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Key Words
• base
• exponent
• power
What
You’ll Learn
To recognize exponential
patterns in tables, graphs,
and equations, and to solve
problems involving exponential
growth and decay
• exponential growth
And Why
A variety of real-life
situations such as
population growth, the
increase in the value of an
antique, radioactive decay, and
drug absorption can be modelled
by exponential growth or decay.
• growth factor
• growth rate
• exponential decay
• decay factor
• decay rate
• initial value
• exponential relations
• doubling time
• half-life
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Activate Prior Knowledge
4
Expressing a Number in Different Forms
Prior Knowledge for 4.1
The number 125 is in standard form.
exponent
3
The exponent form of 125 is written
5 base
with a base and an exponent.
The expanded form is the number written as repeated multiplication.
For example, the expanded form of 125 is 5 5 5.
53 is a power of 5.
It is read “five to the
power three.”
Example
The edge length of a cube is 8 units.
Express the area of one face of the cube and
the volume of the cube in exponent, expanded,
and standard form.
8 units
8 units
8 units
Solution
For the area of a square: A where s is the side length
For the volume of a cube: V s 3, where s is the edge length
A88
Area of a face (square units):
A 82
V888
Volume of the cube (cubic units): V 83
Exponent form Expanded form
s2,
82 is read “eight
squared” and 83 is
read “eight cubed.”
A 64
V 512
Standard form
✓ Check
1. Copy and complete this table.
Exponent Form
Expanded Form
Standard Form
34
49
222
2. Domingos writes 81 in two different exponent forms. Could both be correct? Explain.
3. Soil is sold in cubic yards.
The dimensions of a cubic yard container are 36 by 36 by 36.
Express the volume of this container in exponent form
and sketch the container.
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Expressing a Percent as a Decimal
Prior Knowledge for 4.1
Percent means “per hundred.”
This hundredths grid is made up of 100 small squares.
The grid represents 1 whole. There are 65 shaded squares.
You can describe the shaded portion of the grid in these ways:
65
• 65%, 100
, or 0.65 of the grid is shaded.
You can describe the white portion of the grid in these ways:
35
• 35%, 100
, or 0.35 of the grid is white.
Example
According to a 2006 report, Canada holds about 60% of the world’s investable oil reserves.
Write 60% as a decimal to 2 decimal places, and as a decimal to 1 decimal place.
Solution
To write 60% as a decimal, divide 60 by 100.
60
60
0.60 to 2 decimal places
0.6 to 1 decimal place
100
100
✓ Check
1. Write each percent as a decimal.
a)
8%
b)
80%
c)
88%
d)
108%
2. Write each percent as a decimal to 2 decimal places and to 1 decimal place.
6% of GST on total sales
c) 5.4% financing rate for a car
a)
81% on a test
d) 0.3% meat protein
b)
3. Athletes are often described as “giving 110%.”
Write 110% as a decimal.
b) Explain how you can represent 110% using hundredths grids.
a)
4. Brad contributes 4.5% of his annual salary to the pension plan with his company.
Write this percent as a decimal.
b) Brad’s salary this year is $30 000.
Determine how much money Brad contributes this year to his pension plan.
Explain your method.
a)
Activate Prior Knowledge
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Linear Relations
Prior Knowledge for 4.1
A linear relation can be represented using a table of values, a graph, and an equation.
y
For the equation: y 3x 2
8
y = 3x + 2
x
y
2
4
1
1
0
2
1
5
2
8
First
differences
Slope: rise 6
run
6
6
4
(1) (4) 3
2
2 (1) 3
2
3
When x 0, y 2;
so, the y-intercept is 2.
2
x
523
–2 0
–2
853
2
4
6
–4
The slope of the line, the first differences,
and the coefficient of x in the equation all equal 3.
The y-intercept is the constant term, 2, in the equation.
Example
A customer pays $10 to join a video store’s movie club, then pays $2 for each movie
rented that day. This situation is represented by the equation C 10 2n,
where C is the total cost in dollars and n is the number of movies rented.
a) Create a table of values for the cost of up to 5 movies. Graph the data.
b) What is the vertical intercept? What does it represent?
Solution
a)
b)
20 C
Movies
Cost ($)
1
12
2
14
3
16
4
18
8
5
20
4
16
C = 10 + 2n
12
0
1
2
3
4
n
5
Use a broken line to
show the trend and
find the intercept.
The vertical intercept
is 10. This is the cost
in dollars of joining
the movie club.
✓ Check
1. Create a table of values including first differences and draw a graph for each relation.
a)
y 2x 4
b)
y 12 x 2
c)
y 3x 4
2. What do the negative signs in each equation in question 1 tell you about the graph?
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Modelling Exponential Growth
Mille feuilles is a French dessert made from nearly
1000 layers of pastry separated by a custard filling.
The layers are created by repeatedly rolling out and
folding pastry dough. Each time the dough is folded,
the number of layers grows, or increases.
We can fold a sheet of paper to model this growth.
Investigate
Modelling Growth by Repeated Multiplication
Work with a partner.
You will need a sheet of paper and grid paper.
➢ Fold the paper in half and count the number of layers.
Start this table. Record the number of layers.
Number of folds
Number of layers
1
➢ Fold the paper in half again.
Record the number of folds and the number of layers.
Continue this process until you can no longer fold the paper.
➢ Plot Number of layers against Number of folds.
Describe the growth in the number of layers.
➢ What pattern do you notice in the table?
Use the pattern to extend the table to 10 folds.
➢ How many times does the paper have to be folded to make
at least 1000 layers? Justify your answer.
Reflect
➢ Does it make sense to join the points on the graph? Explain.
➢ Each time you fold the paper in half, what happens to the
number of layers?
➢ Suppose you know the number of folds.
How could you determine the number of layers?
4.1 Modelling Exponential Growth
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Connect the Ideas
2 5 214
4
2 424
2 44
24
32
We use multiplication to represent
repeated addition of the same number.
5 terms in a sum
Similarly, we use exponents to represent
repeated multiplication by the same
number.
10
5
4
2 424
2 44
24
32
2 214
5 factors in a product
32
We read 25 as “2 to the 5th.”
2 is the base, 5 is the exponent, and 25 is the power.
Some patterns of growth involve repeated multiplication by a number
greater than 1. Because repeated multiplication can be represented
by an exponent, this type of growth is called exponential growth.
A horse breeder originally bought 3 mares.
She bred the mares, and kept the best 2 daughters of each mare.
These daughters were bred, and their best 2 daughters were kept,
and so on.
We can model the growth in the number of mares in each generation
in different ways.
Use a table
Generation 0
represents the initial
3 mares.
152
Generation
Pattern
Number
of mares
0
3
3
1
3
32
3
21
6
2
322
3 22
12
3
3222
3
23
24
4
3 2 2 2 2 3 24
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2
2
2
2
We repeatedly
multiply by 2. So,
2 is the growth
factor.
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There is a pattern in the growth.
We start at 3 and multiply by 2 for each generation.
For example, the number of mares in the 4th generation is 3 24.
Use an equation
So, the number of mares, M, in the nth generation is given
by the equation M 3 2n.
To find the number of mares in the 6th generation, substitute n 6.
Press: 3 2 ^ 6 M 3 26
3 64
192
There are 192 mares in the 6th
Growth in the Number of Mares
generation.
200
The graph shows the number of
mares in the initial group of mares
and in the next 6 generations.
Use a graph
With each generation, the graph
curves up more rapidly.
The initial number
of mares is the
y-intercept.
Number of mares
Chapter 04
160
120
80
40
0
3
4
5
2
Generation
1
Since a fractional number of
generations is not meaningful,
the points are joined with a broken curve.
6
Practice
1. Evaluate without a calculator.
a)
62
b)
72
c)
43
d)
25
b)
1.353
c)
3 56
d)
(1 0.06)3
e)
104
f)
81
2. Evaluate.
a)
212
4.1 Modelling Exponential Growth
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3. Kylie is making raffle tickets for a school fundraiser.
She cuts a sheet of paper in thirds.
Kylie stacks the three pieces on top of each other
and cuts the stack in thirds.
She continues the process of stacking and cutting several more times.
Step 1
Step 2
Step 3
Make a table of values to show the number of pieces of paper
at the end of each step until step 5.
b) Graph the data in the table.
c) Does the number of pieces of paper in each step grow exponentially?
