2
16
·--
3
64
--
2 7 -3 3 4 -5
5.9
0.9
{x E IR, x 2},
2.5, y-intercept none,
asymptote x = 2
d) e) f)
2. a) b) c)
3. a)
units b)
units
4. a)
>
x-intercept
b)
reflection in the x-axis
10 3
1.95 4.17
$28 000,
0 2.4
99
10-'o
* 1 x 17
1.89 2.62 -4.15 -2.58 5.06 0.33
194 2.8
1), x -1, x "" 1 (x1J ) , x > 0
c) 2 log (x + y}, x > - y, x "" 0
13. a) _ ± b) 1, extraneous root _ 2_ c) 7, extraneous root 1
3
3
14. Different; the domain of the graph of g(x) is
{x E IR, x > 0}, while the domain of f(x) is {x E IR}.
15. a) P(t) = 38 000(1.12)' b) 6.1 years c) 8.5 years
16. a) y = 2x + x2 + 1; {x E IR}, {y E IR, y 1.9 }
5. a) b)
6. a)
b)
7 . a)
when t = b)
years
8. a) decibels b)
W/m2
9. a) b) c) =
10. a )
b)
c)
d)
e)
11. a)
h b)
h
12. a) log (x +
>
b) log
V1=2'
\/
f)
c) y = (2x + 2 )(x2 - 1 ) ; {x E IR}, {y E IR, y :?: - 3.04 }
Course Review, pages 479-483
1. a) An even function is symmetric with respect to the y-axis.
An odd function is symmetric with respect to the origin.
Substitute -x for x in f(x). If f( -x) = f(x), the function
is even. If f( -x) = -((x), the function is odd.
2. Answers may vary. Sample answer: A polynomial function
has the form f(x) = a,x" + a,_ 1 x" ­1 + . . . + a 1 x + a0•
For a polynomial function of degree n, where n is a positive
integer, the nth differences are equal (or constant).
3. f(x) extends from quadrant 3 to quadrant 4; even
exponent, negative coefficient
g(x) extends from quadrant 2 to quadrant 1 ; even exponent,
positive coefficient
h(x) extends from quadrant 3 to quadrant 1; even exponent,
positive coefficient
4. 4
5. y = - 2(x - 3 )4 + 1
b)
=
17. a) i) C(n) 35 + n, 0 :=; n :=; 200
ii) R(n) = 2.5n, 0 :=; n :=; 200 b) Window variables:
x E [0, 200], Xscl 10, y E [0, 500], Yscl 50
c) (23.33, 58.33); Kathy makes a profit if she sells 24 or
more cups of apple cider. Kathy loses money if she sells
23 or fewer cups of apple cider.
d) P(n) = l .Sn - 35
6. f(x): x-intercept 0, y-intercept 0, (x E IR}, (y E IR}
g(x) : x-intercepts - 2 and 1 , y-intercept 2, (x E IR}, {y E IR}
Window variables: x E [ -20, 20], Xscl 2, y E [ -200, 1 00],
Yscl 20
7. a) 138 b) 1 8; 342 c) The graph is increasing for 1 < x < 3.
e) $265
18. a) linear; neither
8. a) y (x + 3)(x + l )(x - 2)(x - 5) b) y = - (x + 5)(x ­1 )2
9. For x < 0, the slope is positive and decreasing. For x > 0,
=
b) periodic; even
the slope is negative and decreasing.
z
J
10. a) 4x + 6x - 5x + 2 = 2x2 + 4x + 4 + 6_ ,
2x - 1
2x - 1
x ,;, .l
2
2xJ - 4x + 8 2x2 + 4x + 4 + ____!i_ , x 2
,;,
x-2
x-2
c) xJ - 3x2 + 5x - 4 x2 - 5x + 15 + - 34 ' x ,;, -2
x+2
x+2
V1=cos( )
=
b)
_
_
=
d) 5x4 - 3xJ + 2x2 + 4x - 6 = 5K - 8
e) {x E IR}, {y E IR}
·b - 9x2 .
<
1
19. a) y =
, x E IR, - 1 - x < 0, 0 < x <
- 3 ,
x
3
1 ; ( X E IR, X 9}, (y E IR, y 0}
(y E IR} b) y "'"
"'"
x-9
20. a) Window variables: x E [ - 20, 20], Xscl 2,
y E ( - 200, 1 00], Yscl 20
b) [- 5, - 0.65) or (7.65, oo )
{
=
}
+ lOx - 6 x ,;, -1
'
x+ l
11. a) (x - l )(x + 2 )(x + 3 ) b) (x - 3 )(x + l ) (2x + 5 )
c) ( x - 2 )(x2 - 5 x + 1 ) d) (x2 + x + l )(x2 - x + 1 )
12. a) 345 b)
�;
13. a) No. b) No.
3 -2,
1 0, 3
3 d) -2,
2
14. a) - 3, 3 -2, - 1 , 4 c) -2
15. a) Answers may vary. Sample answer:
b)
y = k (x + 3)(x + l ) (x - 1 )2 ; y
y = - (x + 3)(x + l ) (x ­1 )2
2 (x + 3 ) (x + l )(x ­ 1 )y=3
'
=
2(x + 3)(x + l ) (x - 1 ) 2,
b)
Answers
•
MHR
579
f) i)
<)
-3, -1 < X < 1, X > 1
x -3 or x > 4 b) -2 < x < l2
-2 or -1 :s x 5
c) x
17. a) X = 2, y = 0 b) X = -3, y 1
X = -3, X = 3, y = 0 d) y 0
18. a) x = -4, y = 0 y-intercept .!.
4
d) X <
16. a) <
:s
:S
-
C)
i)
Ill )
Y1:1t(X+ )
l
I
X=-
=
=
1 9.
ii)
20. a) 11. b) c)
4
> - b)
21. a) < 4
22. a)
increasing for x < 2 and x > 2
{x E IR, x 2}, {y E IR, y 0}
-3, y = 1 y-intercept _.3!., x-intercept 1
c) i) x
iv)
v)
iv)
v)
d)
i i)
=
ii)
Y1:(X-1)/(X+ )
(
=
n
1
t
Y1:(2X+3)/CSX+U
i)
nn
)
"'""
y
v)
ii)
increasing for x < 0, decreasing for x > 0
{x E IR, x 0}, {y E IR, y > 0}
580
MHR
•
v:.22222222
x=s
b) R(t) =
0; The chemical will not completely dissolve.
0 :s t < 36)
23. a) 3rt b) 3
4
24. a) 30° b) 202.5°
25 15rt + 72
4
26. a) - v; b) -1 c) V3 d) -1
27. a) 1 + VJb) V3 - 1
2Vl 2Vl
Vi1 + 10V6
30
30.
8
31. a) period rt, amplitude 3, phase shift rad to the right,
vertical translation 4 units upward 2
b) {x E IR}, {y E IR, 1 :s y :s 7}
rt 3rt
3rt
.
b) x-mtercepts g'
32. a x-mtercept
l
S
c) t E IR,
{
Advanced Functions
•
Answers
.
)
X= .712319
iv)
v)
:S
29 .
decreasing for x < _.!5_ and x > _.!.5 {x E IR, x _.!.},
5
{y IR, y t}
e) x = 0, = 0 no intercepts
E
7 -2
x or x 235 x < -4 or -1 < x 3 or x 5
0
increasing for x < -3 and x > -3
{x E IR, x -3}, {y E IR, y 1}
3
.
.
') x = - 5,1 y = 52 ") y-mtercept
3 , x-mtercept
-2
iv)
:S
v)
"' :fc=
x=-s
y = 0 ii) y-intercept -
increasing for x < -3 and -3 < x < 3, decreasing for
3 < x < 9 and x > 9
{x E IR, X -3, X 9}, {y E IR, y - 112 , y > o}
positive increasing slope for x < -3, positive decreasing
slope for -3 < x < 2, negative decreasing slope for
2 < x < 7, negative increasing slope for x > 7
=
J
_____..,;, (
9,
Y=1
i)
00
=
iv)
decreasing for x < -4 and x > -4
{x E IR, x -4}, {y E IR, y 0}
b) x = 2, y 0 i ) y-intercept 2
iv)
v)
x = -3, x
Y:O
A
X:.7BS39B16
V:3
c) asymptotes x =
49 .
0, x = rt, x = 2rt
a)
33.
34. a)
d)
n
=1S7Q7963
4rt 5rt b) 5rt 3rt c) ' 5rt 3rt
3 ' 3 6 ' 2' 6 ' 2 6 6 ' 2
.
1'L 1,u
b) y =
50. a)
'
1.199 sin (113.091x) - 0.002
<) r::::
=.015
d)
b)
512 3 d) log 1 1 y = x
0
g(x): x-intercept 0, y-intercept 0, asymptote x = -1
Window variables: x E [ - 2, 10], y E [ - 2, 3]
1,
=
=
lffi, X > 0}, {y E lffi} ; g(x): {x E X > -1}, {y E lffi}
38 = 6561 b) a& = 75 c) 74 = 2401 d) a& = 19
a) 8 b) 1 c) l d) 5 e) 1 f) -1
2
a) 81 b) 5 c) 16 807 d) 3 e) -7 f) -3
40. 62 min
41. a) alkaline, 1.585 10-8 mol!L b) 3.1
a ) x = 1.29 b) x = 0.79 o r x 1.16
a) 3 b) l c) -2 d) l
2
3
44. a) 3.7004 b) 0.9212 c) 2.6801 d) 0.0283
a) 3 b) -2
X = 1.6180
a)
19.7 min b) approximately 7.03 mg
lffi ,
b) ((x): {X E
37. a)
38.
39.
X
42 .
43.
45.
46.
