Practice
Period:
Date of Exam:
Start Time:
End Time:
Teacher:
Course:
Number of Pages:
?
January ?rd, 2008
8:30 am
10:30 am
Mr. A. Cecchini
MHF 4U1 – ?
? (including cover)
Practice
Name: _____________________________________________
Marks per Category
Exam Mark Breakdown
Part A
Part B
Part C
Part D
Communication
Communication
Knowledge
Application
Thinking
20%
30%
30%
20%
/?
Practice
Knowledge
/?
Application
/?
Thinking
Totals by Category
Instructions: 1.
2.
3.
4.
5.
/?
/?
/?
Put your full name on the line above.
Count the number of pages to ensure all are present.
Show full solutions where asked.
Non-graphing calculators may be used – no sharing will be allowed.
Check over answers when finished.
/?
/?
MHF 4U1 – Practice Exam
Page 2 of 12
Part A: Communication (? marks)
Instructions:
Explain and justify all answers in space provided (marks as indicated)
Practice
1. Using the function f(x) = x2, graphically explain the difference between instantaneous and
average velocity.
6 − 5(3) x +1
2. Simplify the function f ( x) =
and then state the domain and range
3
3. Simplify the function y = 4(2)x
π⎞
⎛
4. Write two equivalent expression to y = sin ⎜ x − ⎟
4⎠
⎝
5. Using graphs shown, determine the value of f(g(1)).
f(x)
g(x)
Practice
6. Does y = x4 ever have a negative slope. Explain.
7. Determine the inverse of y = log5 x
8. Complete the following chart
#
A
Equation
y=
y = 3 log 5 4 x − 1
C
y = −2(3) 2−x + 5
#
Equation
B
C
Vertical Shift
Horizontal
Stretch
Vertical
Stretch
3
+3
4 − 2x
B
A
Horizontal
Shift
Practice
y = 2 sin 4 x + 1
⎛1
⎞
y = 3 cos⎜ (θ − π )⎟ − 2
⎝2
⎠
⎡ ⎛
π ⎞⎤
2
y = tan ⎢3⎜θ + ⎟⎥ + 1.5
3
4 ⎠⎦
⎣ ⎝
Phase
Shift
Displacement
Period
Amplitude
MHF 4U1 – Practice Exam
Page 3 of 12
Part B: Knowledge, Understanding & Skills (? marks)
Practice
Instructions:
Answer questions #? – ? in space provided (1 mark each = ? marks)
There are no marks given for rough work on these questions.
9. Determine the exact equation of the following, in form f(x) = k(x - a)(x - b)(x - c)(x + d),
given the information below.
a)
f(x) = k(x – 1)(x + 2) and goes through point (-3, 6)
____________
b)
g(x) = k(x + 2)(x – 2)(x + 3) and g(1) = -24
____________
10. Determine the remainder when (x3 – 2x2 + 3x + 4) ÷ (x + 2)
⎛1⎞
11. Determine the exact value of 3 − 2⎜ ⎟
⎝8⎠
−1
____________
−2
3
____________
12. Is x - 3 a factor of x5 – 4x3 + x2 – 3
____________
Practice
13. Evaluate 3 − − 5 + 3 − 9
____________
14. List point(s) of discontinuity in the graph of y =g(x) below.
____________
15. Write, 4 = log 2 16 , in exponential form
____________
16. Evaluate h(3) given, h( x) = −3 x −2 log 3 ( x 2 + 18 )
____________
x(2 x 2 − 5 x)
, if it exists
2x − x3
Practice
17. Evaluate;
lim
x →∞
18. Given f ( x) = −3x − 2 , g ( x) = ( x + 1) 2 + 2 , m( x ) =
____________
1− x
determine;
2
a) f(x) + g(x)
____________
b) m(f(x))
____________
c) f(x) ⋅g(x)
____________
d) f(x)/m(x)
____________
e) g[f(2)]
____________
MHF 4U1 – Practice Exam
Instructions:
Page 4 of 12
Answer questions #? – ? in the space provided (marks as indicated)
19. Use long division to express the given function f(x) = (x2 + 3x – 5)/(x – 1) as the sum of a
polynomial
x 3 −8
express f(a+1) in simplest form.
20. Given, f ( x) =
x−2
21. Complete the following chart
#
A
B
C
D
Practice
Equation
Vertical
Asymptotes
Horizontal
Oblique
Intercepts
x-axis
y-axis
y = ( x − 2)( x + 1)( x − 3)
1
+3
x +1
x +1
f ( x) = 2
x −4
x2 − 9
g ( x) =
x+2
y=
Practice
22. Sketch the following functions using most appropriate technique.
2
1
b) g ( x) = (− x − 1) 2 + 2
a) h(x) = − x + 1
5
2
c) m(x) = (2x – 1)(x + 3)
d) y = (x – 3)2(x + 5)
Practice
MHF 4U1 – Practice Exam
Page 5 of 12
e) f(x) =- (x - 2)3(x + 3)
f) y =
1
( x − 2) 2 + 1
Practice
h) m(x) =3( ½ )x - 3
g) f(x) = log3(x+1) – 2
Practice
i) g(x) = sin 3x - 5
10
Practice
10
5
−π
j) y = 2 tan (x – π) + 2
5
π
2π
3π
4π
−π
−π/2
π /2
−5
−5
−10
−10
π
3π /2
2π
5π /2
3π
7π/2
4π
MHF 4U1 – Practice Exam
Page 6 of 12
Part C: Application (? marks)
Instructions:
Answer all question fully in space provided. Show all step (marks as indicated)
Practice
23. Create a function that has a vertical asymptote at x = 3 and x = -2, x-intercept at -1 and
horizontal asymptote at y = -5
24. If sin x = −
3π
2
< x < 2π determine the exact value of cos (2x).