Explain how you know.
d) Suppose you know the step number.
How could you determine the number of pieces of paper
at the end of that step?
a)
4. Under ideal conditions, some biological populations
grow exponentially over time.
For each table:
i) By what factor is each number in the second column multiplied
to get the next number?
ii) What is the growth factor?
a)
Day
Number of fruit flies
b)
Hour
0
30
0
50
1
120
1
150
2
480
2
450
3
1920
3
1350
5. Refer to part a of question 4.
Use a table, equation, or graph to predict the number
of fruit flies on day 6. Explain your choice.
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Number of bacteria
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Equations for exponential growth involve the product of a number and a power.
In the power, the base is the growth factor. The exponent is a variable.
Example
A small garden centre propagates tulip bulbs. The gardener begins with 100 bulbs.
Each bulb produces several new bulbs. Three of these are kept for the next round.
Write an equation to represent the number of new bulbs, B, in the nth round
Solution
Start with 100, the initial number of bulbs.
Multiply by the growth factor, 3, for each round.
Round
Number of new bulbs
0
100
100
1
100 3
100 31
2
100 3 3
100 32
3
100 3 3 3 100 33
In each round, the number of new bulbs is 100 times a power of 3. The exponent
is the round of propagation. So, the number of new bulbs, B, in the nth round is
given by the equation: B 100 3n.
6. Assessment Focus A botanist starts with 3 plants. She takes 5 cuttings from each plant
to start new plants. Later, she takes 5 cuttings from each new plant, and so on.
a) Draw a graph of the number of new plants in each of the first 5 rounds of cuttings.
b) Write an equation to model the number of new plants, P, in the nth round of cuttings.
c) How would the graph and equation change in each scenario? Explain.
i) The botanist starts with 10 plants.
ii) In each round, the botanist takes 2 cuttings from each plant.
7. Take It Further The number of bacteria
in a laboratory colony is recorded
over several hours. Is the growth
exponential? Justify your answer.
Hour
0
1
2
3
4
5
6
Number of
bacteria
100
141
199
280
396
560
790
Create a number pattern that represents exponential growth.
How do you know that the pattern represents exponential growth?
Represent the pattern using a table of values, an equation, and a graph.
4.1 Modelling Exponential Growth
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Making Comparisons
Graphic organizers are useful tools for identifying similarities and
differences. They can help you gain a stronger understanding of
each item being compared and of the relationships between them.
Venn diagram
Even numbers
8
26
1000
Multiples of 3
6
15
48
57
125
17
Scalene
triangles
Equilateral
triangles
Number of sides
3
3
Number of equal sides
0
3
Number of angles
3
3
Number of equal angles
0
3
Each loop represents a set.
Place items belonging to a set inside its loop.
Place items belonging to more than one set
in the space where the loops overlap.
Place items that do not belong to any set
outside the loops.
Matrix
Record the items being compared at the top
of the matrix. Record ways they are being
compared in the left column.
Complete the matrix to show similarities
or differences.
➢ Copy the Venn diagram. Record six other numbers in your
Venn diagram. Explain your strategy.
➢ Create a matrix to compare triangular prisms and triangular
pyramids. Explain your strategy.
➢ How are a Venn diagram and a matrix the same?
How are they different?
➢ After Section 4.2, use a Venn diagram or matrix to compare
exponential growth and exponential decay.
Here are some characteristics you could use.
Complete the last
part after you
finish Section 4.2.
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•
•
•
•
Growth factor
Decay factor
Increasing or decreasing
Ways to recognize
growth or decay
CHAPTER 4: Exponential Relations
•
•
•
•
•
Value of base
Equation
Table
Graph
Real-life examples
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Modelling Exponential Decay
Most automobiles depreciate, or lose value, over time.
Rashan purchases a new car for $25 000.
Suppose its value decreases by 15% each year.
So, each year, the car is only worth 85% of its value the preceding year.
We can model the decay, or decrease, in the value of the car by repeated multiplication.
Investigate
Modelling Decay by Repeated Multiplication
Work in a group of 4.
You will need a container with 100 pennies and grid paper.
Copy the following table.
Trial number
Number of pennies left
0
100
➢ Shake the container and empty it onto a desk.
Remove the pennies that land heads up.
Record the trial number and the number of pennies
that are left in the table.
➢ Put the remaining pennies back in the container.
Repeat the previous step until there are no pennies left.
➢ Plot Number of pennies left against Trial number.
Describe the decrease in the number of pennies.
Reflect
➢ What fraction of the pennies would you expect to remove
on each trial? Explain.
➢ How close were your actual results to the expected results?
Explain.
➢ Explain how this experiment may be modelled by
repeated multiplication.
4.2 Modelling Exponential Decay
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Connect the Ideas
Repeated multiplication by a positive number less than 1 produces a
decreasing pattern called exponential decay.
Suppose that in Investigate, you started with 64 pennies.
On each trial, you remove one-half of the pennies.
This is equivalent to repeatedly multiplying by 12 .
We repeatedly
multiply by 12 .
So, 12 is the
decay factor.
We can model the decay in the number of pennies
in different ways.
Use a table
Trial number 0
represents the initial
64 pennies.
Use an equation
Trial
number
0
Pattern
Number of
pennies left
64
64
1
64 2
64 3
64 4
64 1
2
1
2
1
2
1
2
64
64 (
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
)
64 ( )2
64 ( )3
64 ( )4
1
32
16
8
4
1
2
1
2
1
2
1
2
There is a pattern in the decay.
We start at 64 and multiply by 12 for each trial.
For example, after 4 trials, there are 64 ( 12 )4 pennies.
So, the number of pennies, P, after n trials is given
by the equation P 64 ( 12 )n.
Use a graph
The graph shows the number
of pennies in the initial
set of pennies and
after the first 6 trials.
With each trial, the graph
curves down less rapidly.
Since a fractional number
of trials is not meaningful,
the points are joined
with a broken curve.
Number of Pennies Left
after Each Trial
70
Number of pennies left
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60
50
40
30
20
10
0
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1
2
3
4
5
Trial number
6
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Compare the equation for exponential growth from Section 4.1 with
the equation for exponential decay on the opposite page.
Exponential Growth
M3
Both equations involve
the product of a
number and a power
with a variable
exponent.
Exponential Decay
P 64 ( 12 )n
2n
Initial value
Growth factor
(greater than 1)
Initial value
Decay factor
(between 0 and 1)
The equation y abx models both exponential growth and decay.
a is the initial value.
b is the growth or decay factor (b > 1 for growth, 0 < b < 1
for decay).
y is the value after x periods of exponential growth or decay.
Practice
1. Evaluate without a calculator.
a)
( 12 )6
b)
( 15 )3
c)
( 17 )2
d)
( 101 )5
d)
4.9 0.796
2. Evaluate. When necessary, round to the nearest hundredth.
a)
( 34 )7
b)
0.258
c)
296 ( 17 )3
3. Kylie cuts a piece of paper with area 486 cm2 in thirds. She stacks the pieces on top of each
other, and cuts the stack in thirds. She stacks and cuts the pieces of paper several more times.
Step 1
Step 2
Step 3
How does the area of the top of the stack change in each step? Explain.
b) Make a table of values and a graph to show the area of the top of the stack
at the end of each step until step 5.
c) Explain why the decrease in the area of the top of the stack is exponential.
What is the decay factor?
d) Compare this problem to question 3 in Section 4.1 Practice. Explain how the same
situation can involve both exponential growth and exponential decay.
a)
4.2 Modelling Exponential Decay
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4. For the table at the right:
By which factor is each number in the second column
multiplied to get the next number?
b) Determine the decay factor.
c) Determine the area in step 8.
Justify your answer.
a)
Step number
Area (cm2)
0
500
1
100
2
20
3
4
4
0.8
The decay factor may be given as a percent.
Example
A rubber ball drops from a height of 200 cm and bounces
several times.
After each bounce, the ball rises to 80% of its previous height.
a) Write an equation to represent the height of the ball,
H centimetres, after the nth bounce.
b) Determine the height of the ball after the 5th bounce.
Solution
H 200(0.8)n
means the same as
H 200 (0.8)n.
160
As a decimal, 80% 0.8.
So, after each bounce, the ball’s height is 0.8 times its previous height.
a) To determine the height after any bounce,
start at 200 and multiply by 0.8 for each bounce.