47.
c)
=
h =
V=2.5
V1:2"( 'X/lT)-2C05(
c)
)
V: "1.5
V1:2"( ·Xttrl*2<05( X)
X:3.1 15927
d)
V=1
t .lli
Y1:(2"( "XIlT))/(2co5( X))
5 1 . a)
1
53. )
=
2x2 + 5x + 22 b) 2x2 3x - 2 c) -5 d) 1 17
a ) 1 b) l c) does not exist d) 1l
4
4
a f(g(x) ) Vx'+l, {x E lffi, x 2!: -1};
g(f(x)) = Vx + 1, { x E lffi, x 0 }
b) f(g(x) ) = sin (x2 ), {x E lffi } ; g(f(x)) = sin2 x, {x E lffi }
c) f(g(x) ) = l x2 - 6 1 , {x E lffi } ; g(f(x)) = I xl 2 - 6, {x E lffi }
d) f(g(x)) 21Jx+ {x E lffi } ; g(f(x)) 3(2x + 1 ) + 2, {x E lffi }
e) f(g(x)) = 6
+ x + 6, {x E lffi, x 2!: 3};
g(f(x)) Vx2 + 6x + 6, {x E lffi, x -3 - V3,
x -3 + VJ) f) f(g(x)) = (x + 1) log 3, {x E lffi } ;
g(f(x)) = 3 1ogx+ {x E lffi, X > 0}
54. a y = _1_
b) {x E lffi, x > 0 c) neither
Vx
approximately
a
-4.43) or ( -3.11, -1.08) or (3.15,
(
52 .
4.04
3 229 660
+
2::
=
=
31,
=
2::
)
1,
_
56. )
d) Answers may vary. Sample answer: The graph would
decrease faster because the sample would be decreasing
at a faster rate.
48. a ) d =
years b) approximately
)
~
X:3.1 15927
a ) log7 49 = b) log. c = b c) log8
a) f(x): x-intercept asymptote x
2
V1=2"( "XIlf)+2C05(
=3.1 15927
V:1.187559B
135.6 cm/s
35.
36.
35 000(0.82)' b) $12 975.89
3.5 years
e) Answers may vary. Sample answer: The graph would
decrease faster.
V=1.19
:.QH
a) y =
c) approximately
V1:1/(C05( -(lT/2)))
b) approximately [
-co,
co
)
-1.69, 2]
Answers
•
MHR
581
)
57. a) h ( t = 20 sin
b)
(��) + 22
V1:(1TXli1S
,.-------,
k--u
V=22
V=.B377SBO
X=
u
X=i.2131BS3
V3=2os;n((1TXl11Sl+22
V=22
c)
X=30
c) The period of h ( O) is 2 n rad. The period of h(t) is
58. al W(t) =
b) (t E IR, t
3V'l0o + 25t
0}, ( W E
71., w
30 s.
30}
2
d)
59. a) C(t) =
b)
+
" "'"J'
V1:( -2X-5ll+2
I
16 X 220 + 53.12
V1= - OX+2l>-5
=
Determine Equations of Quadratic Functions, page 486
1 . a) f(x) = 2(x - l )(x ­ b) f(x) = - (x + 2)(x
c) f(x) = l .S(x
d) f(x) 0.5(x +
5)
+ 6)(x + 1)
- 1)
3)(x - 0.5)
Determine Intervals From Graphs, page 487
1. a) x-intercepts -2 and 2; above the x-axis for
c) approximately
31 years
P R E R EQ U I S ITE SKILLS A P P E N D I X A N SWERS
Angles From Trigonometric Ratios, page 484
1. a) 8 . 8 b)
c)
d)
e)
f)
1 ° 136.5° 70.0° -40.9° 75.7° -74.4°
Apply the Exponent Laws, pages 484-485
_
+6
65
- 15
4x6 - 4x4 +
3 - 4x2
81 1024 1 6
0 m-5n2, m, n 0
0
l _ c) 7 + 1._ d) .l..x6
- + 2x - x3
x
x3 9 81x54
x6 25
I
2. a) 9x2
xi + xi b) x + 9x4 ­l Ox
c)
8x3 - 8 x d ) V2x
+ l Ox - 20
1 - c) d) e) 125
3. a)
b) 4. a) 20x9y7 b) b 3 c3 , a , b, c i= c)
i=
d) xy -\ x, y i=
1. a) l_ b)
Apply Transformations to Functions, pages 485-486
1 . a) vertical translation b) vertical stretch c) horizontal
compression d) vertical reflection e) horizontal translation
f) horizontal reflection g) horizontal translation h) horizontal
stretch i) vertical translation j) vertical stretch
2 . a) vertical stretch by a factor of and horizontal reflection
in the y-axis
b) vertical translation downward by units and horizontal
compression by a factor of
3
±
3
c) horizontal translation left by 2 units and vertical reflection
in the x-axis
d) vertical compression by a factor of , horizontal
t
compression by a factor of .l, and horizontal reflection
in the y-axis
5
582
MHR
•
Advanced Functions
•
Answers
- 2 < x < 2; below the x-axis for x < - 2 and x > 2
and above the x-axis for
<x<
b) x-intercepts
and x > below the x-axis for x <
and < x <
3;
-3, 0,
3;
-3
0
-3
3
Distance Between Two Points, page 487
f) V82.
e)
1. a)
b) V65 c) V74 d)
3VS 3VS
3Vl
Domain and Range, page 488
1. a) (x E IR}, (y E IR}
b) (x E IR}, (y E IR}
-1}
c) (x E IR}, (y E IR, y
d) (x E IR}, ( y E IR, y
4}
(y E IR , y
e) (x E IR I x
f) (x E IR, x 2}, (y E IR, y 0}
g) (x E IR, x i= -2), (y E IR, y i=
h) (x E IR, x i=
(y E IR, y i=
-5},
=
0}
0)
0)
1},
Equation of a Line, pages 488-489
1. a) y = 2x + 1 b) y
-4x + 4
2. a) y = 3x - 1 b) y = - x - 3
3. a) y = Zx + l b) y = -x + 8
4. a) y =
2
-4x +24 b) y = 32 x ­2
Evaluate Functions, pages 489-490
-7 b) 35 c 5 d) _.§2.
e) 39.22 1 f) n3 +3n2 - 4n - 7
27
73
- 7 x6 + 3x4 - 4x2 - 7
2. a) - 1 b) 20 c) 2 d) x2 + 4x
3. a) 0 b) 2 c) 1 d)
4. a) 5 b) 2 c) 10 d) 0
1. a)
)
g) - 2 x + 27x2 + 1 2x
h)
- 1
5. a) true b) false c) true d) false
0
Exact Trigonometric Ratios of Special Angles,
pages 490-491
V3
1
. 30° = 2' cos 30° = 2' tan 30°
1 . sm
=
V3
.
1
sm 2 1 0° = -2, cos 2 1 0° = -2, tan 2 1 0°
csc 2 10° = -2, sec 2 1 0° = .
1 cos 330°
sm 3 30° = -2,
csc 330° = - 2, sec 330°
=
V3
'
1
'
V3
cot 2 1 0° = V3,
=
1
V3 '
cot 330° = - V3
2' tan 330° = -
=
'
V1=HK+3)
Cl.
1
' csc 30° = 2 ,
V3
sec 30° = _1_, cot 30° = V3
V3
V3
.
1
1
2. sm 1 50° = 2' cos 150° = - 2, ran 150° = V3 '
cot 1 50° = - V3,
csc 1 50° = 2, sec 1 50° = -
'
2. a)
V=2
K=1
b) {x E IR, x
<)
:::0:
-3), (y E IR, y ::=o: 0)
d) {x E IR, x :::0: 0), {y E IR, y ::=o: - 3 )
e)
Yes. The graph of the inverse passes the vertical line rest.
Graphs and Transformations of Sinusoidal Functions
Using Degree Measure, page 495
1. a) Window variables: x E [ - 360, 360], Xscl 30, y E [ - 5, 5]
Factor Quadratic Expressions, pages 491 -492
1. a) (x - 2)(x + 2) b) (y ­1 0)(y + 10) c) (2n - 7) (2n + 7)
d) (5m - 9)(5m + 9) e) ( 1 - 6x)( 1 + 6x) f) (2y - 3x)(2y + 3x)
2. a) (x - 5 )(x + 4) b) (y + 5 ) (y - 2) c) (n - 9)(n + 4)
d) (m + 3 )(m + 6) e) (x - 6)(x - 5) f) (c - 6)(c + 4)
g) ( 1 6 - y)(1 + y) h) (x + 4y)(x + Sy) i) (c - 7d) (c + 4d)
3. a)
(x - 2)(3x + 4) b) (2c - 1 ) (c + 4) c) (4m
d) (y + 1 )(5y + 3 ) e ) (n + 1 ) (3 n - 2 ) f ) ( 6 x +
g ) ( 3 x + 4y)(x - 3y) h ) (5x - 4)(x - 2 ) i ) (4x
j ) (p + q)(2p - q )
- 3)(m - 2 )
1 )(x - 3 )
+ 3)(x + 5 )
amplitude 3, period 360°, no phase shift, vertical translation
1 unit downward
b) Window variables: x E [- 360, 360], Xscl 30, y E [ - 5, 5]
Finite Differences, pages 492-493
1 . a) linear b) quadratic c) linear d) neither e) quadratic f) neither
Graph an Exponential Function, pages 493-494
1 . a)
b) {x E IR), {y E IR, y > 0), asymptote y
c) i) 1 .4 ii) 0.2
=
0
Graph an I nverse, page 494
amplitude 1, period 1 80°, phase shift 90° to the left,
no vertical translation
"M
amplitude 4, period 360°, phase shift 60° to the right,
vertical translation 2 units downward
2. Window variables: x E [ - 360, 360], Xscl 30, y E [ -4, 4]
a)
b)
{x E IR), {y E IR, y
::=o:
V1=<os(K)+1
2)
amplitude 1 , period 360°, no phase shift, vertical translation
1 unit upward
b)
d) (x e IR, x
::=o: 2), (y e IR)
e) Since the graph of the inverse does nor pass the vertical
line rest, r- i is not a function.
amplitude 2, period 360°, phase shift 90° to the left,
no vertical translation
Answers
•
MHR
583
3. •••••• ••••••
b)
t
n
1
amplitude 1 , period 1 20°, phase shift 30° to the left,
vertical translation 1 unit downward
Identify Linear, Quadratic, and Exponential Growth
Models, page 496
1. a)
•
:- :- :·
:
:·
,,
,,
:·
,,
:· :·
n
•
8
4
16
5
32
c) exponential
d) Window variables: x E [0, 7], y E [0, 40], Yscl 5
•
•
:·
): ): ) ):- ):• ): )(•
•
3
•
.
..
•
•
4
•
.
..
:·
•
2
•
.
..
b)
)(- ): )(-): ): ): ): )(- ):•):•)
)():
):
):•
):
):
):..
