4
Chapter Review
FREQUENTLY ASKED Questions
Q:
How can you identify an exponential function from
• its equation?
• its graph?
• a table of values?
A:
The exponential function has the form f (x) 5 b x, where the variable
is an exponent.
Study
The shape of its graph depends upon the parameter b.
• See Lesson 4.5.
• Try Chapter Review
y
10
–10
y = 2x
()
x
y = 1 10
2
Questions 9 and 10.
y
x
0
10
0
–10
If b . 1, then the curve increases as
x increases.
Aid
x
10
If 0 , b , 1, then the curve
decreases as x increases.
In each case, the function has the x-axis (the line y 5 0) as its horizontal
asymptote.
A differences table for an exponential function shows that the differences
are never constant, as they are for linear and quadratic functions. They are
related by a multiplication pattern.
x
NEL
y 5 3x
0
1
1
3
2
9
3
27
4
81
5
243
6
729
First
Differences
Second
Differences
2
6
18
54
162
486
4
12
36
108
324
Exponential Functions
265
Study
Aid
• See Lesson 4.6, Examples 1,
2, and 3.
• Try Chapter Review
Questions 11 and 12.
Q:
How can transformations help in drawing the graphs of
exponential functions?
A:
Functions of the form g (x) 5 af (k(x 2 d )) 1 c can be graphed by
applying the appropriate transformations to the key points and asymptotes
of the parent function f (x) 5 b x, following an appropriate order—often,
stretches and compressions, then reflections, and finally translations.
In functions of the form g (x) 5 ab k(x2d ) 1 c, the constants a, k, d, and c
change the location or shape of the graph of f (x). The shape of the graph
of g(x) depends on the value of the base of the function, f (x) 5 b x.
• a represents the vertical stretch or compression factor. If a , 0, then the
function has also been reflected in the x-axis.
• k represents the horizontal stretch or compression factor. If k , 0, then
the function has also been reflected in the y-axis.
• c represents the number of units of vertical translation up or down.
• d represents the number of units of horizontal translation right or left.
Study
Aid
• See Lesson 4.7, Examples 1,
2, 3 and 4.
• Try Chapter Review
Questions 13 to 17.
Q:
How can exponential functions model growth and decay? How
can you use them to solve problems?
A:
Exponential functions can be used to model phenomena exhibiting repeated
multiplication of the same factor.
Each formula is modelled after the exponential function
y 5 ab x
The amount
in the future
The present The growth/decay The number
factor: if growth, of periods of
value or
then b . 1; if
initial
growth or
decay, then
amount
decay
0,b,1
When solving problems, list these four elements of the equation and fill
in the data as you read the problem. This will help you organize the
information and create the equation you require to solve the problem.
Here are some examples:
Growth
Decay
Cell division (doubling bacteria,
t
1 t
Radioactivity or half-life: N(t) 5 100a b H
2
yeast cells, etc.): P(t) 5 P0 (2)D
266
Chapter 4
Population growth: P(n) 5 P0 (1 1 r) n
Depreciation of assets: V(n) 5 V0 (1 2 r) n
Growth in money: A(n) 5 P(1 1 i) n
Light intensity in water: V(n) 5 100(1 2 r) n
NEL
Chapter Review
PRACTICE Questions
Lesson 4.2
Lesson 4.4
22
If x . 1, which is greater, x or x ? Why?
b) Are there values of x that make the statement
x22 . x 2 true? Explain.
1. a)
2
2. Write each as a single power. Then evaluate. Express
answers in rational form.
a) (27) 3 (27) 24
(22) 8
(22) 3
(5) 23 (5) 6
53
b)
c)
4210 (423 ) 6
(424 ) 8
1 7
e) (11) 9 a b
11
(23) 7 (23) 4 23
b
f) a
(234 ) 3
d)
Lesson 4.3
answers in rational form.
a) (5x) 2 (2x) 3; x 5 22
8m25
b)
;m54
(2m) 23
2w(3w22 )
c)
; w 5 23
(2w) 2
(9y) 2
d)
; y 5 22
(3y21 ) 3
e) (6(x24 ) 3 ) 21; x 5 22
(22 x22 ) 3 (6x) 2
1
f)
;x5
21 3
2(23x )
2
8. Simplify. Write each expression using only positive
3. Express each radical in exponential form and each
power in radical form.