Write the equation y abx as H abn.
The initial height is 200 cm, so a 200.
The decay factor is 0.8, so b 0.8.
The equation is H 200(0.8)n.
b) Substitute n 5 in the equation H 200 (0.8)n.
H 200 (0.8)5
. 65.5
H
The height of the ball after the 5th bounce is about 65.5 cm.
CHAPTER 4: Exponential Relations
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5. Dalibor buys a new car for $25 000.
Suppose that each year, the car is worth 85% of its value
from the preceding year.
a) Write an equation to determine the value of the car, V dollars,
after n years.
b) What is the value of the car after 5 years?
c) Draw a graph to show the initial value of the car and its value at the end
of each of the first 5 years.
d) How would the graph and equation change in each scenario?
Explain your thinking.
i) Dalibor buys the car for $20 000.
ii) The car is worth 75% of its value in the preceding year.
6. Assessment Focus Consider the following equations.
y 100 (0.75)x
y 100 (1.25)x
a) Which equation models exponential growth?
Which equation models exponential decay?
How do you know?
b) Pose a problem that can be solved using the exponential decay equation.
Solve the problem.
7. Take It Further Jamie has a string of licorice lace that is 200 cm long.
A friend passes by and Jamie gives away 14 of the licorice.
Another friend passes by, and Jamie gives away 14 of
the licorice she has left.
Jamie repeats this process with several more friends.
Determine the length of the licorice lace after Jamie
has shared it with 10 friends.
Explain the strategy you used to solve the problem.
Create a number pattern that represents exponential decay.
How do you know that the pattern represents exponential decay?
Represent the pattern using a table of values, an equation, and a graph.
4.2 Modelling Exponential Decay
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Butterfly Exponentials
Materials
• TI-83 or TI-84 graphing calculator
• grid paper
This butterfly has exponential curves built into its design.
y
26
24
A
22
20
B
18
C
16
Try this puzzle after
you have completed
Section 4.5.
14
12
F
D
10
8
6
E
4
2
x
–12 –10 –8
To match a curve to
an equation, try
substituting specific
values for x. Graph the
equation and compare
its shape to the curve.
Notice that
. (1.05)1
0.952 162
–6
–4
–2 0
2
4
6
8
10
12
Match each labelled curve in the butterfly to one of the equations
shown below. Use the graphing calculator to help you.
y 12(1.15)x
y 8(1.08)x
y 18(2)x
y 8(1.05)x
y 8(0.952)x
y 6(0.94)x
➢ Sketch the labelled curves on the grid paper.
➢ Determine the equations for the other curves in the butterfly.
Use a graphing calculator to help you check.
Explain your strategy.
Sketch the new equations on grid paper.
CHAPTER 4: Exponential Relations
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Exponent Rules
We can use repeated multiplication to simplify expressions involving exponents.
Investigate
Finding Patterns in Powers
Work with a partner.
➢ Copy and complete each table.
Multiplying Powers
Original expression
Powers in expanded form
Simplified expression
23 22
(2 2 2) (2 2)
25
34 31
75 73
107 102
Power of a Power
We call a power whose
base is also a power a
“power of a power”.
Original
expression
Powers in expanded form
Simplified
expression
(23)2
23 23 (2 2 2) (2 2 2)
26
(34)1
(75)3
(107)2
Dividing Powers
Original expression
Powers in expanded form
1
23 22
222 冫
2冫
22
22
冫
2 冫
2
1
34
75
73
Simplified expression
1
21
1
31
107 102
➢ Compare exponents in each original expression with the exponent
in the simplified expression. What patterns do you notice?
Reflect
➢ Extend each table with three more examples. Predict the exponent
of the simplified expression without expanding the original
expression. Explain your strategy.
4.3 Exponent Rules
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Connect the Ideas
When we multiply or divide powers with the same base, the answer is
also a power of that base.
In the comparisons that follow, refer to the powers of 3 in this table.
Multiplying
powers
31
32
33
34
35
36
37
38
3
9
27
81
243
729
2187
6561
By evaluating
32
34
By expanding
9 81
729, or 36
32
1
32
33 14
3 4244
3 3 3
3
34
2 factors
4 factors
144
3 34424
33
343
3
44
(2 4) factors
36, or 729
Multiplication Rule
bm bn bm n
Power of a power
To multiply powers with the same base,
keep the base and add the exponents.
By evaluating
(32)4 94
6561, or 38
By expanding
(32)4 1442443
32 32 32 32
4 factors
(3 3) (3 3) (3 3) (3 3)
123 123 123 123
2 factors
2 factors
2 factors
2 factors
144444424444443
(4 2) factors
38, or 6561
Power of a Power Rule
(bm)n bm n
Dividing powers
To determine a power of a power, keep
the base and multiply the exponents.
By evaluating
36 34 729 81
9, or 32
By expanding
6 factors
1
14
444244
443
Chapter 04
1
1
1
36 34 冫
3冫
3冫
3 冫
3 33
冫
3冫
3冫
3 冫
3
1 424
1
14
1
14
3
4 factors
32
33
1
(6 4) factors
32, or 9
Division Rule
bm bn bm n, b 0
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To divide powers with the same base,
keep the base and subtract the
exponents.
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Practice
1. Write each product as a single power.
74 711
c) 103 102 10
a)
e)
(2.3)5
(2.3)9
25 22
d)
( 35 )13 ( 35 )14
f)
814
() ()
means the same as
( ) ( )
3 13 3 14
5
5
b)
3 13
5
87
3 14
5
2. Write each quotient as a single power.
a)
d)
89 82
114 11
2
2
e) ( )13 ( )2
3
3
b)
1.511
1.56
c)
f)
319
315
514 59
3. Write each power of a power as a single power.
(52)3
d) (0.84)6
a)
b)
e)
(84)2
(795)3
f) (47)3
c)
2 2 25
(( 3 ) )
4. Sadaf is asked to simplify 68 63.
She is not sure whether the answer is 611 or 624.
Explain how Sadaf can use the definition of exponents
to determine the correct answer.
5. Carlos made these mistakes on a test.
➢ He simplified 34 38 as 912.
6
➢ He simplified 82 as 14.
8
a) What mistake is Carlos making?
b) Use the definition of exponents to explain how
Carlos can get the correct answers.
6. Astronomers estimate that there are about 1011 galaxies
in the universe. They also estimate that each galaxy
contains about 1011 stars. About how many stars are there
in the universe?
7. Write each expression as a single power, and evaluate
without a calculator.
1
1
a) ( )3 ( )2
2
2
c)
(2.3)4 (2.3)3
b)
d)
2 3
(( 101 ) )
53
514
58
32 35
34
33 by evaluating each side.
b) Which side is easier to evaluate? Explain.
8. a) Verify that
4.3 Exponent Rules
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9. Assessment Focus
a)
Write each expression as a single power.
i)
b)
1.434
ii) 1.426
175 173
iv)
172
8
vi) 763 4
76
5113 5125
4 7
iii)
(( 67 ) )
v)
910
(92)3
( )
Choose three expressions from part a.
Explain your work, and how you know your answer is correct.
The exponent rules can be used as shortcuts.
Example
The table shows the first 10 powers of 2.
21
22
23
24
25
26
27
28
29
210
2
4
8
16
32
64
128
256
512
1024
Use the table to evaluate 32 16 without multiplying or dividing.
Solution
32 16 25 24
29
512
Use the table to represent each number as a power of 2.
Use the exponent rules to simplify the expression.
Use the table to evaluate the power.
10. Use the table of powers of 2 in the Guided Example.
Evaluate each expression without multiplying or dividing.
1024
a) 16 16
b)
c) 45
128
11. Take It Further Simplify each algebraic expression.
a)
t 3 t 14
b)
(a7)5
c)
s12
s5
You have simplified expressions involving powers by adding, multiplying,
or subtracting exponents. Create and simplify an expression
that involves each operation with the exponents.
How do you know that you have simplified correctly?
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To avoid dividing by 0,
we assume s ≠ 0.
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Zero and Negative Exponents
Exponents that are positive integers represent repeated multiplication by the same number.
This definition does not apply to powers such as 20 or 23.
It does not make sense to multiply 2 by itself 0 times or 3 times.
Investigate
Defining Zero and Negative Exponents Using a Pattern
Work with a partner. You will need a scientific calculator.