):
):
)(): )(- ):- ):· ):· )f. ):· ):· ):·):· )r
2
•
t
•
•
e) t =
2"
x
±R;
Inverses, pages 496-497
+
1
x
5 funcnon
.
b) y = ---;
1. a) y = --; funcnon
·
3
1
8
2
14
c) y =
3
20
not a function
4
26
5
32
e)
:
6
not function d) y
y
= x
=
+ 1 ; function
±Vx ;
8;
Primary Trigonometric Ratios and the CAST Rule,
pages 497-498
c) linear
d) Window variables:
x
8
. e = 17' cos e =
1. a) Sin
E
Vs tan oll =
. ll
c) Sin
v = -3,
[0, 7], y E [0, 40], Yscl 5
15
12
5
b) cos e = - TI, ran e = - 12
17
8
5
Vs d) Sin
. ll
v'89'
cos llu = v'89
o=
2
-
2. a) positive, 0.2867 b) positive, 0.9397 c) negative, - 0.5736
d) positive, 0.2867 e) negative, - 0.43 84 f) negative, -0.8192
e) t =
2. a)
x
Product ofTwo Binomials, page 498
1 . a) 2a2 + a - 1 5 b) 3m2 + 14m + 8 c) 2 2 + 2xy ­1 2y2
d) 1 0m2 - 1 1 mn
6n + 2
b)
c) quadratic
d) Window variables:
x
E
n
t
1
1
2
4
3
9
4
16
5
25
+
3n2
Quadratic Functions, pages 498-499
1. a) vertex (0, 2); opens upward; vertical stretch factor 1
b) vertex (0, - 1 ); opens downward; vertical stretch factor 1
[0, 7], y E [0, 40], Yscl 5
c) vertex (0, 0); opens upward; vertical stretch factor 2
584
MHR
•
Advanced Functions
•
Answers
d) vertex ( -4, 0); opens upward; vertical stretch factor 3
V1:)(
X\]
d) vertical asymptote x
V1:1
e) vertex (1, 4); opens downward; vertical stretch factor 2
=
H+12l
Reciprocal Trigonometric Ratios, pages 499-501
25 , cot e = - .2.
7
24
3
Vi3
Vi3
,
sec e = c) csc e = - V5s' cot e = V5s d) csc e = 3
2
2 . a) 1 .3270 b) -2.4586 c) - 7 . 1 853 d) - 1 .0724 e) - 1 .70 1 3
f ) - 1 .0038
3. a) 29° b) 105° c) - 80° d) 23°
1. a) esc e = 1.Z sec e = 1.Z b) sec e = -
15 '
8
�v
f) vertex ( -2, 3 ); opens upward; vertical stretch factor 0.5
V1=0.5(
8
Simplify a Radical Expression, page 501
1 . a) 2V3 b) 2VS c)
g) vertex ( - 1 , -5); opens downward; vertical stretch factor 1
V1: '(X+i)>-S
d) -
1
1 . a) 2
vertex (3, 1 ); opens upward; vertical stretch factor 1
4
V1=0.2S(H-
\U
v;
e) 5 + VS f) 3 - V6
Simplify Expressions, page 501
1 . a) 5x + 20 b) 7y ­30 c) 8x - 18 d) 4w + 1 1
e) 3 5 ­6y f) 7x2 - 2x + 4
g) 4x2 + 6x - 46 h) 5a2 - 16a + 20
Slope, page 502
h)
-4, horizontal asymptote y = 0
3
11
2
3
3
b) - 5 c) 3 d) 4 e) -9 f) 4 e) - 2 f) - 2
Slope and y-intercept o f a Line, page 502
1. a) slope 4; y-intercept 1 b) slope 1; y-intercept - 2
c) slope 3; y-intercept 5 d) slope - 7; y-intercept 3
e) slope 3; y-intercept 0 f) slope 0; y-intercept - 8
g) slope 5; y-intercept 2 h) slope 4 ; y-intercept ­3
i) slope - 3.5; y-intercept 2.5 j) slope 1 ; y-intercept 0
k)
Reciprocal Functions, page 499
1. Window variables: x E [ -6, 6], y E [ - 6, 6]
a) vertical asymptote x = 4, horizontal asymptote y = 0
l
b) vertical asymptote x = - 3, horizontal asymptote y = 0
slope 4; y-intercept 4
I)
slope 6; y-intercept 4
Solve Quadratic Equations, page 503
1. a) - 4, -5 b) 1 , 2 c) 3, - 1 0 d) -3, - 5
2. a)
- 1,
3
b)
l2 c) :t l5 d ) 0, ±9
1 , 1 b) l ' l c) 1 ,
3
2 2 5
1 :t V7
- 1 :t VU ; 0.5 or ­0.9 b) --; 1 .2 or -0.5
4. a)
3
5
- 1 :!: ViS
4 :!: v'2
; 1 .4 or -2.4
c) ---; 2.7 or 1.3 d)
2
2
- 5 :t V97 ; 0.4 o r - 1 .2
1 VU
; 0.9 or - 0.5 f)
e) :t
12
5
3. a)
Trigonometric Identities, page 504
1. Answers may vary.
c) vertical asymptote x = 1 , horizontal asymptote y = 0
Use Long Division, page 504
1. a) 28 R3 b) 156 R3 c) 146 R 1 7 d) 127 Rl l
Answers
•
MHR
585
I G l o s sa ry
argument
a function.
The first element of an ordered pair. In the ordered
pair (x, y), x is the abscissa. See ordinate.
abscissa
absolute value
number line.
1 - sl
=
The distance from zero of a number on a
A piecewise function written as
Jxl, where y = x for x 0 and y = -x for x < 0.
acceleration The rate of change of an object's velocity with
respect to time.
acute angle
arithmetic sequence A sequence that has a common
difference between consecutive terms.
1, 4, 7, 10, . . . is an arithmetic sequence.
astronomical unit (AU) The distance from Earth to the Sun,
or approximately 1 50 000 000 km.
s
absolute value function
y =
The value(s) assigned to the variable(s) in
An angle that measures less than 90°.
algebraic expression An expression that includes at least
one variable.
2t, 3x2 + 4x - 5, and 2"' are algebraic expressions.
algebraic modelling The process of representing a
relationship by an equation or a formula, or representing
a pattern of numbers by an algebraic expression.
The height of a geometric figure. In a triangle, an
altitude is the perpendicular distance from a vertex to the
opposite side.
altitude
amplitude Half the difference between the maximum and
minimum values of a periodic funtion. Jal for functions in
the form y a sin x or y = a cos x. See period.
=
angle in standard position The position of an angle when
its vertex is at the origin and its initial ray is on the positive
x-axis. The angle is measured counterclockwise from the
initial ray. For example, 45° is shown in standard position.
y
asymptote A line that a curve
approaches more and more closely
but does not intersect. The line
may be a horizontal asymptote,
a vertical asymptote, or a linear
oblique asymptote.
X
average rate of change The rate of
change that is measured over an interval
on a continuous curve. It corresponds to
the slope of the secant between the two endpoints
of the interval.
average rate of change of y with respect to x
y
=
For a function
f(x), the average rate of change of y with respect to x
.
.
y
x
over the mterval x E [a, b] ts -
=
((b) - ((a)
.
b-a
A vertical mirror line that reflects every
point on the graph of a function onto another point on the
graph of the function. See line of symmetry.
axis of symmetry
base function For various functions, the simplest form of
the function that others are related to by tranformations.
y x2 is the base function for quadratic functions.
=
The number that is repeatedly multiplied.
In 3\ the base is 3.
base (of a power)
A polynomial with two terms.
Sx - 2 is a binomial.
binomial
X
break-even point
are equal.
angular velocity (or speed) The rate at which the central
angle is changing, often measured in radians per second.
arc
A part of the circumference of a circle.
586
MHR
•
Adva nced Functions
•
Glossary
The point at which costs and revenue
The greatest volume that a container can hold,
usually measured in litres, millilitres, or kilolitres.
capacity
The system developed by Rene
Descartes for graphing points as ordered pairs on a grid,
using two perpendicular number lines. Also referred to as
the Cartesian plane, the coordinate grid, or the x-y plane.
Cartesian coordinate system
A rule that tells which
trigonometric ratios are positive
in each quadrant.
CAST rule
An angle formed
by two radii of a circle.
central angle
change of base formula
b > 0, b * 1
chord (of a circle)
log& m =
s
sme
T
y
A
all
0
c
tangent
X
cosine
log m
, m > 0,
Iog b
--
A line segment with its endpoints on a circle.
The set of all points in the plane that are equidistant
from a fixed point called the centre.
circle
The factor by which a variable is multiplied.
In the term 3x2, the coefficient is 3.
coefficient
cofunction identities Relationships between pairs of
trigonometric functions involving pairs of complementary
angles that are true for all values of the variable. For
- x and cos x = sin
-x .
example, sinx = cos
(I )
(I )
common difference The difference between consecutive
terms of an arithmetic sequence.
For 2, 5, 8, 1 1 , . . . , the common difference is 3.
common factor Any factor that two or more numbers, or
two or more terms of a polynomial, share.
2 is a common factor of 4, 6, and 1 8 .
3 x i s a common factor o f 3x2 - 1 2x.
common logarithm
A logarithm with base 1 0 .
common ratio The ratio o f consecutive terms o f a
geometric sequence.
For 2, 6, 1 8, 54, . . . , the common ratio is 3.
The property that the order of the
operands in an expression does not matter. For example,
for a, b E IR, a + b = b + a and ab = ba.
commutative property
composite function A function made up of (composed of)
other functions. The composition of f and g is defined by
f(g(x)) and read as "f of g of x" or "f following g of x." In
the composition (f(g)(x)), first apply the function g to x, and
then apply the function f to the result.
A trigonometric expression
that depends on two or more angles.
compound interest Interest that is calculated at regular
compounding periods and then added to the principal for
the next compounding period. The future amount, A, of an
investment with initial principal P is A = P(1 + i)", where i is
the interest rate per compounding period, in decimal form,
and n is the number of compounding periods.
concurrent lines
Two or more lines that have one point in
common.
congruence The property of being congruent. Two geometric
figures are congruent if they are equal in all respects.
conjecture A generalization, or educated guess, made from
available evidence.
constant function
constant rate of change The rate of change of a relation is
constant when the average rate of change between any two
points on the relation is equal to the average rate of change
between any other two points on the relation. Linear
relations have a constant rate of change.
A term that does not include a variable.
A numerical term.