,where
2
3
Practice
25. Given h(x) = 2x -1 and g(x) = x2 +1 sketch f(x) = g(x) + h(x)
Practice
26. Factor fully
a) x3 – 27
b) 6x3 – 17x2 + 11x – 2
MHF 4U1 – Practice Exam
27. Solve
Page 7 of 12
a) -x2 –5x - 6 < 0
b) x3 – 3x2 = 4x - 12
Practice
c)
2+ x 2
< x +1
3
−5
d) x3 – 9x2 < 24 – 26x
e) 4x4 – 2x3 – 16x2 = 8x
Practice
28. How long, to nearest tenth of a year, does it take money to double at a rate of 5.25% /a
compounded annually?
29. Functions r(x) = -x2 + 30x amd c(x) = 17x + 36 are the estimated revenue and cost functions
for the manufacture of a new product. Given profit is revenue minus cost;
Practice
a) Determine the average profit function, AP( x) =
P( x)
x
b) Express the average profit function in a different form.
c) What are the break even quantities?
MHF 4U1 – Practice Exam
Page 8 of 12
30. When ax3 + bx2 + 4x + 1 is divided by x – 1, the remainder is 12. When it is divided by x + 2,
the remainder is -20. Find the values of a and b.
Practice
31. Analyze and sketch the following
a) f ( x) =
2 x 2 − 5x − 7
x−3
b) g ( x) =
x2 − 9
x + 4x 2 − x − 4
3
Practice
Practice
MHF 4U1 – Practice Exam
Page 9 of 12
Part D: Thinking & Inquiry (? marks)
Practice
Instructions: Answer all questions in full in the space provided (marks as indicated)
Provide labeled diagrams where appropriate.
32. Write a cubic function h(x), that has a y-intercept of -6 and an x-intercept of -3, given that
h(1) = 0 and h(x) ≤ 0 when x > -3.
33. A sperical hailstone grows in a cloud. The hailstone maintains a spherical shape while its
radius increases at a rate of 0.5 mm/min.
a) express the radius, r in millimeters, of the hailstone, as a function of time, t in minutes.
b) express the volume, V, in cubic millimetres, of the hailstone in terms of r.
c) Determine v[r(t)]
Practice
d) What is the volume of the hailstone 1h after it begins to form.
34. Bob earns $19.45/h operating a fork lift at Home Depot. He recieves $0.64/h more for
working the evening shift, as well as $0.39/h more for working weekends.
a) Write a function that describes Bob’s pay.
b) What function shows his evening shift premium?
c) What function show his weekend premium?
d) What function represents his earnings for the night shift on Saturday?
e) How much does Bob earn working 11h on Satruday evening, if he earns time and half
on that days rate for any hours more than 8 hours work?
Practice
35. The table below documents the world population since 1750 in 50 year intervals. Enter the
data into your graphing calculator and answer the questions below;
Interval
Population (in billions)
0
0.79
1
1.19
2
1.78
3
2.68
4
4.03
5
6.06
a) Perform an exponential regression to determine equation that fits data. Give your
equation in the form P(y) = aby + c where y is the calendar year AD.
b) Using your equation from part a, predict the world’s population, to nearest tenth, in
2071.