Each rectangle in the diagram models the power of 2 to the left of it.
23 = 8
22 = 4
21 = 2
2? = 1
2? =
2? =
2? =
➢ Go down the list of powers. How do the values of the powers
change? How do the exponents change?
➢ What do you think the last four exponents in the list of powers
should be? Explain your thinking. Then use a calculator to confirm
your predictions.
➢ Write the numbers in each pair as powers of 2.
The numbers in each
pair are reciprocals.
2, 12
4, 14
8, 18
What do you notice?
Reflect
Repeat the procedure for powers of 3.
➢ Which power has a value of 1? Does this make sense? Explain.
➢ How are the powers for a number and its reciprocal related?
4.4 Zero and Negative Exponents
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Connect the Ideas
Zero and
negative
exponents
The patterns in Investigate suggest definitions for zero and negative
exponents.
Zero exponent
For any non-zero base b,
b0 is equal to 1;
that is, b0 1.
Negative exponent
For any non-zero base b,
bn is the reciprocal of bn;
that is, bn b1n
To evaluate a power with a negative exponent using paper and pencil,
rewrite the power with a positive exponent.
For example, 43 413
Negative exponents
represent reciprocals.
Since 43 64,
43 1 .
1
444
1
64
64
When you evaluate a power with a negative
exponent on a calculator, the answer is given
as a decimal.
On a TI-83 or TI-84 graphing calculator,
you can press d 1 to convert the
decimal to a fraction.
Extending the
exponent rules
The definitions of zero and negative exponents allow us to extend the
exponent rules from Section 4.3 to any integer exponent.
For example, 23 23 23 3
20
1
This is reasonable because any non-zero number divided by itself is 1.
Similarly, 23 25 23 5
22
212
14
1
1
1
2冫 冫2 冫2
This is reasonable because 23 25 2冫 冫2 冫2 2 2
1
1
22
14
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Practice
1. Write as a whole number or as a fraction without an exponent.
a)
23
b)
130
c)
62
d)
103
e)
2.50
2. Evaluate as a decimal.
Round to the nearest thousandth where necessary.
a) 2.52
b) 82
c) 12.7(1.99)
d) (1 0.08)5
e)
0.95(1.780)
3. a) Evaluate the powers in each pair without a calculator.
42 and 42
ii) 34 and 34
b) Explain the pattern in the answers.
i)
iii)
91 and 91
iv)
104 and 104
In an equation representing an exponential pattern, the initial value is determined using a zero
exponent. Future values are determined using positive exponents.
Past values are determined using negative exponents.
Example
Since the late 1990s, the number of hybrid cars sold worldwide
has nearly doubled every year. The number of hybrid cars, C, sold can be
modelled by the equation C 250 000(2)n, where n is the number of years
after 2005.
Determine the number of cars sold in 2005 and 2001.
How do you know that your answers are correct?
Solution
Use the formula: C 250 000 2n.
2005 is 0 years after 2005.
Substitute: n 0
C 250 000(2)0
250 000
250 000 cars were sold in 2005.
2001 is 4 years before 2005.
Substitute: n 4
C 250 000(2)4
15 625
15 625 cars were sold in 2001.
The answer is correct because it
agrees with the initial value given
in the equation, which is 250 000.
250 000 cars were sold in 2005.
Since the number of cars doubles
every year, we could divide 250 000
by 2 four times to determine the
number of cars in 2001:
250 000 15 625
24
So, the answer is correct.
4.4 Zero and Negative Exponents
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4. The value of the maple syrup produced in Ontario, V million dollars, can be represented
by the equation V 1.25(1.0866)t, where t is the number of years since 1970.
a) Determine the value of the maple syrup produced in 1970 and 2000.
b) Suppose the model applies to years before 1970.
Determine the value of the maple syrup produced in 1960.
5. Use the exponent rules to write each expression as a single power, then evaluate.
23 28
e) 32 36
35 33
f) 23 22
a)
84 84
g) (22)3
b)
c)
4 1
, Janine reasons as follows:
(
5)
4 1
( 5 ) is the reciprocal of ( 45 )1.
( 45 )1 45 , so ( 45 )1 54
6. a) To evaluate
Is Janine correct? Justify your answer.
b) Write as a fraction or an integer without using exponents.
i)
( 32 )1
ii)
( 107 )1
iii)
( 15 )3
iv)
( 23 )2
7. Assessment Focus
Evaluate without using a calculator:
1
i) 20
ii) 92
iii) ( )2
3
b) The equation B 1000 2t models the number
of bacteria in a colony t hours from now.
i) Determine the value of B when t 0.
What does this value represent?
ii) How many bacteria were in the colony 3 h ago?
Justify your answers.
a)
8. Take It Further Jason bought a used car for $7200.
Suppose this model of car depreciates so that each year
it is worth 82% of its value the previous year.
a) How much would the car be worth 5 years from now?
b) When Jason bought the car, it was 6 years old.
What was the original price of the car?
Suppose a classmate is having difficulty understanding what a negative
or zero exponent represents. Write an explanation for your classmate.
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75 75
h) (85)0
d)
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Exponential Relations
Equations of the form y ab x represent exponential relations.
When a 1, the equation y ab x becomes y b x.
You will investigate how the value of b affects the graph of y b x.
Investigate
In the equation of an
exponential relation,
the variable x is an
exponent.
Graphing y ⴝ bx
Work with a partner. You will need a TI-83 or TI-84 graphing calculator.
Use these
window settings.
➢ Graph the equations in Set A on the same screen.
Sketch and label the graphs in your notebook.
Set A (b > 1)
Set B (0 < b < 1)
y 2x
y ( 12 )x
y 3x
y ( 13 )x
y 4x
y ( 14 )x
➢ Compare the graphs.
How are they similar? How are they different?
➢ The graphs have one point in common.
What are the coordinates of this point?
➢ Do any of the graphs ever intersect the x-axis?
Use TRACE and the left arrow key to examine how y changes as x
decreases. Are the values of y ever zero or negative? Explain.
➢ Repeat the previous steps for the equations in Set B.
Reflect
➢ Why do all the graphs have a point in common?
➢ Describe the graph of y bx for b > 1 and 0 < b < 1.
What happens if b 1?
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Connect the Ideas
We can make a table of values and graph y 2x and y ( 12 )x.
The graph of
y ⴝ 2x
The graph of y 2x curves upward, slowly at first,
and then more rapidly.
All exponential relations of the form y bx, b >1, have this shape.
y ⴝ 2x
x
3
23 2
22 1
21
1
23
1
22
1
21
0
20 1
1
21 2
2
22 4
3
23 8
8
y
y = 2x
1
8
1
4
1
2
6
4
2
x
–3
The graph of
y ⴝ ( 12 )x
–2
–1 0
1
2
3
The graph of y ( 12 )x curves downward, rapidly at first,
and then more slowly.
All exponential relations of the form y bx, 0 < b < 1, have this shape.
1
x
y ⴝ ( 2 )x
3
( 12 )3 23 8
( 12 )2 22 4
( 12 )1 21 2
( 12 )0 1
( 12 )1 12
( 12 )2 14
( 12 )3 18
2
1
0
1
2
3
8
y=
y
x
( )
6
4
2
x
–3
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–2
–1 0
1
2
3
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The graphs of y 2x and y ( 12 )x illustrate the following properties
of the graph of y bx, b > 0 and b ≠ 1.
The graph of
y ⴝ bx
➢ When b > 1, the graph curves up to the right.
So, the graph illustrates exponential growth.
➢ When 0 < b < 1, the graph curves down to the right.
So, the graph illustrates exponential decay.
➢ The graph passes through the point (0, 1),
because when x 0, y b0 1.
So, the graph has a y-intercept of 1.
➢ The graph gets very close to the x-axis but does not intersect it.
➢ The graph does not have a maximum or minimum value.
Practice
1. Match each equation with its graph. How do you know?
a)
y 4x
y 1.2x
10
y 10x
y
b)
y ( 14 )x
B
y 0.9x
C
10
y ( 23 )x
y
A
8
8
B
6
6
4
4
C
A
2
2
x
x
–4
–2
0
2
4
–4
–2
0
2
4
2. a) Determine whether each equation represents exponential growth or exponential decay.
How do you know?