In x2 + Sx - 1, the constant term is ­1 .
constant term
A one-to-one pairing of all ordered pairs
of real numbers with all points of a plane. Also called the
Cartesian coordinate plane.
coordinate plane
A variable used to measure the
strength of a relationship when regression is used to find
an equation that approximates data.
correlation coefficient
cosecant ratio
cosine law
The reciprocal of the sine ratio.
See law of consines.
cosine ratio In a right triangle, for LC, the ratio of
the length of the side adjacent to LC and the length
of the hypotenuse.
A
cos LC = BC
AC
-Z_j
h ypote
C
ad'Jacent
opposite
B
The reciprocal of the tangent ratio.
cotangent ratio
compound angle expression
coterminal angles
The following relationships are
true for all values of x and y.
sin (x + y) = sin x cos y + cos x sin y
sin (x - y) = sin x cos y - cos x sin y
cos (x + y) = cos x cos y - sin x siny
cos (x - y) = cos x cos y + sin x siny
not true.
compound angle identities
A function of the form f(x) = b, where
b E IR.
counterexample
Angles that have the same terminal arm.
An example that shows that a conjecture is
cubic equation
A polynomial equation of degree 3.
cubic function
A polynomial function of degree 3 .
curve o f best fi t A curve that approximates the distribution
of points in a scatter plot.
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cylinder A three-dimensional shape with two parallel faces
that are congruent circles, and a curved surface connecting
the two circles.
II
The individual members of a set.
elements
A method of solving a system of
equations by addition or subtraction of the equations to
eliminate one variable.
elimination method
A logarithmic scale used to compare
decibel scale
sound levels.
A function f(x) such that f(a) > f(b) for
decreasing function
all a < b.
-
The greatest exponent of the
variable in any one term.
The degree of x3 + 6x2 1 is 3 .
degree of a polynomial
I n a relation, the variable whose value
depends on the value of the independent variable. On a
coordinate grid, the values of the dependent variable are
on the vertical axis.
In d = 4.9tl, d is the dependent variable.
dependent variable
The amount by which an item decreases
in value over time.
depreciation
A line segment joining two non-adjacent vertices
of a polygon.
diagonal
diameter
A chord that passes through the centre of a circle.
A function has a discontinuity at x = a if it
is not continuous at x = a. For example, f(x) = .!.
X has a
discontinuity at x = 0.
discontinuity
=
A function is discontinuous at x
a discontinuity at x a.
discontinuous
distributive property
a(b + c) = ab
dividend
In 30
+
ac.
=
The behaviour of the y-values of a function
as x approaches +co and x approaches -co.
end behaviour
a if it has
The property that, for all a, b, c E IR,
A number, or expression, being divided.
5 = 6, 30 is the dividend.
+
.
divisor A number, or expression, that is dividing
into another.
.
In 4x2 + l Ox + 25 , 2X + 5 IS the d"! VISOr.
2X + 5
The set of numbers for which a
function is defined. The set of all first coordinates of the
ordered pairs in a function.
domain of a function
equation A mathematical sentence formed by two
equivalent expressions.
5x - 3 = 2x + 6 is an equation.
equilateral triangle
A triangle with all sides equal.
equivalent algebraic expressions Expressions that are equal
for all values of the variable.
7t + 3t and l Ot are equivalent algebraic expressions.
equivalent equations
evaluate
Equations that have the same solution.
To determine a value for.
A function f(x) that satisfies the property
f( -x) = f(x) for all x in its domain. An even function is
symmetric about the y-axis. See odd function.
even function
To multiply, usually applied to polynomials.
4(n - 3 ) expands to 4n - 12.
expand
A raised number in a power that indicates
repeated multiplication of the base.
In (x + 3 )2 = (x + 3 ) (x + 3 ), the exponent is 2.
exponent
Exponential decay occurs when a
quantity decreases exponentially over time.
exponential decay
An equation that has a variable in
an exponent.
3x = 81 is an exponential equation.
exponential equation
exponential form A shorthand method for writing
repeated multiplication.
43 is the exponential form for 4 X 4 X 4.
A function of the form y = abX, where
a of. 0, b > 0, and b -:f. 1 .
f(x) = 3 (2"') is an exponential function.
exponential function
exponential growth Exponential growth occurs when a
quantity increases exponentially over time.
double root A solution of a polynomial equation that
occurs twice. For example, in (x - 2f(x + 1 ) = 0, 2 is a
double root.
exponential regression A method of determining the
exponential equation of a curve that fits the distribution
of points on a seatter plot.
dynamic geometry software Computer software that allows
the user to plot points on a coordinate system, measure line
segments and angles, construct two-dimensional shapes, create
two-dimensional representations of three-dimensional objects,
and transform constructed figures by moving parts of them.
expression
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G lossary
A mathematical phrase made up of numbers
and/or variables.
x2 + x - 5, 2x, and 3n are expressions.
Roots that occur in a solurion but that do
not check in the orginal equation and so are invalid.
extraneous roots
extrapolate Estimate values lying outside the range of the
given data. To extrapolate from a graph means ro estimate
coordinates of a point beyond those that are plotted.
II
In a relation, the variable whose
value determines that of the dependent variable. On a
coordinate grid, the values of the independent variable are
on the horizontal axis.
In d = 4.9r, t is the independent variable.
independent variable
Two expressions related by an inequality symbol
( > , <, , .;; , of. ) .
4x + 5 > 8 i s a n inequality.
inequality
A number or an algebraic expression that is multiplied
by another number or algebriac expression ro give a product.
The factors of 12 are 1 , 2, 3, 4, 6, and 12.
The facrors of 4x1 + Bxy are 4, x, and x + 2y.
factor
A polynomial P(x) has ax - b as a facror if
factor theorem
and only if P
(%)
=
0, where a, b E IR and a i= 0.
A group of functions with a common
characteristic. For example, a family of polynomial
functions has the same zeros.
family of functions
Differences found from successive y-values
in a table of values with evenly spaced x-values. See first
differences and second differences.
finite differences
In a relation between two variables, the
difference between successive values of the second variable
for regular steps of the first variable.
first differences
y = 2x + l
X
y
-2
-3
-1
-1
1
3
1
0
2
frequency
5
First Differences
-
1
-
( - 3)
=2
1 - (- 1 ) = 2
3 - 1 =2
5 - 3 = 2
The number of cycles per unit of rime.
A relation in which each element in the domain
(or x-value) has only one corresponding element in the
range (or y-value).
y = (3x - Sf is a function.
function
A number in the sequence
. . . , -3, -2, - 1 , 0, 1, 2, 3, . . . . Represented by the set
symbol 7l..
integer
integral zero theorem If x - b is a facror of the polynomial
function P(x) with leading coefficient one and remaining
coefficients that are integers, then b is a facror of the
constant term.
The distance from the origin of rhe Cartesian
coordinate plane ro the point at which a line or curve
crosses a given axis. See x-intercep t and y-intercep t.
intercept
interest The amount earned on an investment or savings
alternative, or the cost of borrowing money.
The rate, as a percent, ar which an investment
or savings alternative increases in value, or rhe cost of
borrowing money, expressed as a percent.
interest rate
interior angle
An angle that is inside a polygon.
interpolate Estimate values for a relation that lie between
given data points.
interval A set of real numbers having one of these forms
where a, b E IR: x > a, x a, x < a, x ,;; a, a < x < b,
a < x ,;; b, a :S x < b, or a ,;; x ,;; b
interval notation Representations of intervals using
brackets: (a, oo ) , [a, oo ) , ( - oo, a], (a, b), (a, b], [a, b), or [a, b].
invariant point
m
transformation.
The rime in which rhe mass of a radioactive
substance decays to half its original mass.
half-life
hypotenuse
instantaneous rate of change The rate of change that is
measured at a single point on a continuous curve.
Instantaneous rate of change corresponds ro the slope of
the tangent line at that point.
The longest side of a right triangle.
D
A point that corresponds ro an object point
under a transformation.
image point
increasing function A function f(x) such that
f(a) < f(b) for all a < b.
A point that is unchanged by a
1
The inverse function ( of a function f,
if it exists, is found by writing the function in the form
y = f(x), exchanging x and y, and then solving for y.
inverse function
inverse relation The relation formed by interchanging the
domain and the range of a given relation.
inversely proportional
If two variables, x and y, are
inversely proportional, then y
=
' where k is a constant.
irrational number A real number that cannot be expressed
in the form !!:.. , where a and b E 71. and b i= 0.
b
isosceles triangle A triangle with exactly two equal sides.
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law of cosines The relationship between the lengths of the
three sides and the cosine of an angle in any triangle.
a2 = b2 + c2 - 2bc cos LA
law of sines The relationship between the sides and their
opposite angles in any triangle.
sin A sin B
sin C
a
c
b
The coefficient of the greatest power
of x in a polynomial P(x).
leading coefficient
like terms Terms that have exactly the same variable(s)
raised to exactly the same exponent(s).
3x2, -x2, and - 7x2 are like terms.
The line that best describes the distribution
of points in a scatter plot. The line passes through, or close
to, as many of the data points as possible.
line of best fit
line of symmetry
A mirror line that reflects an object
onto itself.
The part of a line that joins two points.
line segment
line symmetry A graph has line symmetry if there is a line
x = a, called the axis of symmetry, that divides the graph
into two parts such that each part is a reflection of the other
in the line x = a.
linear equation An equation that represents the relationship
between two variables that have a linear relationship and
form a straight line when graphed.
linear function A function of the form f(x)
where m and b are constants.
=
Growth represented by a linear equation and
a straight-line graph. Arithmetic sequences and simple
interest show linear growth.
A method for determining the linear equation
that best fits the distribution of points on a scatter plot.
linear regression
A relationship between two variables that
forms a straight line when graphed.
linear relation
local maximum The point on a function that has the
greatest y-value on some interval close to the point.
The point on a function that has the least
y-value on some interval close to the point.
local minimum
logarithm The logarithm of a number is the value of the
exponent to which a given base must be raised to produce the
given number. For example, log 81 = 4, because 34 = 8 1 .
3
logarithmic equation An equation that has a variable in
a logarithm.
log x = log 3 + log 5 is a logarithmic equation.
2
2
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logarithmic scale A scale that uses powers of 10, for
example, the decibel scale.
II
mathematical model A description of a real-life situation
using a diagram, a graph, a table of values, an equation, a
formula, a physical model, or a computer model.