3
a) !x 7
b) y
7. Evaluate each expression for the given values. Express
8
5
c)
exponents. All variables are positive.
3
a) !27x 3y 9
( !p) 11
d) m1.25
b)
4. Evaluate. Express answers in rational form.
2 23
5
16 20.5
b) a
b
225
(81) 20.25
c)
3
!
2125
a) a b
e)
5
6
(!
232) ( !
64) 5
a) a ( a
23
b0.8
b) 20.2
b
2
)
f ) " ( (22) )
3 2
25
d) d d
(d
e)
11
2
23 2
)
7
22 2 22
( (e ) )
5
c)
c ac 6 b
c
2
7
2
mn
3
2
2
11
(x4 ) 2 2
( (2x0.5 ) 3 ) 21.2
f)
"x6 ( y 3 ) 22
(x 3y) 22
Lesson 4.5
6
5. Simplify. Write with only positive exponents.
3
2
c)
m 2n 22
4 216 6 26
"
x (x )
e)
3
3
d) ( !227) 4
a6b5
Å a8b3
d)
f)
( (f
1 6
26 5 21
) )
9. Identify the type of function (linear, quadratic, or
exponential) for each table of values.
a)
b)
x
y
25
238
0
245
0
23
2
215
5
42
4
15
10
97
6
45
15
162
8
75
20
237
10
105
x
y
6. Explain why !a 1 b 2 !a 1 !b, for a . 0 and
b . 0.
NEL
Exponential Functions
267
c)
d)
x
e)
y
x
y
1
13
22
2000
2
43
21
1000
3
163
0
500
4
643
1
250
5
2 563
2
125
6
10 243
3
x
f)
y
x
y
40
0.2
210.8
21
20
0.4
29.6
0
10
0.6
27.2
1
5
0.8
22.4
2
2.5
1
3
1.25
1.2
4
⫺8 ⫺4 0
⫺4
x
4
8
⫺8
Lesson 4.6
11. For each exponential function, state the base
function, y 5 b x. Then state the transformations that
map the base function onto the given function. Use
transformations to sketch each graph.
x
a)
26.4
b)
c)
d)
1 2
y5a b 23
2
1
y 5 (2) 2x 1 1
4
y 5 22(3) 2x14
21
y5
(5) 3x29 1 10
10
12. The exponential function shown has been reflected
x
⫺8 ⫺4 0
⫺4
4
7.2
10. Identify each type of function (linear, quadratic, or
4
8
62.5
22
exponential) from its graph.
y
a)
8
y
c)
8
in the y-axis and translated vertically. State its
y-intercept, its asymptote, and a possible equation
for it.
10
⫺8
y
8
y
b)
6
4
8
4
⫺8 ⫺4 0
⫺4
x
4
8
2
⫺8 ⫺6 ⫺4 ⫺2 0
x
2
4
6
8
⫺8
268
Chapter 4
NEL
Chapter Review
Lesson 4.7
15. The value of a car after it is purchased depreciates
according to the formula
13. Complete the table.
V(n) 5 28 000(0.875) n
Exponential Initial
Growth
Growth or
Value (y- or Decay
Decay?
intercept) Rate
Function
a)
V(t) 5 100(1.08)t
b)
P(n) 5 32(0.95) n
c)
A(x) 5 5(3) x
5
8
where V(n) is the car’s value in the nth year since it
was purchased.
n
d) Q(n) 5 600a b
14. A hot cup of coffee cools according to the equation
t
1 30
T(t) 5 69a b 1 21
2
where T is the temperature in degrees Celsius and t is
the time in minutes.
a)
b)
c)
d)
e)
f)
NEL
Which part of the equation indicates that this is
an example of exponential decay?
What was the initial temperature of the coffee?
Use your knowledge of transformations to
sketch the graph of this function.
Determine the temperature of the coffee, to the
nearest degree, after 48 min.
Explain how the equation would change if the
coffee cooled faster.
Explain how the graph would change if the
coffee cooled faster.
a)
b)
c)
d)
e)
f)
What is the purchase price of the car?
What is the annual rate of depreciation?
What is the car’s value at the end of 3 years?
What is its value at the end of 30 months?
How much value does the car lose in its first year?
How much value does it lose in its fifth year?