1
i) y 3x
ii) y 1.25x
iii) y ( )x
iv) y 0.8x
5
b) Make a table of values for x between –3 and 3, then graph each equation.
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3. Assessment Focus
Graph y 4x and y ( 14 )x on the same grid for x from 3 to 3.
b) Compare the two graphs. How are they similar? How are they different?
1
c) How do you think the graphs of y 5x and y ( )x would compare
5
to the graphs in part a? Justify your answer.
a)
4. Graph the equation y 2x on a TI-83 or TI-84 graphing calculator.
Use TRACE and the arrow keys to examine values of y for large
positive values of x and large negative values of x. What do you notice?
b) Explain why the graph of y 2x does not have a maximum or a minimum value.
a)
To determine whether a given relation is exponential, make a table of values
and look for a growth factor or decay factor.
Example
Determine whether the graph
is exponential.
Justify your answer.
If it is exponential,
what is the growth factor?
10
y
8
6
4
2
x
–1 0
Solution
Select at least four evenly spaced points.
The points with x-coordinates
0, 1, 2, and 3 lie on grid markings.
This makes it easier to read them.
Use these points to make a
table of values.
10
1
2
3
4
1
2
3
4
y
8
6
4
2
x
y
0
1
1
2
2
2
3
4
2
8
2
The y-coordinates are repeatedly
multiplied by 2.
The relation is exponential with growth factor 2.
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x
–1 0
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5. Which relations are exponential? Justify your answers.
a)
x
y
c)
1
2
10
20
10
3
30
4
40
b)
5
50
d)
y
x
1
0
1
2
3
y
1
5
1
5
25
125
10
8
8
6
6
4
4
2
2
y
x
–2
–1 0
1
2
3
0
1
2
3
x
4
6. Take It Further Use a TI-83 or TI-84 graphing calculator.
a)
Graph these three equations on the same screen.
y 3(2)x
y 4(2)x
y 2(2)x
Sketch and label the graphs.
Use these window settings.
Compare the 3 graphs.
How are they the same?
How are they different? Explain.
c) Repeat parts a and b for these equations:
y 3(0.5)x
y 4(0.5)x
y 2(0.5)x
d) How does the value of a affect the graph of y abx? Explain.
b)
How does the value of b affect the graph of y bx?
Include examples in your explanation.
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Mid-Chapter Review
4.1 1. Evaluate without using a calculator.
a)
73
b) 28
c)
122
4.3 6. Use the definition of exponents as
repeated multiplication to check your
answers for question 5.
2. Nguyet folds a 5-page letter in half several
times.
a) Complete the table for 5 folds.
Number of folds
Number of layers
0
5
Graph the data in the table.
c) Write an equation to represent the
number of layers, L, after n folds.
d) Determine the number of layers
after 8 folds.
b)
4.2 3. Evaluate without a calculator.
a)
( 16 )2
b)
( 12 )6
c)
( 34 )4
4. A ball drops from a height of 300 cm.
On each bounce, the ball rises
to 60% of its previous height.
a) Create a table of values and a
graph to show the height of the ball
after each of the first four bounces.
b) Write an equation to represent the
height of the ball, H centimetres,
after the nth bounce.
c) Determine the height of the ball
after the eighth bounce.
5. Simplify and evaluate
without using a calculator.
Explain your strategies.
a) 102 104
b) (23)2
723
721
19
1.5
e)
1.518
c)
176
d)
f)
4
( 33 )2
2
57 54
56 52
CHAPTER 4: Exponential Relations
4.4 7. a) Copy and complete the last four
equations in the pattern. Write your
answers as fractions.
43 64
42 16
41 4
40 ?
41 ?
42 ?
43 ?
b) Write the next two equations in the
pattern. Explain your strategy.
8. Evaluate without a calculator.
Write your answers as fractions.
a) 32
b) 26
c) 104
d) 120
9. Student attendance, A, at City High’s
football games has been dropping
9 n
according to the equation A 1200( 10
),
where n is the number of years since
2000.
a) How many students attended games
in 2000? How do you know?
b) What is the decay factor?
c) What was the attendance in 1999?
2005?
4.5 10. a) Use a table of values to graph each
equation for x from 2 to 2.
1
i) y 3x
ii) y ( 3 )x
b) Describe each graph.
How are they similar?
How are they different?
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Applications of Exponential Growth
Asha works part-time at a pizzeria. One of her jobs is to make pizza dough.
After Asha mixes the dough, she waits until it doubles in volume before using it.
Investigate
The Time for a Volume of Pizza Dough to Double
The graph shows how the volume of a batch
of pizza dough grows exponentially over time.
➢ What is the
initial volume
of the dough?
What is the
volume of the
dough when the
initial volume
doubles?
How much time
does it take for
the initial volume
to double?
Volume of Pizza Dough
4.0
3.6
3.2
Dough volume (L)
Chapter 04
2.8
2.4
2.0
1.6
1.2
0.8
0.4
➢ How much time
does it take for
0
10 20 30 40 50 60 70 80 90
the dough to
Time (min)
double in volume
from 2 L to 4 L? How do you know?
➢ Compare this time to the time you calculated earlier.
What do you notice? Explain.
➢ What is the doubling time for the pizza dough? Explain.
Reflect
➢ Compare your answers with a classmate.
If you have different answers, try to find out why.
➢ Does the time for the volume to double depend on the points
used? Explain.
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Connect the Ideas
Growth rate as
a percent
In many situations involving exponential growth, the growth rate
is given instead of the growth factor. The growth rate
can be expressed as a percent increase.
From the article, the wolf population increases by 20% each year.
So, the annual growth rate is 20%.
Wolf Reintroduction a
“Great Success”
F
ive years ago, 25 Canadian grey wolves were released in a
conservation area in Montana, where they had become extinct.
According to Dr. Ed Barton of the US Fishery and Wildlife
Service, the program has been a great success. Since reintroduction, the wolf population has increased dramatically, with
an annual growth rate of nearly 20%.
To determine the corresponding growth factor,
we reason like this.
The whole wolf population in any year is 100%.
The next year, it increases by 20%.
100% 20% 120%
That is, each year, the population is 120% of the population from the
preceding year, or 1.2 times the population the preceding year.
So, an annual growth rate of 20% has growth factor 1.2.
When the growth rate is given as a percent increase,
the growth factor b is given by b 1 r,
where r is the percent expressed as a decimal.
The table shows the growth of the wolf population over the 5 years.
Year
0
1
2
3
4
5
Population
25
30
36
43
52
62
1.2 1.2 1.2 1.2 1.2
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The population is
rounded to the nearest
whole number.
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Growth rate in
terms of doubling
time
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The growth rate can also be described in terms of doubling time,
the time it takes for a quantity to double in value.
We can estimate the doubling time from the table.
The initial population is 25 wolves.
In Year 4, the population is 52, which is just more than 2 25 50.
So the doubling time is just under 4 years.
We get a more accurate estimate of the
the doubling time from a graph.
The initial population is 25 wolves.
Look for a time when the population is
2 25 wolves 50 wolves
This happens at about 3.8 years.
So, the doubling time is about 3.8 years.
In Investigate, you discovered that the
doubling time is the same no matter which
two points you choose on the graph.
In the first year, the population is 30 wolves.
Look for a time when the population is
2 30 wolves 60 wolves.
Growth in Wolf Population
Number of wolves
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40
20
0
1
2
3
Year
4
5
Growth in Wolf Population
Number of wolves
Chapter 04
60
40
20
0
1
2
3
Year
4
This happens at about 4.8 years.
4.8 years 1 years 3.8 years.
The doubling time is the length of time it takes for a quantity
growing exponentially to double in value. The doubling time
applies to all times during the growth, not just at the starting value.
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Practice
1. Copy and complete the table.
5%
Growth rate
8%
1.5%
50%
1.12
Growth factor
1.03
2. Sushil thinks that if a quantity grows by 30% each year,
the growth factor should be 0.3 instead of 1.3.
Explain why Sushil is incorrect.
3. Determine the doubling time for each exponential graph.
Can you verify your answer? Explain your thinking.
a)
10
b)
y
16
c)
y
y
40
8
12
6
8
4
4
30
20
–1 0
10
x
2
1
2
0
x
3
2
4
x
6
0
10
4. Latreese measures the volume of a batch of cinnamon dough every 30 min.
She divides each measurement by the preceding measurement
to determine the growth factor.
a) Copy and complete the table showing Latreese’s measurements.