The process of describing a real­
life situation in mathematical form.
mathematical modelling
mean The sum of a set of values divided by the number
of values in the set.
II
n! (n factorial)
For any positive integer n, a short form for
the product n X (n - 1 ) X . . . X 2 X 1 .
5! = 5 X 4 X 3 X 2 X 1
= 1 20
natural logarithm
natural number
1 , 2, 3, 4, . . . .
A logarithm in base e.
The set,
N,
of positive integers:
non-linear relation A relationship between two variables
that does not form a straight line when graphed.
mx + b,
linear growth
590
logarithmic function The inverse of an exponential
function. A function of the form f(x) = log. x, where a > 0,
a * 1, and x > 0.
y = log. x means aY x.
G l ossary
oblique asymptote
nor vertical.
oblique triangle
An asymptote that is neither horizontal
A triangle that is not right-angled.
An angle that measures more than 90°, but
less than 1 80°.
obtuse angle
obtuse triangle
A triangle containing one obtuse angle.
odd function A function that satisfies the property
f( -x) = -f(x) for all x in its domain. An odd function is
symmetric about the origin. See even function.
If a polynomial function has a factor (x - a)
that is repeated n times, then x = a is a zero of order n. For
example, the function f(x) = (x - 2 )(x + 1 )2 has a zero of
order 2.
order (of a zero)
A pair of numbers, such as (3, 8), used to
locate a point on a graph.
ordered pair
ordinate The second element in an ordered pair. In the
ordered pair (x, y), y is the ordinate. See abscissa .
origin The point of intersection of the x-axis and the y-axis
on a coordinate grid. It is described by the ordered pair (0, 0).
outlier A data point that does not conform to the pattern of
the other data.
point of tangency
II
The point that is common to two
The point of intersection of a tangent
and a curve.
The graph of a quadratic
relation, which is U-shaped and
symmetric about a line of symmetry.
parabola
point symmetry
X
A constant that can
assume different values but does not change the form of the
expression or function. In y = mx + b, m is a parameter
that represents the slope of the line and b is a parameter
that represents the y-intercept.
parameter
A number that represents a fraction or ratio with a
denominator of 100.
34 as a percent 1s
. 34 0/
;o .
1 00
percent
perfect square A whole number that can b e expressed a s
the square o f a whole number.
25 is a perfect square.
perfect square trinomial A trinomial that can be factored as
the square of a binomial.
a2x2 + 2abx + b2 = (ax + b)2
perimeter
point of intersection
non-parallel lines.
The distance around a two-dimensional shape.
period The magnitude of the interval of the domain over
which a periodic function repeats itself.
y
A graph has point symmetry about a point
(a, b) if each part of the graph on one side of (a, b) can be
rotated 1 80° to coincide with part of the graph on the other
side of (a, b).
polar coordinates A method of locating
a point in a plane using its distance, r,
from the origin and its angle in
standard position.
0
polygon A two-dimensional closed
shape whose sides are line segments.
polyhedron A three-dimensional object with faces that
are polygons.
polynomial
An expression of the form
a, X11 + a" _ 1X11 - I + . . . + a 1 X + a0, where an, a,, _ I ' . . . ,
1
a0 E IR, a,. * 0, and n E ?L, n > 0.
3x4 + 2x3 + 5x2 - x + 1 is a polynomial.
polynomial function A function of the form
P(x) = a,.x" + . . . + a x2 + a1 x + a0, where n is a
2
whole number.
power An abbreviation for repeated multiplication.
The power 53 means 5 X 5 X 5 and has value 125.
A function of the form f(x)
0 and n is a positive integer.
power function
a
*
power law of logarithms
x > 0, n E IR
=
ax", where
logbx" = nlogb x, b > 0, b
primary trigonometric ratios
* l,
The sine, cosine, and
tangent ratios.
prime number
and
1.
principal angle
periodic function A function that repeats itself over an
interval of its domain.
A logarithmic scale used to measure the acidity or
alkalinity of chemical solutions. pH = -log [H+ ], where [H+ ]
is the concentration of hydronium ions, in moles per litre.
pH scale
The horizontal translation of a trigonometric
function. See period for diagram.
phase shift
A point P on a curve where the curve
changes from concave upward to concave downward or
from concave downward to concave upward.
point of inflection
A number with exactly two factors-itself
The least positive coterminal angle.
prism A three-dimensional shape with two parallel,
congruent polygonal faces. The prism is named according to
the shape of these two faces, for example, triangular prism.
The likelihood of an event occurring.
number
of favourable outcomes .
P(event) =
number of possible outcomes
probability
product law of logarithms
b * l, m > 0, n > 0
log& (mn) = logb m + logb n, b > 0,
proportion An equation that states that two ratios
are equal.
3
X 1s a proport10n.
=
S 80
·
.
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A polyhedron with one base in the shape of a
polygon and the same number of lateral triangular faces as
there are sides in the base.
radical equation
Pythagorean identity
The relationship sin2x + cos2x = 1 ,
which i s true for all values o f x.
radical expression
In a right triangle, the square of the
length of the hypotenuse is equal to the sum of the squares
of the lengths of the other two sides.
radical function
pyramid
Pythagorean theorem
y
One of the four
regions formed by the
intersection of the x-axis
and the y-axis.
quadrant
Q
An equation that
can be written in the form ax2 + bx + c
where a, b, and c E IR and a 'I 0.
=
1
X
Quadrant 3
quadratic equation
Quadrant 4
=
- b ::t: Vb2 - 4ac .
2a
A relationship between two variables
defined by an equation of the form y = ax2 + bx + c, where
a, b, and c E IR and a 'I 0. Irs graph is a parabola.
A polygon with four sides.
quartic function
A polynomial of degree four.
quintic function
A polynomial of degree five.
The result of a division.
3
30x
.
In
. the quot1ent.
= 5 x- , 5 x- 1s
quotient function
g (x) 'I 0.
f(x)
g (x)
. of the form q(x) = -- ,
A functwn
sinx , which is
The relationship tan x = cos
x
true for all values o f x .
quotient identity
b > 0, b 'I 1 , m > 0, n > 0
Iogb (';; ) = logbm - logb n,
arc of length
one radius
The measure of the
angle formed by rotating
the radius of a circle
through an arc equal in
length to the radius. There
are 2 7t radians in one complete revolution (360°).
In the diagram, IJ measures 1 rad.
radian
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radical sign The symbol F, which indicates the principal
or non-negative square root of an expression.
An expression under a radical sign.
random number A number chosen from a set of numbers in
such a way that each number has an equally likely chance of
being selected.
The set of all second
coordinates, or ordinates, of the ordered pairs of a relation.
range of a function or relation
rate A special type of ratio that compares two quantities
with different units.
8.4 L/100 km is a rate.
•
rate of change.
See instantaneous rate of change and average
A comparison of quantities with the same unit.
3 cans of water to 1 can of juice is 3 : 1 .
ratio
An equation that contains one or more
rational expressions.
3- 5 is a rational equation.
-=
x2 - 4
rational equation
An exponent that is a rational number.
l
1
5 2 has a rational exponent of 2.
7
quotient law of logarithms
A function that has a variable in a radicand.
is a radical function.
f(x) =
rational exponent
quotient
7
An expression that contains a radical.
2'1(3 and 3vx are radical expressions.
rate of change
quadratic function
quadrilateral
8 is a radical equation.
The difference between the greatest and the
least values in a set of data.
The zeros, or solutions, of a quadratic
equation of the form ax2 + bx + c = 0, where a 'I 0, are
.
=
range of data
0,
quadratic formula
g1ven by x
Vx+T + 3
radicand
Quadrant
Quadrant 2
An equation that has a variable in the
radicand.
Glossary
g (x)
,
h (x)
where g (x) and h (x) are polynomials and h(x) 'I 0.
1- is a rational function.
f(x) -A function of the form f(x) =
rational function
=
2x ­5
rational number
form
A number that can be expressed in the
E 7l. and b 'I 0.
f, where a and b
rational zero theorem
If P(x) is a polynomial function with
%
integer coefficients and x = is a zero of P(x), where a and
b E 7l. and a 'I 0, then
• b is a factor of the constant term of P(x)
• a is a factor of the leading term of P(x)
• ax - b is a factor of P(x)
real numbers All the rational and irrational numbers,
represented by the set symbol R
1
reciprocal function A function, g(x) = -- defined by
f(x)
1 =1 l. f f(a) = b .
g(x) = -
f(a)
b
reciprocal trigonometric ratios
cotangent ratios.
cosecant = _.1_, secant
sme
=
The cosecant, secant, and
11-. , cotangent tangent
cosme
=
Two non-zero numbers that have a product of one.
are reciprocals.
reciprocals
An expression that
has no like terms.
Sx + 3 - x + 2 in simplest form is 4x + 5 .
simplest form of an algebraic expression
sine law
zJ
See law of sines.
relation A relationship between variables that can be
represented by a table of values, a graph, or an equation.
In a right triangle,
for LC, the ratio of the length
of the side opposite LC and
hypme
the length of the hypotenuse.
C
sin LC = AB
AC
remainder theorem When a polynomial P(x) is divided by
x ­b, the remainder is P(b), and when it is divided by
A function
that is used to model periodic data.
x and
regression A method for determining the equation of a curve
or line that best fits the distribution of points on a scatter plot.
ax ­b, the remainder is P
(%) , where a, b E
71.. ,
and a
* 0.
A constraint o n the value(s) o f a variable. For
example, in vx, the restriction is x ;;;. 0; in f(x ) = .l, the
X
restriction is x * 0.
restriction
Richter scale A logarithmic scale used to measure the
magnitude of earthquakes.
An angle that measures 90°.
right angle
right triangle
roots
A triangle that contains a 90° angle.
The solutions of an equation.
II
scalene triangle
A triangle with no sides equal.
A graph showing two-variable data by means
of points plotted on a coordinate grid.
scatter plot
A line passing through two different points on
a curve.
secant
secant ratio
The reciprocal of the cosine ratio.
The differences between consecutive first
differences in a table of values with evenly spaced x-values.
See first differences.
second differences
sector A part of a circle bounded by two
radii and an arc of the circle.
segment A part of a circle bounded by
a chord and an arc of the circle.
sequence
An ordered list of numbers or terms.
sine ratio
A
a d.Jacent
oppmi•
B
y
sinusoidal function
sinusoidal regression A
statistical process that determines
a sinusoidal function that best
represents data.
slope A measure of the steepness
of a line. The slope, m, of a line
containing the points P(xp y 1 )
and Q(x , y ) is
2 2
y
b.