16. Write the equation that models each situation. In each
case, describe each part of your equation.
a) the percent of a pond covered by water
lilies if they cover one-third of a pond now and
each week they increase their coverage by 10%
b) the amount remaining of the radioactive isotope
U238 if it has a half-life of 4.5 3 109 years
c) the intensity of light if each gel used to change
the colour of a spotlight reduces the intensity of
the light by 4%
17. The population of a city is growing at an average rate
of 3% per year. In 1990, the population was 45 000.
a) Write an equation that models the growth of the
city. Explain what each part of the equation
represents.
b) Use your equation to determine the population
of the city in 2007.
c) Determine the year during which the population
will have doubled.
d) Suppose the population took only 10 years to
double. What growth rate would be required for
this to have happened?
Exponential Functions
269
8.
a)
b)
c)
d)
9. a)
10. a)
11.
12.
13.
14.
15.
15361
During the 29th year the population will double.
17 years ago
{n [ R}; {P [ R | P $ 0}
82 °C
b) 35 °C
c) after 25 min
C 5 100(0.99) w.
100 refers to the percent of the colour at the beginning.
99 refers to the fact that 1% of the colour is lost during
every wash.
w refers to the number of washes.
b) P 5 2500(1.005) t
2500 refers to the initial population.
1.005 refers to the fact that the population grows 0.5% every year.
t refers to the number of years after 1990.
c) P 5 P0 (2) t
2 refers to the fact that the population doubles in one day.
t refers to the number of days.
a) 100%
d) 226
b) P 5 80(2t )
e) 13.6 h
c) 5120
f ) {t [ R | t $ 0}; {P [ R | p $ 80}
a) V 5 5(1.06t )
b) $0.36
c) $0.91
a) I 5 100(0.91d )
b) 49.3%
a) P 5 100(0.01a ) b) 6 applications
approximately 2.7%
8. a) 3xy 3
c)
1
e) 2
1
2 2
m n
b
a
9. a) quadratic
b) linear
10. a) exponential
b)
x9
y
e) exponential
f ) exponential
c) exponential
d) x 9
f)
c) exponential
d) exponential
b) quadratic
1
11. a) y 5 x; horizontal stretch by a factor
2
of 2 and vertical translation
of 3 down
y
8
4
1
b) y 5 2x; vertical stretch of ,
4
reflection in the y-axis, and vertical
translation of 1 unit up
2. a)
b)
7
5
4
2
c) y 5 3x; reflection in the x-axis, vertical
stretch by a factor of 2, horizontal
compression by a factor of 2, and horizontal
translation of 2 left
5.
4
4
⫺8 ⫺4 0
⫺4
y
x
4
y = ⫺2(3)2x+4
d) y 5 5x; reflection in the x-axis,
y = ⫺ 1 (5)3x⫺9 + 10
1
10
vertical compression of ,
y
10
10
horizontal compression by a factor
x
of 3, horizontal translation of
3 units right, and vertical translation ⫺30⫺20 ⫺10 0
10 20
⫺10
of 10 units up
⫺20
4
d) "m 5
1
c) 2
15
e) 264
d) 81
f) 2
1
a) 1
c)
b) b
d) d
c
1
6
13
2
12. y-intercept 5 2
asymptote: y 5 1
equation: y 5 2x 1 1
13.
e) e14
f)
1
f
6.
2
⫺8
11
b) "y
125
4. a)
8
15
b)
4
x
y = 1 (2)⫺x + 1
4
⫺4
c) p 2
8
()
⫺4 ⫺2 0
⫺2
1
x 2 . x 22, if x . 1; If x . 1, then x 22 5 2 will be
x
less than one and x 2 will be greater than one.
1
x 22 . x 2, if 21 , x , 1 and x 2 0, then 2 will be greater
x
2
than one and x will be less than one.
1
(7) 21 5 2
c) 50 5 1
e) 112 5 121
7
(22) 5 5 232
d) 44 5 256
f ) (23) 3 5 227
3. a) x 3
8
x
y
Chapter Review, pp. 267–269
b)
4
y= 1 2 ⫺3
2
⫺8
t
a)
x
⫺8 ⫺4 0
⫺4
16. a) P 5 200(1.753 )
b) 200 refers to the initial count of yeast cells.
1.75 refers to the fact that the cells grow by 75% every 3 h.
t
refers to the fact that the cells grow every 3 h.
3
17. a) It could be a model of exponential growth.
b) y 5 4.25x may model the situation.
c) There are too few pieces of data to make a model and the
number of girls may not be the same every year.