Time (min)
Volume (L)
Growth factor
0
1.4
2.0 .
1.4 1.43
30
60
90
120
2.0
2.9
4.2
6.0
2.9 .
2.0 1.45
Latreese says that the volume is not growing exponentially because
the values she calculates for the growth factor are not equal to each other.
Do you agree? Explain.
c) The dough must double in volume before Latreese can use it.
Approximately how long will this take?
b)
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We can use technology to determine the doubling time.
Example
An antique costs $600 and is increasing in value by 12% each year.
a) Write an equation to represent the value of the antique, V dollars,
after n years.
b) Estimate the value of the antique after 5 years.
c) Estimate the time it takes for the antique to double in value.
Solution
a) Use the equation V abn.
The initial value is 600, so a 600.
The growth factor b is 100% 12% 112%.
So, the equation is V 600(1.12)n.
b) Substitute: n 5
V 600(1.12)n
V 600(1.12)5
. 1057.41
V
The value of the antique after 5 years is about $1060.
c) We can use a graphing calculator to generate the graph.
Each year, the value
increases by a factor
of 1.12.
Press r. Move the cursor as close as possible to y 1200.
. 6.1
Read the corresponding x-value: x So, the doubling time is about 6.1 years.
5. A rare stamp was worth $125 in 2005.
It was predicted to increase in value by 8% each year.
a) Write an equation that models this situation.
b) Estimate the value of the stamp in 2012.
c) Estimate the length of time it takes for the value
of the stamp to double.
What strategies and tools did you use?
Explain the reasons for your choices.
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6. The table shows the growth in the world’s population from 1900 to 2000.
Year
1900
1910 1920
1930 1940 1950 1960
1970 1980 1990 2000
World Population (billions)
1.65
1.75
2.07
3.70
1.86
2.30
2.52
3.02
Graph the data. Does it appear to be growing exponentially? Explain.
b) Use the graph to predict the world’s population in 2050.
c) Why may the world’s population continue to increase?
Why may it level off?
d) Use the graph to pose and solve a problem
about the growth of the world’s population.
a)
7. Assessment Focus The number of motor vehicles
in the world, M million, can be modelled
by the equation M 50(1.061)t,
where t is the number of years since 1946.
a) What do the numbers in the equation represent?
b) Graph the equation.
c) Use the graph to pose and solve
two problems. Explain your solutions.
8. Take It Further In 1990, the average ticket
for an NHL game cost about $25.
Since then, ticket prices have increased by
about 7.5% each year.
To predict the price in 2000,
Zdravko reasons like this.
The price increases by 7.5% each year, so after
10 years, it will have increased by 7.5% ten times,
for an overall increase of 75%
So, the price in 2000 is: $25 1.75 $45
Is Zdravko correct? Justify your answer.
Use a graphic organizer such as a Frayer model or concept map to
describe exponential growth.
Explain the reason for your choice of graphic organizer.
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Applications of Exponential Decay
Radioactive substances are found in nuclear power plants,
cancer treatment centres, and, in very small amounts,
in most smoke detectors.
Investigate
The atoms of
radioactive substances
are unstable. They
break down over time
in a process called
radioactive decay.
Simulating Radioactive Decay
Work as a whole class. One student is the recorder.
Each remaining student needs a regular die.
You will need grid paper.
➢ The class represents a radioactive substance. To begin,
all students stand up. Each standing person, except the recorder,
represents a radioactive atom. The recorder counts and records
in a table the number of students who are standing.
Trial number
Number of students standing
0
➢ Each standing person rolls her or his die.
Anyone who rolls a “6” sits down. A student who sits down
represents an atom that has undergone radioactive decay. The
recorder counts and records the number of students still standing.
➢ Repeat the previous step until 2 or fewer students are still standing.
➢ Work individually. Graph the data in the table.
Reflect
➢ What fraction of students would you expect to remain standing
on each trial? Explain.
➢ Does the graph appear to represent exponential decay?
Justify your answer.
➢ Divide the number of standing students in each trial by the
number of standing students in the previous trial. Do these
quotients support your answers to the previous questions?
Why or why not?
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Connect the Ideas
In many situations involving exponential decay, the decay rate is given
instead of the decay factor.
Decay rate
as a percent
The decay rate can be given as a percent decrease.
Maya takes a 250-mg dose of penicillin.
Each hour, about 40% of the penicillin in her bloodstream
is eliminated. So, the decay rate is 40%.
To determine the decay factor, we reason as follows.
The mass of penicillin in any hour is 100%.
The next hour, it decreases by 40%.
100% 40% 60%
0.6 1 0.4, where
0.4 is the decay rate
of 40% expressed as
a decimal.
40% of the penicillin is
eliminated each hour.
So, 60% of the
penicillin remains in
the bloodstream.
The mass of penicillin remaining in her
bloodstream each hour is 60%, or 0.6 times,
the amount in the previous hour.
So, a decay rate of 40% is equivalent to a decay factor of 0.6.
When the decay rate is given as a percent decrease,
the decay factor b is given by b 1 r,
where r is the percent expressed as a decimal.
The table shows how the mass of penicillin in Maya’s bloodstream
decreases over the first 5 h.
Time (h)
0
1
2
3
4
5
Mass of penicillin (mg)
250
150
90
54
32.4
19.4
0.6
Decay rate in
terms of half-life
184
0.6
0.6
0.6
0.6
The decay rate can also be described in terms of a quantity’s half-life,
the time required for a quantity to be reduced by a factor of one-half.
We can estimate the half-life from the table.
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1
2
250 mg 125 mg
After one hour, the amount of penicillin is 150 mg.
After two hours, the amount of penicillin 90 mg.
So the half-life is between 1 h and 2 h.
Elimination of Penicillin
from Maya’s Bloodstream
We can get a more accurate estimate
of the half-life from a graph.
The initial dose is 250 mg.
Look for a time when the mass of
penicillin is 12 250 mg 125 mg.
This happens at about 1.4 h.
So, the half-life of penicillin is
about 1.4 h.
Amount of penicillin (mg)
Chapter 04
300
250
200
150
100
50
0
1
2
3
4
Time (h)
5
The half-life is the same no matter which two points you choose
on the graph. For example, notice that it also takes about 1.4 h
for the amount of penicillin to drop from 200 mg to 100 mg.
The half-life is the time required for a quantity decaying
exponentially to be reduced by a factor of one-half. The half life
applies to all times during the decay, not just at the starting value.
Practice
1. Determine the half-life for each exponential graph. Can you verify your answer?
Explain your thinking.
a)
10
b)
y
10
c)
y
10
8
8
8
6
6
6
4
4
4
2
2
2
x
0
2
4
6
8
y
x
x
0
2
4
6
8
0
10
20
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2. A capacitor is an electronic device that stores energy.
The table shows how the voltage on a capacitor changes with time.
Time (s)
0
1
2
3
4
5
6
7
8
9
10
Voltage (Volts)
9.0
7.7
6.9
6.0
5.2
4.6
4.0
3.5
3.1
2.7
2.3
Most DVD players
and digital clocks
have a small
capacitor for storing
energy during brief
power outages.
Graph the data in the table.
b) Does the voltage appear to be decreasing exponentially? Explain.
c) Pose and solve a problem using the graph in part a.
d) Estimate the half-life of the voltage using either the table or the graph.
Did you use the table or graph? Why?
a)
Elimination of Caffeine
from the Blood
3. Assessment Focus After you drink a cup of coffee,
the caffeine levels in your blood drop slowly over time.
The graph shows the percent P of caffeine
remaining in your blood after t hours.
a) Show that the amount of caffeine in the blood
decays exponentially. What is the decay rate?
b) What is the half-life of caffeine in the blood?
c) Pose and solve two problems that are based on the graph.
Caffeine in blood (%)
Chapter 04
100
80
60
40
20
0
2
4
6
8
Time (h)
When we know the decay rate and the initial value,
we can use the equation y abx to model the exponential decay.
Example
Suppose a computer that cost $1000 decreases in value by 17% each year.
a) Write an equation to determine the value of the computer,
C dollars, after n years.
b) Estimate the value of the computer after 5 years.
Solution
a) Use the equation C abn.
The initial value is 1000, so a 1000.