Yz - y , , x i' x1•
m=
=
b.x x - x 1 2
2
y
- ---
,tJ)
<..lTx2, y2J
P(x ,
y,
'---v---'
0
Xz - xl
X
t
A linear
mx + b, where m is the
slope and y-intercept form of a linear equation
equation written in the form y
slope and b is the y-intercept.
=
To find the value of a variable in an equation.
When 2x = 16 is solved, x = 4.
solve
standard form of a linear equation A linear equation written
in the form Ax + By + C 0, where A, B, and C E 71.., A and
B are not both zero, and x and y E R
=
standard form of a quadratic function
written in the form y
=
ax2 + bx +
c,
A quadratic function
where a * 0.
substitution method A method of solving a system of
equations by solving one equation for one variable and
then substituting that value into the other equation.
subtend An arc, AB, of a circle can subtend an angle at the
centre, LAOB, or at the cirumference, LACB.
similar figures Figures having corresponding angles equal
and corresponding lengths proportional.
simple interest Interest calculated only on the original
principal using the simple interest formula I = Prt.
simplest form The form of a fraction or ratio that has no
common factors.
Glossary
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The sum of two or more functions
can be found by adding the y-coordinates of the function for
each x-coordinate.
superposition principle
The number of square units needed to cover
the surface of a three-dimensional shape.
surface area
A function that has either line
symmetry or rotational symmetry.
symmetric function
a
II
vertex
A point at which two sides of a polygon meet.
A quadratic equation
expressed in the form y = a(x - h )2 + k, where a, h, and
k E IR and a * 0. The vertex of the parabola is (h, k).
vertex form of a quadratic function
In a right triangle, for LC, the ratio of the
length of the side opposite LA and the length of the side
adjacent to LA.
A
tan LC AB
BC
tangent ratio
=
hypore
oppo<ire
B
a d.Jacem
tangent to a curve A line that intersects a curve at exacrly
one point.
C
A number or a variable, or the product or quotient of
numbers and variables.
3x2 - 5 has two terms: 3x2 and - 5 .
term
The ray o f a n angle i n standard position that
is not on the positive x-axis.
terminal arm
A mapping of points on a plane onto points
on the same plane.
transformation
A transformation that maps an object onto its
image so that each point in the object is moved the same
distance in the same direction.
translation
An equation involving one or more
trigonometric functions of a variable.
cosx - 2 sin x cos x = 0 is a trigonometric equation.
trigonometric equation
A function involving a primary
trigonometric ratio or a reciprocal trigonometric ratio.
y = 2.8 cos is a trigomometric function.
trigonometric function
}
trigonometric identity A trigonometric equation that is true
for all values of the variable for which both sides of the
equation are defined.
trinomial
594
unlike terms Terms that have different variables,
or different powers of the same variable.
2x2, -x, and 5y are unlike terms.
A polynomial with three terms.
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G l ossary
The point where the axis of symmetry
of the parabola intersects the parabola.
vertex o f a parabola
A test for determining whether a given
graph represents a function. If a vertical line intersects the
graph more than once, then the relation is not a function.
vertical line test
A transformation where (x, y) on the
graph of y = f(x) is transformed into (x, ay) on the graph
of y = af(x). The stretch is an expansion when a > 1 and
a compression when 0 < a < 1 .
vertical stretch
The amount of space that an object occupies,
measured in cubic units.
volume
II
The values of x are positive and increasing
in magnitude without bound; may be written x -7 +oo.
x approaches +oo
The values of x are negative and increasing
in magnitude without bound; may be written x -? - oo .
x approaches -oo
The distance from the origin of the point where
a line or curve crosses the x-axis.
x-intercept
II
The distance from the origin of the point where
a line or curve crosses the y-axis.
y-intercept
A value of x (or the independent variable)
that results in a y-value (or value of the dependent variable)
of zero.
zero of a function
I n d ex
Abscissa, 3 14, 4 1 7
Absolute value, 255, 383
Acidity, 349
Addition
cosine, 228-229
functions, 4 1 6-423
sine, 230
Al-Khwarizmi, 104
Alchemy, 376
Alkalinity, 349
Alternating current, 425
Amplitude
cosine, 253-257
sine, 253-257
tangent, 253-257
Angles and trigonometric ratios, 250
Angular velocity, 207, 2 1 0
Aperture, 392
Arc, 204, 206
Arcsin, 2 1 9
Asymptote, 146
exponential functions, 3 1 4
graphing calculator, 1 5 6
inverse function, 3 1 6
rational functions, 1 6 8
reciprocal functions, 149-150,
1 5 8-160
Average rate of change, 1 , 53-6 1 ,
76-77
equations, from, 60-6 1
exponential functions, 3 1 0-3 1 3
graphs, from, 56-57
secant, 54-55
sinusoidal functions, 290-295
slope of secant, 54-55
table of values, from, 58-59
Axis of symmetry and transformation,
47
Bacterial culture, 22
Base, 7, 325
changing, for logarithms, 344-346
changing to solve an equation,
366-367
powers, 371-372
powers, change of, 365
Binary codes, 348
Binomial
polynomial functions, 84-85, 86
product, 201
Boyle's law, 1 8 9
Bracket notation, 7
See also Interval notation; Intervals
Break-even point, 422
Career connections
actuary, 449
atmospheric scientist, 2 8 9
computer engineering, 227
electrical engineer, 369
environmental engineer, 93
mathematical modelling, 1 76
meteorologist, 289
mining engineering technology, 29
nuclear medicine technologist, 322
water resources management, 93
Cartesian coordinate system, 2 1 0
CAST rule, 200, 2 1 3 , 283
Catenary, 475
Central angle, 202, 204
Change of base formula, 344
Chromatic scale, 463
Coefficient of determination, 395
Coefficients, 4
leading, 4
Cofunction identities, 22 1 , 238
Combined functions
inequalities, 450-456, 473
modelling with, 461-468, 473
Common logarithms, 325
Commutative law, 426, 438
Complex number, 1 12
Composite functions, 439-446, 472
evaluating, 443
Compound angle expression, 228
Compound angle formulas,
230-2 3 1 , 237, 238, 244
Compound interest, 344, 34 7-348
Compound interest formula, 400
Compression, 309
logarithmic functions, 333-336
sine, 255
Computer algebra system (CAS)
change degree measure to radian
measure, 205
divide a polynomial by a binomial,
95
evaluate trigonometric ratios, 2 1 2
find a remainder, 98
find zeros of a function, 1 78
solve polynomial inequalities, 136
solve quadratic trigonometric
equations, 284
Congruent, 222
Constant function, 5
Constant term, 4
Constructive interference, 426
Consumer price index (CPI), 72
Continuity, 41
Correlated angle identities, 22 1
Cosecant, 200, 249
graphs, 261-266, 300
Cosine, 200, 249
addition, 228-229
amplitude, 253-257
domain, 253-257
graphs, 252-257, 261-266, 300
maximum, 253-257
minimum, 253-257
period, 253-257
range, 253-257
subtraction, 229
transformation, 271-272
y-intercept, 253-257
zeros, 253-257
Cost function, 72, 84, 1 90, 420-422,
425, 459
Cotangent, 200, 249
graphs, 26 1-266, 300
Cotes, Roger, 206
Counterexample, 236
Crests, 462
Critical values of rational inequalities,
181
Cubic functions, 5
family, 1 1 5-1 1 6
graph, 1 6
number o f factors, 96
zeros, 1 15-1 1 6
Curves o f best fit, 4 70
Decibels, 350, 3 5 1-352, 3 92
Degree
functions, 4
polynomial functions, 1 9-20, 30-38
power functions, 5
Degree measure, 202-207
del Ferro, Scipione, 1 04
Demand function, 1
Destructive interference, 426
Differences
functions, 4 1 6-423, 472
logarithms, 3 8 2
Direct current, 425
Discontinuity, 1 8 8
Discriminant, 1 12, 1 37
Displacement, 459
Distance formula, 20 1
Diving time, 1 8 7
Division, 8 2
polynomial functions, 84-85, 8 6
Domain, 2 , 146
cosine, 253-257
exponential functions, 3 1 4
I ndex
•
MHR
595
inverse function, 3 1 6
power functions, 8
reciprocal functions ' 149-150,
Expressions
factoring, 82
simplifying, 82
Extraneous roots, 374
logarithmic equations, 3 8 9
1 5 8- 1 60
sine, 253-257
tangent, 253-257
transformation, 4 7
Double angle formula, 238
Earthquakes, 352, 354
Electrical resistance, 1 90
End behaviour
polynomial functions, 1 5-17, 1 9-20
power functions 6 8 9
quartic function; , l 6 1 7
Equations, 2
average rate of change, 60-61