MHF 4U1 – Practice Exam
Page 10 of 12
Formula Page – MHF 4U1
m = slope =
rise Δy y 2 − y1
=
=
run Δx x 2 − x1
⎛1⎞
a −n = ⎜ ⎟
⎝a⎠
n
1
an
or
a
x b = b xa
Practice
A f = Ao (base)
Growth/decay formula:
t
ti
where Af = end
Logarithms properties Basic: a) log b 1 = 0
b) log b b = 1
or
( x)
a
b
Ao = original
a) log a x r = r log a (x)
b) log a xw = log a x + log a w
⎛x⎞
c) log a ⎜ ⎟ = log a x − log a w
⎝ w⎠
Operational:
c) log b bx = x
d) b log b x = x
Formulii that make use of logarithms:
a
r
Radian/Degrees conversion formulas
Arc angle formula
θ=
⎛ I
M = log⎜⎜
⎝ Io
⎛ I
L = 10 log⎜⎜
⎝ Io
⎞
⎟⎟
⎠
⎞
⎟⎟
⎠
where θ = angle in radians
360° = 2π rad
[ ]
pH = − log H +
a = arc length
180° = π rad
or
Practice
xº
Exact trigonometric
equations for
common angles
0º
x
30º
45º
60º
90º
π
π
π
π
6
4
1
3
2
3
2
1
2
1
2
0
1
3
-
0
Sin x
0
Cos x
1
Tan x
0
π
1
2
2
1
3
2
1
3
− x) = cos x and cos(
π
− x) = sin x
Co-function relationship
sin(
Supplemental relationship
sin θ o = sin(π − θ o ) and cos θ o = − cos(π − θ o )
Positional relationship
sin(− x) = − sin x and cos(− x) = cos x
Compound formulas
2
2
cos( x + y ) = cos x cos y − sin x sin y
cos( x − y ) = cos x cos y + sin x sin y
sin( x + y ) = sin x cos y + cos x sin y
sin( x − y ) = sin x cos y − cos x sin y
Practice
Double angle formulas sin(2 x) = 2 sin x cos x
cos( 2 x) = cos 2 x − sin 2 x
2
= 2 cos x − 1
Quotient Identities
tan θ =
sin θ
cos θ
tan 2 θ =
Reciprocal identities
csc θ =
1
sin θ
sec θ =
Pythagorean identities
cos 2 x + sin 2 x = 1 or
tan(2 x) =
2 tan x
1 − tan 2 x
sin 2 θ
cos 2 θ
1
cos θ
cos 2 x = 1 − sin 2 x or
cot θ =
1
tan θ
sin 2 x = 1 − cos 2 x
MHF 4U1 – Practice Exam
Page 11 of 12
Answers
1. Secant gives the average slope between two points. Tangent gives instantaneous slope at one
point.
2. f(x) = 2 – 5(3)x or f(x) = -5(3)x + 2
Domain: x ∈ R
Range: y < 2
3. y = 2x+2
4. As function is periodic answers may from; y = sin(x + 7π/4) or y = cos(x + π/4)
5. f(g(1)) = 4
6. Yes. When x<0 graph decrease and thus has a negative slope. Illustrate with sketch.
7. y=5x
8. See table below
Practice
#
A
B
C
#
A
B
C
Equation
Equation
V Shift
+3
-1
+5
Displacement
+1
-2
1.5
H Strecth
-½
¼
-1
Period
π/2
4π
2π/3
V Strecth
-3
+3
-2
Amplitude
2
+3
2/3
Practice
a) f(x) = 3/2(x-1)(x+2)
r(x) = -18
-7 2/3 or -7.67
no
4
x = -4, 1
24 = 16
-1/2
-2
a) x2-x-2 b) (3x+3)/2
1
19. f ( x) = ( x + 4) −
x −1
20. f(x) = a2 + 4a + 7
21. See table below
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
#
A
B
C
D
H Shift
+2
n/a
+2
Phase Shift
n/a
+π
-π/4
Equation
b) g(x) = 2(x+2)(x-2)(x+3)
c) -3x3 – 8x2 – 16x – 8
Vertical
n/a
-1
x = -2, +2
X = -2
Horizontal
n/a
y =3
y=0
n/a
d) (-6x-4)/(x-1)
Oblique
n/a
n/a
n/a
y= x - 2
e) 51
x-axis
2, -1, 3
-4/3
-1
3, - 3
Practice
22. Transformations technique is best used for #a,b,g,h,i, zeros technique for #c,d,e and
reciprocal technique for #f. Check sketches using graphing calculator.
23. f ( x) =
− 5( x + 1) 2
( x − 3)( x + 2)
y-axis
+6
4
-1/4
-4.5
MHF 4U1 – Practice Exam
24.
25.
26.
27.
28.
Page 12 of 12
cos2x – sin2x = 1/9
f(x) = x (x + 2)
a) (x - 3)(x2 + 3x + 9)
b) (x - 2)(3x - 1)(2x - 1)
a) x < - 3 or x > -2 b) x = -2, 2, 3 c) x > - 3
d) x < 2 or 3 < x < 4
t = 13.5 seconds
Practice
29. a) AP( x) =
− x 2 + 13 x − 36
( x − 4)( x − 9)
b) AP( x) =
x
x
or
e) x = 0 x = ½
AP( x) = (− x + 14) −
50
c) 4, 9
x
30. a = 4, b = 3
31. Table below gives some characteristics. Check actual sketches on graphing calculator.
#
A
B
Equation
Vertical
x=3
x = -1, 1, 4
Horizontal
n/a
y =3
Oblique
y = 2x + 1
n/a
x-axis
-1, 1/2
-3, +3
32. h(x) = -2(x + 3)(x – 1)2
4 3
πr
3
4
c) v(t ) = π (0.5t ) 3
3
Practice
b) v(r ) =
33. a) r(t) = 0.5t
d) V(60) = 113097 mm3 or 113 cm3
34. a) p(h) = r(h) + e(h) + w(h)
d) p(h) = 20.48h
35. a) P ( y ) = 0.79(1.503)
y −1750
50
b) e(h) = 0.64h
c) w(h) = 0.39h
e) p(h) = 20.48(11) + 20.48(11-8) = $286.72
b) P(2071) =10.81 billion
Practice
y-axis
7/3
9/4