The decay rate is 17%, so the decay factor b is:
100% 17% 1 0.17 0.83
So, the required equation is C 1000(0.83)n.
b) Substitute n 5 in the equation C 1000(0.83)n.
C 1000(0.83)5
. 393.90
C
The value of the computer after 5 years is about $390.
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4. A cancer treatment centre has a 100-mg sample of radioactive iodine.
Each day, about 8.3% of the sample decays.
a) Write an equation to represent the mass of iodine left after t days.
b) Estimate the mass of iodine left after 8 days.
c) Estimate the half-life of the iodine.
Explain your strategy.
5. A class of 26 students carried out the Investigate in this section to model radioactive decay.
They decided that their data could be modelled by the equation N 26( 56 )t,
where N is the number of people standing and t is the number of trials.
a) Explain the formula.
b) Estimate the number of people still standing after:
i) 5 trials
ii) 10 trials
6. Refer to your own data from the Investigate.
How well does the equation in question 6 model your data?
Explain. Why might it be different?
b) Change the equation, if necessary, to better model your data.
Plot the equation on a TI-83 or TI-84 graphing calculator.
Sketch the graph.
c) Use the TRACE feature.
Estimate the number of trials needed so that only one-half
of the class remains standing.
Estimate the number of trials needed so that only one-quarter
of the class remains standing.
d) After how many trials will only one person be left standing? Explain why your results
in the Investigate may be different.
a)
7. Take It Further Equations for exponential decay may be written
with a negative sign in the exponent. Explain why the equations
y 35(0.5)x and y 35(2)x are equivalent.
Use a Venn diagram or matrix to compare exponential growth
and exponential decay.
Include new concepts that you have studied since Section 4.2.
Explain your choice of organizer.
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Exponential Model for an Experiment
The results of an experiment can sometimes be modelled using an exponential relation.
You can use the model to interpret the data and make predictions.
Exploring the Cooling Curve for Water
Inquire
Work in a small group.
If the top of the can is
sharp, you can put
masking tape around it.
You will need a clock or stopwatch that shows minutes and seconds,
an empty soup can, a laboratory thermometer, a stir stick, and hot water.
You will also need grid paper, a TI-83 or TI-84 graphing calculator, or a spreadsheet program.
1. Collecting the data
Record the room temperature.
b) Copy the table.
a)
Measurement
number
Time
(min)
0
0
1
5
Water
temperature (°C)
Temperature
difference (°C)
Place the soup can on a book or pad of paper.
Ask your teacher to fill the can with hot water.
Stir the water with the stir stick once or twice.
Put the thermometer in the water,
being careful not to touch the can.
Record the water temperature.
d) Record the water temperature
every 5 min for 45 min.
Stir the water before each reading.
c)
During the 5-min intervals,
work on question 2.
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2. Graphing the data
Calculate the temperature difference; that is,
the difference between the water temperature
and the room temperature.
Record each temperature difference in your table.
b) Use the data in your table.
Plot Temperature difference against Time.
a)
3. Analysing the data
This graph is called
a cooling curve.
Work with your group to answer these questions.
a) Describe the change in the data for temperature difference
over time.
When did the temperature difference change most rapidly?
When did it change most slowly?
b) On the x-axis of the graph, below each measurement, write the measurement number.
Each measurement corresponds to a 5-min interval.
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Use the equation T abx to model the data, where
T is the temperature difference and x is the measurement number.
i) Recall that a represents the initial value.
What is the value of a in this experiment?
What does the value of a represent in this experiment?
ii) Recall that b is the decay factor.
The mean of the data
To determine b, divide each temperature difference
is the sum of the
numbers divided by
by the preceding temperature difference,
the number of
then find the mean of the quotients.
numbers.
What does the value of b represent in this experiment?
iii) Use a and b to write an equation that models your data.
d) Graph the equation from part c on the same grid as the data.
e) Use the table or the graph of the equation to determine how long it takes
for the temperature difference to be reduced by a factor of one-half.
Explain your reason for deciding whether to use the table or graph.
c)
Reflect
➢ Is an exponential relation a good model for the results from your
experiment? Justify your answer.
➢ How do you think your graph and equation would change in
each scenario? Justify your answers.
· You start with cooler water.
· You use an insulated mug instead of a soup can.
· You start with a smaller volume of water.
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Comparing Linear, Quadratic,
and Exponential Relations
According to a folktale from India, the inventor of the game of chess
asked to be rewarded for his invention. He wanted two grains of rice
on the first square of the chessboard, four grains on the second square,
eight grains on the third square, sixteen grains on the fourth square, and so on.
Investigate
1
2
3
4
5
6
Comparing Reward Schemes on a 3 by 3 Chessboard
Work with a partner to calculate the reward on a 3-by-3 chessboard,
and compare it to other reward schemes.
You will need a scientific calculator and grid paper.
7
8
9
A chessboard is
8 squares by
8 squares.
In each formula, R is
the number of grains
of rice on square n of
the chessboard.
➢ Copy and complete these three tables.
For each reward scheme, use the given equation to determine
the number of grains of rice, R, on each of the nine squares
of the chessboard.
Linear Reward
R ⴝ 2n
Quadratic Reward
R ⴝ n2
Exponential Reward
R ⴝ 2n
n
R
n
R
n
R
1
2
1
1
1
2
2
2
2
➢ Choose a vertical scale, and plot the three sets of data
on the same graph.
Reflect
➢ Which reward scheme has the slowest growth?
Why does it have the slowest growth?
➢ Which reward scheme has the fastest growth?
Does it have the fastest growth for all values of n? Explain.
➢ Do you think the inventor of the game of chess could have been
given the reward he wanted? Explain.
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Connect the Ideas
We can use tables of values, graphs, or equations
to compare linear, quadratic, and exponential relations.
A linear relation has constant first differences.
The graph is a straight line.
The equation has the form y mx b.
Linear relations
y ⴝ 2x
x
y
3
6
2
4
1
2
0
0
1
2
2
4
3
6
4
First differences
3
4 (6) 2
2
2 (4) 2
1
y
0 (2) 2
202
–3
422
–2
–1 0
–1
642
1
x
3
2
–2
–3
–4
A quadratic relation has constant second differences.
The graph is a parabola.
The equation has the standard form y ax2 bx c.
Quadratic
relations
y ⴝ x2
x
y
3
9
2
4
1
1
0
0
1
1
2
4
413
3
9
945
First
differences
4 9 5
1 4 3
0 1 1
101
8
Second
differences
y
7
6
3 (5) 2
1 (3) 2
5
1 (1) 2
4
312
3
5 3 2
2
1
–3
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–2
–1 0
1
2
x
3
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An exponential relation has constant growth or decay factors.
The graph is a curve that approaches the x-axis in one direction and
grows in the other direction.
The equation has the form y abx.
Exponential
relations
y ⴝ 2x
x
y
3
1
8
2
1
4
1
1
2
0
1
1
2
2
4
8
7
Growth factor
6
1
4
1
8
2
1
2
1
4
2
1
1
2
2
3
212
2
422
1
5
4
842
3
y
–3
–2
–1 0
1
2
x
3
8
Practice
1. Classify each relation as linear, quadratic, or exponential.
How do you know?
a) d 7t 56
d) A 100r2
S 43(0.8)x
e) y 0.93(2.5)x
b)
C 25r
f) d –4.9t2 3.2t – 5.9
c)
2. Which relations are exponential? Justify your answer.
N 17(0.78)b
d) T 6.3冑苶
d
R 23w
e) d 4.1(5)w
a)
b)
V 9.0(1.5)t
f) L 7.2t 2
c)
3. Look at the pattern in each table.
Determine whether the relationship between x and y is linear,
exponential, or quadratic. Explain your answer.
a)
b)
c)
x
0
1
2
3
4
y
0.5
1.5
4.5
9.5
16.5
x
0
1
2
3
4
y
0.5
1.1
2.4
5.3
11.7
x
0
1
2
3
4
y
0.5
3.5
6.5
9.5
12.5
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4. a) Graph the three data sets in question 3
on the same grid.
b) Explain the shape of each graph.
Use the graphs to verify your answers to question 3.
5. Assessment Focus Three water storage tanks
are allowed to drain. For each tank, an equation
giving the height, h metres, of the water
in the tank after t hours is given.