exponential functions, 3 1 3
finite differences, 1 7- 1 8
instantaneous rate of change ' 6 9
line, 2
polynomial functions ' 1 7- 1 8 ,
f(x)
=
a sin[k (x - d)] + c' 270-274,
94-1 0 1
polynomial inequalities, 1 32-138,
141
trigonometric equations, 2 8 5
Family of cubic functions, 1 1 5-1 1 6
Family o f functions, 1 14
Family of polynomial functions, 1 14
Fathom™, 3 96-399, 46 1-462
Ferris wheel, 292-293
Ferris, George Washington Gale' Jr. '
transformations, 221-223
Escape velocity, 1 8 9
Even degree, 1 1 , 24
Even-degree polynomial functions 36
Even function and symmetry, 36 : 414
Exponent, 7
laws, application of, 362
Exponent laws, 308, 362
Exponential equations
equivalent forms, 364-367
modelling using, 3 93-403
solution strategies, 3 6 1
techniques for solving, 370-374
Exponential functions, 7, 307,
292
Finite differences, 2, 20
equation, 1 7- 1 8
polynomial functions, 17-18, 20-2 1
. differences, 1 8
FirSt
exponential functions, 3 12-3 1 3
Force, 1 5 5
Frequenc 258, 463
Function notation' 2
Functions
addition, 4 1 6-423
circle, 4
degree, 4
differences, 4 1 6-423, 472
evaluating, 82
product, 429-434, 472
quotient, 429-434, 4 72
sinusoidal, 4 1 6
subtraction, 4 1 6-423
sum, 4 1 6-423, 472
superposition principle, 4 1 7-423
3 1 0-3 1 7, 356
asymptote, 314
average rate of change, 3 1 0-3 1 3
domain, 3 1 4
equations, 3 1 3
first differences, 3 1 3
graphs, 3 0 8
growth models, 4 1 4
instantaneous rate of change,
3 1 0-3 1 3
inverse, 3 1 4
range, 3 1 4
rate o f change, 3 1 0-3 1 3
x-intercept, 3 1 4
y -intercept, 3 1 4
Exponential growth, modelling,
Gap, 1 8 8
Geometer's Sketchpad® The
9
l
203-204, 242, 290-2 1 , 3 4-3 1 5 ,
3 3 1 , 401-402, 439, 465-467
Geostationary orbit, 209, 2 1 0
Gradian measure, 209
Gravity, 72, 321
364-365
Exponents, unknown, 341
Advanced Functions
=
(ax + b)
1 68-173, 1 92-1 93
(ex + d) '
a cos[k (x - d)] + c, 270-274,
f-stop, 38 7, 392
Factor theorem, 94-1 0 1 , 140
factoring by grouping, 99
polynomial equations, 105, 106
Factoring
expressions, 82
grouping, by, 99
intervals, 30-3 1
logarithmic equations, 3 8 9
polynomial equations, 1 0 5 , 146
polynomial functions' 30-32,
220-224
•
f(x)
300
30-38, 75-76
MHR
=
300
power functions, 8
quadratic functions, 82
quartic functions, 1 1 7
rational functions, 1 9 3
transformation 47 48
Equivalent trigon met;ic expressions'
596
f(x)
•
I ndex
Half-life, 370
Harmonic mean, 1 84
Heil, Jennifer, 467
Hertz (Hz), 258, 463
Hole, 1 8 8
Horizontal asymptote, 1 46, 2 5 1
rational inequalities, 1 79-1 80
Identity, 221
If and only if, 95
Illumination, 1 77
intensity, 1 8 5
Inequalities, 8 1
combined functions, 450-456, 473
graphs, 1 23-124
graphs, combined, 453
illustrating, combined, 4 5 1
rational equations, 1 77-1 83
revenue functions, 123, 128
solving, 147
technology, 123-129
x-intercepts, 125
Infinity, 7
Instantaneous rate of change, 1 ,
65-70, 77, 3 0 1
equations, from, 6 9
exponential functions, 3 1 0-3 1 3
graphs, from, 67-68
sinusoidal functions, 290-295
table of values, from, 68-69
Integral, 97
Integral zero theorem, 97
Intensity of illumination, 1 77' 1 85 '
189
Intensity of sound, 1 86-1 8 7, 350,
35 1-352, 354, 392, 463
Interval notation, 7
table of bracket intervals' 8
Intervals
factoring, 30-3 1
graphs, 83
inverse function, 3 1 6
polynomial functions, 30-3 1
polynomial inequalities, 1 26-127
reciprocal functions, 1 5 8-160
Inventory control, 454-456
Inverse
exponential functions, 3 1 4
sine, 2 1 9
Inverse function, 3 1 0-3 1 7, 3 5 6
asymptote, 3 1 6
domain, 3 1 6
exponential, 3 1 0-3 1 7
graphs, 308
intervals, 3 1 6
range, 3 1 6
x-intercept, 3 1 6
y-intercept, 3 1 6
Inverse notation, 265-266
Inverse square law, 1 8 6
Investment and optimization, 400
See also Compound interest
Leading coefficient, 4
negative, 24
positive, 24
Levers, 1 4 8 , 1 5 5
Line. See Linear functions
Line symmetry
polynomial functions, 30
power functions, 6
Linear functions, 1 , 5
equation, 2
graph, 1 6
growth models, 4 1 4
rate of change, 2 5 1
reciprocal, 148
Linear inequalities, 132
Local maximum
polynomial functions, 15-17, 1 9-20
quartic functions, 1 6- 1 7
Local minimum
polynomial functions, 15-17, 1 9-20
quartic functions, 1 6- 1 7
Logarithmic equations
extraneous roots, 3 8 9
factoring, 3 8 9
modelling using, 393-403
solution strategies, 3 6 1
techniques for solving, 3 8 7-390
Logarithmic functions, 307, 323-327
compressions, 333-336
evaluating, 324-325
exponents, 324, 325
reflections, 333-336
stretches, 3 3 3-3 3 6
transformations, 331-3 3 7
translations, 332-336
Logarithmic scales, 349
Logarithms, 323-327, 356
approximation, 326-327
differences, 382
graphs, 330
product, 378-384
quorien , 378-384
sum, 382
Logic, 95
Logistic curve/function, 407
Long division, 82, 84-90, 98
dimension problems, 86-87
remainder, 87-88
Marketing, 432-434
Maximum
absolute, 1 5
cosine, 253-257
global, 1 5
local, 1 5- 1 7, 1 9-20
polynomial functions, 15-17, 1 9-20
quartic functions, 1 6- 1 7
sine, 253-257
tangent, 253-257
Minimum
absolute, 15
cosine, 253-257
global, 1 5
local, 1 5-17, 1 9-20
polynomial functions, 15-17, 1 9-20
quartic functions, 1 6- 1 7
sine, 253-257
tangent, 253-257
Morphing, 242
n factorial,
n!, 20
20
Napier, John, 4 1 2
Nautical miles, 2 0 9 , 2 1 0
Negative infinity, 7
Oblique asymptote, 1 90
Octave, 463
Odd degree, 1 1 , 24
Odd-degree polynomial functions, 36
Odd function and symmetry, 37, 4 1 4
Operational amplifiers, 340, 3 8 5
Optimization o f investment, 400
Order of polynomial functions, 30-38
Ordinate, 259, 3 1 4, 4 1 7
Oughtred, William, 4 1 2
Outliers, 470
Parabola, 1 3
Parameters i n transformations, 42-43
Partial fractions, 1 8 5
Pendulum, 475
Period
cosine, 253-257
sine, 253-257
tangent, 253-257
Periodic functions, 1 3
p H scale, 349, 357
Photoperiod, 297
Point symmetry
polynomial functions, 30
power functions, 6
Polar coordinates, 2 1 0
Polar curves, 2 8 9
Polynomial equations, 8 1 , 104-1 09,
140
factor theorem, 105, 1 06
factoring, 1 05, 146
rational zero theorem, 106
roots, 1 04-1 05, 107-109
x-intercepts, 104-105
zeros, 1 04-1 05
Polynomial functions, 1 , 4, 1 40
application, 22-23
binomial, 84-85, 86
characteristics of, 1 5-24, 74-75
degree, 1 9-20, 30-38
division, 84-85, 86
end behaviour, 15-17, 1 9-20
equation, 1 7- 1 8 , 30-3 8, 75-76
evaluating, 82
factoring, 30-32, 94-1 0 1
families of, 1 1 3-1 1 8, 140-14 1
finite differences, 1 7-18, 20-2 1
form of, 4
graphs, 15-17, 1 9-20, 75-76
graphs, analysing, 33, 34-35
intervals, 30-3 1
line symmetry, 30
local maximum, 15-1 7, 1 9-20
local minimum, 1 5- 1 7, 1 9-20
maximum, 1 5- 1 7, 1 9-20
minimum, 1 5- 1 7, 1 9-20
order, 30-38
point symmetry, 30
rate of change, 251
recognizing, 7
sign, 32
symmetry, 30-3 8, 36-37
x-intercept, 1 5- 1 7, 1 9-20, 30-38
y-intercept, 30-38
zeros, 32, 1 14
Polynomial inequalities, 8 1
factoring, 1 32-1 38, 1 4 1
graphs, 126
intervals, 126-127
Population, 73, 295
scatter plot, 3 94, 406
Power chords, 469
Power functions, 4-1 1 , 74
degree, 5
domain, 8
end behaviour, 6, 8, 9
equations, 8
graphs, 5, 6, 4 1 4
range, 8
symmetry, 8
line, 6
point, 6
volume, 9
Power law of logarithms, 341-346,
356
applications, 363
Powers and base, 371-372
Predator and prey, 295, 439
Primary trigonometric ratios, 200
Probability, 3 76, 434
Product
binomials, 201
functions, 429-434, 472
logarithms, 378-384
symmetry, 429-430
Product law of logarithms, 378-384,
408
Profit function, 1, 73, 1 90, 420-422,
425, 459
Pythagorean identity, 20 1 , 237
Quadratic equations, 146
solving, 362
Quadratic formula, 362, 372-373
Quadratic functions, 1 , 3, 5, 82
equations, 82
graph, 1 6
growth models, 4 1 4
reciprocal, 1 5 7- 1 63, 1 92
solving, 82
I ndex
•
MHR
597
Real numbers, 7
Reciprocal
linear function, 148
quadratic function, 1 57-163
trigonometric equations, 2 8 5
trigonometric expression, 223
Reciprocal functions, 1 46, 1 57-163,
Quartic functions, 5
end behaviour, 1 6- 1 7
equation, 1 1 7
family, 1 1 7
graphs, 1 6
local maximum, 1 6- 1 7
local minimum, 16-1 7
maximum, 1 6-1 7
minimum, 16-17
x-intercept, 16- 1 7
zeros, 1 1 7
Quintic functions, 5
graph, 1 6
Quotient
functions, 429-434, 472
logarithms, 378-384
symmetry, 429-430
Quotient identity, 201 , 234, 237
Quotient law of logarithms,
1 92
asymptote, 1 49-1 50, 1 5 8-160
domain, 1 49-150, 1 58-160
graphs, 148-149, 1 57-1 5 8
intercepts, 1 5 1
intervals, 1 58-160
key features, 1 62-1 63