18. a) exponential decay
b) 32.3%
1.
1
x 1.8
1
5
Let a 5 9 and b 5 16; then "9 1 16 5 "25 5 5 but
a)
Function
Growth/
Decay
y-int
V(t) 5 100(1.08)t
growth
100
8%
32
25%
5
200%
600
237.5%
b) P(n) 5 32(0.95)n
decay
Growth/
Decay Rate
"9 1 "16 5 7 , and 5 2 7 so "a 1 b 2 "a 1 "b
7.
a) 200x 5 5 26400
b)
650
64
m2
c)
3
21
5
2w 3
18
d) 3y 5 5 296
Answers
x 12
2048
5
6
3
16
32
5
f)
3x
3
e)
c)
A(x) 5 5(3)x
growth
d)
5 n
Q(n) 5 600 a b
8
decay
NEL
Temperature (°C)
14. a) The base of the exponent is less than 1.
b) 90 °C
c)
Temperature vs. Time
y
t
y = 90 (21 ) 30 + 21
50
⫺50 0
x
50 100 150 200
Time (min)
d) 44 °C
e) The 30 in the exponent would be a lesser number.
f ) There would be a horizontal compression of the graph; that is, the
graph would increase more quickly.
15. a) $28 000
c) $18 758
e) $3500
b) 12.5%
d) $20 053
f ) $2052
1
16. a) P 5 (1.1) n
3
1
1
refers to the fact that the pond is covered by lilies.
3
3
1.1 refers to the 10% increase in coverage each week.
n refers to the number of weeks.
t
1 4.5 3 10
b) A 5 A0 a b
2
A 0 refers to the initial amount of U238.
1
refers to the half-life of the isotope.
2
t refers to the number of years.
c) I 5 100(0.96) n
0.96 refers to the 4% decrease in intensity per gel.
n refers to the number of gels.
17. a) P 5 45 000(1.03) n c) during 2014
b) 74 378
d) 7.2%
9
a) I 5 100(0.964) n
b) 89.6%
c) As the number of gels increases the intensity decreases
exponentially.
5. a) P 5 2(1.04) n, where P is population in millions and n is the
number of years since 1990
b) 18 years after 1990 or in 2008
6. (d)
7. n 2 0; n must be odd because you cannot take even roots of
negative numbers.
4.
Chapter 5
Getting Started, p. 274
1.
2.
3.
4.
5.
6.
7.
8.
a) c 5 13 m
b) f 5 "57 m
5
12
5
a) sin A 5
, cos A 5
, tan A 5
13
13
12
8"57
8
"57
b) sin D 5
, cos D 5
, tan D 5
11
11
57
a) 67°
b) 43°
a) 0.515
b) 0.342
a) 71°
b) 45°
c) 48°
61 m
25.4 m
Answers may vary. For example,
This question
cannot be solved
with the sine law.
Chapter Self-Test, p. 270
1.
y = ⫺ 21 (32x+4) + 5
⫺10 ⫺8 ⫺6 ⫺4 ⫺2 0
2
4
2.
3.
NEL
a) 2243y5
1
b)
3125a3b
c) 2x
1
d)
4xy4
Are you given any
other sides?
No
Are you given
any other angles?
4
No
Yes
Are you given the
side opposite your
unknown angle?
Yes
Use the sine law to
solve for the
unknown angle.
Use the sine law to
find the angle
opposite this given
side
Yes
2
1
a) 2
125
b) 9
No
Yes
Answers
a) There is a variable in the exponent part of the equation, so it’s an
exponential equation.
b) You can tell by the second differences.
1
c) reflection in the x-axis, vertical compression of , horizontal
2
compression by a factor of 2, and translations of 4 left and 5 up
y
No
Are you given a
side and opposite
angle?
x
2
Use angle sums to
find the unknown
angle.
4
Lesson 5.1, pp. 280–282
1.
2.
3.
4.
5
12
5
, cos A 5
, tan A 5
,
13
13
12
13
13
12
csc A 5
, sec A 5
, cot A 5
5
12
5
17
17
15
csc u 5
, sec u 5
, cot u 5
8
15
8
2
4
a) csc u 5 2
b) sec u 5
c) cot u 5
3
3
a) 0.83
b) 1.02
c) 0.27
sin A 5
d) cot u 5 4
d) 1.41
Answers
651