range, 1 49-1 50, 1 5 8-160
rate of change, 1 5 1-152, 1 6 0-162
x-intercept, 1 5 1
y-intercept, 1 5 1
Reciprocal identities, 237
Reciprocal notation, 265-266
Reciprocal trigonometric functions
and graphs, 26 1-266, 300
Reciprocal trigonometric ratios, 200,
378-3 84, 408
Rad, 205
Radian, 199, 202, 204, 205, 206
Radian measure, 199, 202-207, 244
trigonometric ratios, 250
Radical expression, 363
Radioactive decay, 370
Range, 2, 146
cosine, 253-257
exponential functions, 3 1 4
inverse function, 3 1 6
power functions, 8
reciprocal functions, 149-150,
239
Rectangular prism, volume, 1 3 7
Reflection, 309
Reflections and logarithmic functions,
333-336
Regression analysis, 407
Remainder in long division, 87-88
Remainder theorem, 84-90, 140
application, 8 9
unknown coefficients, 90
verification, 89
Resistance, 1 90
Revenue function, 1 , 73, 420-422,
1 5 8-160
sine, 253-257
tangent, 253-257
transformation, 4 7
Rate of change
exponential functions, 3 1 0-3 1 3
linear functions, 2 5 1
polynomial functions, 25 1
reciprocal functions, 1 5 1- 1 52,
425, 444-445, 459
inequalities, 123, 128
Richter scale, 352, 354, 3 5 7
Roots of polynomial equations,
104- 1 05, 1 0 7-109
Rotation, 222
Rutherford, Ernest, 370
1 60-1 6 2
Rational equations and inequalities,
Scatter plot, 330
population, 394, 406
Secant, 53-6 1 , 200, 249
average rate of change, 54-55
graphs, 26 1-266, 300
slope, 53-61
tangent, 65-66
Second differences, 1 7
Sector, 202, 203
Semi-logarithmic scatter plots, 330
Sign of polynomial functions, 32
Simplification
expressions, 82
radical expression, 363
sin x2 + cosx2 = 1 , 201
Sine, 200, 249
addition, 230
amplitude, 253-257
compression, 255
1 77-1 83
Rational functions, 145, 148,
1 68-1 73, 1 92
asymptote, 1 68
comparing, 1 71-1 73
discontinuity, 1 8 8
equations, 1 86-1 89, 1 93
graphs, 4 1 4
key features, 1 6 9-1 70
Rational inequalities, 1 93
critical values, 1 8 1
horizontal asymptote, 1 79-1 80
quadratic, 1 80-1 8 2
solving, 1 77-1 78, 1 79-1 80
vertical asymptote, 1 79-1 80
x-intercept, 1 79-1 80
y-intercept, 1 79-1 80
Rational zero theorem, 1 00
polynomial equations, 1 06
·
598
MHR
•
Advanced Fu nctions
•
I n d ex
domain, 253-257
graphs, 252-257, 2 6 1 -266, 300
inverse, 2 1 9
maximum, 253-257
minimum, 253-257
period, 253-257
range, 253-257
shift, 255-256
stretch, 25 5
subtraction, 230
transformation, 272-273
translation, 254
y-intercept, 253-257
zeros, 253-257
Sinusoidal functions, 270-275, 426
average rate of change, 290-295
instantaneous rate of change,
290-295
solving, 300
transformations, 250
Sinusoidal regression, 280-2 8 1
Slide rule, 4 12
Slope, 2, 1 4 6
secant, 53-6 1
tangent, 65-70
Slope of secant, 53-6 1 , 76-77
average rate of change, 54-55
slope of tangent, 65-66
Slope of tangent, 65-70, 77
slope of secant, 65-66
Sound, 350, 3 92, 406, 463
Special angles and trigonometric
ratios, 250
Sphere
surface area, 63
volume, 1 0, 63
Stellar magnitude scale, 354
Stretch
sine, 255
logarithmic functions, 333-336
Substitution method, 285
Subtends, 202, 204
Subtraction
cosine, 229
functions, 4 1 6-423
sine, 230
Sum
functions, 4 1 6-423, 472
logarithms, 382
Superposition principle and functions,
4 1 7-423
Surface area of sphere, 63
Sustain, 376
Symmetry
even-degree polynomial functions,
36
even function, 36, 4 1 4
identifying, 3 7
odd-degree polynomial functions, 36
odd functions, 37, 4 1 4
polynomial functions, 30-38
power functions, 8
product, 429-430
quotient, 429-430
transformation, 47
Synthetic division, 97, 98
Tangent, 200, 249
amplitude, 253-257
domain, 253-257
graphs, 252-257, 26 1-266, 300
maximum, 253-257
minimum, 253-257
period, 253-257
range, 253-257
secant, 65-66
slope, 65-70
y-intercept, 253-257
zeros, 253-257
Tartaglia, Niccolo, 1 04
Technology and inequalities, 1 23-129
Temperature, 293-294
Thomson, James, 206
Trajectory, 286
Transformation, 3, 42, 76, 309
See also Compression; Reflection;
Stretch; Translation
axis of symmetry, 4 7
cosine, 271-272
domain, 47
equations, 4 7, 48
equivalent trigonometric identities,
22 1-223
graphs of, 46
logarithmic functions, 3 3 1 -33 7
parameters, 42-43
range, 47
sine, 272-273
sinusoidal functions, 250
symmetry, 4 7
vertex, 47
Translation
logarithmic functions, 332-336
sine, 254
Trigonometric equations
factoring, 2 8 5
quadratic, 283-284
reciprocal, 285
solving, 282-287
Trigonometric expression, reciprocal,
223
Trigonometric functions, 7, 249
Trigonometric identities, 22 1
proving, 236-240
Trigonometric ratios, 200, 244
angles, 250
radian measure, 250
special angles, 2 1 1 -2 1 5 , 250
Trigonometry, 1 99
Troughs, 462
Unit circle, 2 1 3
Unknown coefficients and remainder
theorem, 90
Variable, 4
Velocity, 1 4 8
angular, 2 0 7
Vertex in transformation, 4 7
Vertical asymptote, 1 46, 2 5 1
rational inequalities, 1 79-1 80
Voltage, 258, 289, 425
Volume
power functions, 9
rectangular prism, 1 3 7
sphere, 1 0, 63
X- intercept
exponential functions, 3 1 4
inequalities, 125
inverse function, 3 1 6
polynomial equations, 1 04-1 0 5
polynomial functions, 1 5 - 1 7, 1 920, 30-3 8
quartic functions, 1 6- 1 7
rational inequalities, 1 79-1 80
reciprocal functions, 1 5 1
y = xi, 1 3
y-intercept, 2
cosine, 253-257
exponential functions, 3 1 4
inverse function, 3 1 6
polynomial functions, 30-38
rational inequalities, 1 79-1 8 0
reciprocal functions, 1 5 1
sine, 253-257
tangent, 253-257
Zeros
cosine, 253-257
cubic functions, 1 1 5-1 1 6
polynomial equations, 1 04-1 05
polynomial functions, 32, 1 14
quartic functions, 1 1 7
sine, 253-257
tangent, 253-257
Without bound, 7
I ndex
•
MHR
599
C re d i t s
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•
Advanced Functions
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Credits
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IVY IMAGES; p393 ]. Speijer; p4 1 2 David AyersliStock;
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IVY IMAGES; p429 David RoseliStock; p43 9 Sharon
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Michelizza/iStock; p465, p470 B. Lowry/IVY IMAGES;
p478 David Tanaka
y
Circle
G
sin e =
cos e =
C = 2nr
tan e =
0
Cylinder
r
r
2'.
X
X
Sine Law
c
V = nr2h
SA = 2nrh
B
+ 2 nr2
sin A
=
sin B
_
a_ =
b_
_
sin B
a
A
c
sin A
b
=
sin C
=
c_
_
sin C
c
Cosine Law
Cone
a 2 = b2
+
c2 - 2bc cos A
b2 = a2
+
c2 - 2ac cos B
c2 = a2
+
bc2 - 2ab cos C
Fundamental Identities
sin2 e
Sphere
sec e
cos2 e = 1
tan e =
1
cos e
cot e =
+
=
-
sin e
cos e
1-
-
tan e
Compound Angle Formulas
sin (A ± B ) = sin A cos B ± cos A sin B
4
V = -nr3
3
cos (A ± B) = cos A cos B + sin A sin B
SA = 4nr 2
tan (A ± B ) =
TRIGO N O M ETRY
Double-Angle Form ulas
Angle Measure
sin 2A
1°
cos 2A
=
1 rad
1 80
=
ra d
adjacent
=
2 sin A cos A
cos2 A - sin2 A
1
= 1 - 2 sin2 A
1t
:zd
=
= 2 cos 2 A -
1 8 0o
tan 2A =
Primary Trigonometric Ratios
h yp
tan A ± tan B
+ tan A tan B
1
opposite
sin e = -=--=--­
hypotenuse
oppo.i«
adj acent
cos e = ..,---'---­
hypotenuse
tan e =
opposite
--:­
adj acent
2 ta n A
1 - tan 2 A
csc e = 1sin e
ALG E B RA
ANALYTI C GEOM ETRY
Factoring Special Polynomials
Distance Between Two Points
x2 ± 2xy
+ y2 = (x
± y) 2
+ y) 2
x2 - y2
=
(x - y)(x
x3 ± YJ
=
(x ± y)(x2 + xy
The distance between two points P , (xp y1) and P (x , y )
2 2 2
is P , P
+ y2 )
0, then x
=
-b
b
(x ") (x )
=
-
-
_
Power of a
Product
(xy) "
=
x "y "
Logarithms
y
=
logo
X
is the y-intercept
Power
Quotient
x" + b
l
I
y(x2 - x f + (y , - Y/ ·
For a line through the points P , (xp y1 ) and P (x , y ) ,
2 2 2
Y - Y,
2
Slope: m = x
xI
2
Slope y-intercept form of equation: y = mx + b, where
Vb2 - 4ac
---'="
2a
±
Rules for Exponents
Product
=
Linear Function
Quadratic Formula
I(ax2 + bx + c =
2
0
X
-
xb
- X
a -
b
Rational
ExJ>onent
1
x"
= fiX
aY = x
I
I
b
(x ")
=
x "b
Negative
ExJ>onent
X
-a
=
1
x"
Point - slope form of equation: y - y 1
Quadratic Function
Equation for a parabola with vertex
y
= a (x - p ) 2 + q
=
m(x - x1)
(p, q):
Circle
Equation for a circle centre
(x -
h f + (y - k ) 2
(h, k) and radius
= r2
r:
log 1 0x is usually written as log x.
loga
a= 1
M EASU REMENT
ax = X
log' x
=X
a
Joga
loga (xy)
=
loga X
0
loga b
+ loga y
log x
log
X=
loga
( ) = loga X - loga y
b
loga x"
=
P represents perimeter, C the
A the area, V the volume, and SA
In the following,
circumference,
the surface area.
-
n
Triangle
loga x
P=a+ b+c
A = .!_ bh
2
Tra pezoid
a
h
A = .!.. (a + b)h
2
b