Tank A: h 20(0.95)t
Tank B: h 20 0.5t
Tank C: h 0.0125t2 t 20
a) Classify each drainage pattern as linear,
quadratic, or exponential.
Explain your thinking.
b) What was the initial height of the water in each tank?
Explain how you know.
The graph of an exponential relation may look like part of the graph of a quadratic relation
or other nonlinear relation.
Example
Tell whether curve A and curve B show exponential decay.
Justify your answer.
1.0
0.9
y
(2, 1)
(1, 0.8)
0.8
0.7
0.6
(2, 0.64)
(3, 0.51)
0.5
(3, 0.44)
(4, 0.41)
0.4
(5, 0.33)
0.3
(4, 0.25)
0.2
B
(5, 0.16)
0.1
A
(6, 0.11)
1
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4
5
6
7
8
9
10
11
x
12
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Both curves have the general features of an exponential decay curve.
Create a table of values for each graph.
Curve B
Curve A
To determine the
decay factor,
divide each y-value
by the one preceding it.
Decay factor
x
y
1
0.80
. 0.51
2
0.64
0.64
0.80
. 0.80
0.25
0.51
. 0.49
3
0.51
0.51
0.64
. 0.80
0.16
0.16
0.25
. 0.64
4
0.41
0.41
0.51
. 0.80
0.11
0.11
0.16
. 0.68
5
0.33
0.33
0.41
. 0.80
Decay factor
x
y
2
1
3
0.51
0.51
1.00
4
0.25
5
6
The decay factors are constant in curve B, but not in curve A.
So, only curve B is decreasing exponentially, showing exponential decay.
6. The data in the table shows how the speed of a fibreglass boat increases over time
when the engine is on full throttle.
10
20
30
40
50
60
3.0
4.9
6.3
17.0
30.5
58.5
Time (s)
Speed (km/h)
Graph the data.
b) Tim says graph is an exponential growth curve.
Is he correct? Justify your answer.
a)
7. Take It Further The table gives Canada’s gross national debt between 1970 and 1995.
Year
1970
1975
1980
1985
1990
1995
Gross National Debt (billion $)
36
55
111
251
407
596
Describe the trend in the data.
Estimate the gross national debt in 2000 and 2005.
Use a Venn diagram or matrix to compare exponential relations with
either linear relations or quadratic relations. Include an example of
each type of relation and a real-world situation you can model with it.
Explain the reason for your choice of graphic organizer.
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Exponential Model for a Trend
Researched data can sometimes be modelled using an exponential relation.
You can use the model to interpret the data and make predictions.
Analysing Population Trends
Inquire
In this activity, you will locate data on Canada’s population,
and analyse trends in the data.
You will need a computer with Internet access, grid paper,
and a TI-83 or TI-84 graphing calculator.
1. Retrieving the data
a)
Copy the table.
Continue from 1921 in steps of 10 years until 2001.
Canada’s Population (in thousands)
Year
Actual population
1851
2437
1861
3230
1871
3689
1881
4325
1891
4833
1901
5371
1911
7201
Population predicted by exponential model
1921
1931
Go to www. statcan.ca. Click English. Select Learning Resources from the menu on the
left. Click E-STAT in the yellow box on the right. Then click Accept and Enter.
If you are working from home, you will need to enter the user name and password
assigned to your school. You should see a tale of contents on your screen.
c) Find the People heading. Click on Population and demography.
Under the CANSIM heading, click on Population estimates and projections.
d) Scroll down to the Terminated tables heading,
and click on Table 051-0026.
b)
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e)
f)
g)
h)
i)
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Under Geography, select Canada.
Under Sex, select Both sexes.
Under Age group, select All ages.
Under Reference period,
select From: 1921 To: 1971.
Click Retrieve as individual Time Series.
On the next screen, find SCREEN OUTPUT formats.
Under HTML, table:, select Time as rows.
Click Retrieve Now.
Record the population of Canada every 10 years
from 1921 to 1971 in the Actual population
column in your table.
Click on the Table of contents link under E-STAT
on the navigation bar at the left of the page.
Repeat the steps in part c.
Under Active tables, click on Table 051-0001.
Repeat the steps in part e, except for the Reference
period, which should be set From: 1981 To: 2001.
Record the total population of Canada, in thousands,
for the years 1981, 1991, and 2001 in the column
labelled Actual population in your table.
These data are in
thousands, so you can
copy them directly into
your table.
These data are actual
values, so you must
divide them by 1000
before copying them
into your table.
2. Graphing the data
Graph the data in the first two columns
of your table to make a full page plot
of Canada’s population from 1851 to 2001.
Describe the growth in Canada’s population.
3. Analysing the data
The population of Canada has been growing at an
average of 1.71% per year since 1851. We can use the equation
P 2437(1.0171)n to model the population, P thousand,
where n is the number of years after 1851.
a)
Press o on your
graphing calculator.
Enter the equation
shown at the right.
Explain how the equation
P 2437(1.0171)n models
Canada’s population since
1851.
4.10 Exponential Model for a Trend
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b)
Press y p to
access the TABLE
SETUP editor. Enter the
values shown.
What do these values mean?
c)
Press y s to
show a table of values
for the equation you
entered.
What is the value of y when
x 60?
In the table, the values in the first column represent years since 1851.
So 0 represents 1851, 10 represents 1861, 20 represents 1871, and so on.
The values in the second column represent the population.
Enter these values in the Population predicted by
exponential model column of your table.
e) Plot these new points on your graph, joining
them with a smooth curve. Label the two curves.
f) Suppose you wanted to predict the population at
times between the years for which you have data.
How might you do this? How much confidence
will you have in your predictions? Explain.
g) Suppose you wanted to predict the population at
times after the years for which you have data.
How might you do this? How much confidence
will you have in your predictions? Explain.
d)
Reflect
➢ Is an exponential relation a good model for your data?
Justify your answer.
➢ How could you use the data to make predictions?
Do you think it is easier to make predictions using an equation
or a graph? Explain.
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Chapter Review
What Do I Need To Know?
bn 14
b 4
b4424
b 4
443
b
Exponents that are positive integers represent
repeated multiplication by the same number.
n factors
Zero and Negative Exponents
Zero exponent
Exponent Rules
b0 1
Negative exponent bn 1
bn
, (b ≠ 0)
Multiplication
bm bn bm n
Division
bm bn bm n, (b ≠ 0)
(bm)n bmn
Power of a Power
Exponential growth is a pattern of increase involving repeated multiplication
by a number greater than 1. The number is the growth factor.
There are 100 bacteria in a Petri dish. The number of bacteria doubles every hour.
Table of Values
Graph
Number of
bacteria (N)
0
100
1
200
2
400
3
800
Growth of Bacteria
Number of
bacteria
Time (hours)
(t)
2
2
2
Equation
N 100 2t
800
Initial
value
400
0
Growth factor
(greater than 1)
1
2
3
Time (h)
Exponential decay is a pattern of decrease involving repeated multiplication
by a number between 0 and 1. The number is the decay factor.
A patient takes 250 mg of penicillin. At the end of each hour, only 60% of the penicillin present at the
end of the previous hour remains in the blood stream.
Table of Values
Time (h),
represented
by t
Amount of
penicillin (mg) ,
represented by P
0
250
1
150
2
90
3
54
Graph
Elimination of Penicillin
from the Body
0.6
0.6
0.6
Amount of
penicillin (mg)
Chapter 04
Equation
P 250 0.6t
Initial
value
300
Decay factor
(between 0 and 1)
200
100
0
3
1
2
Time (h)
Chapter Review
199
Chapter 04
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What Should I Be Able to Do?
4.1
1. Evaluate without using a calculator.
a)
34
b)
25
c)
106
d)
(5.2)1
4.4
9. Evaluate without a calculator.
a)
23
b)
1.030
c)
51
10. Evaluate with a calculator.
2. Evaluate.
a)
(2.56)11
b)
3.2
1.063
c)
Round to the nearest hundredth.
a) (1.73)2
b) 2.6(1.05)0
c) 15.3(1.4)12
220
3. A population of 30 rabbits is released
in a nature park. The population doubles
each year for 4 years.
a) Model the growth using a table
of values, a graph, and an equation.
b) What is the population after 4 years?
4.2
11. Simplify using the exponent rules, then
evaluate.
a)
( 15 )
3
b)
( 47 )
2
c)
can be modelled by the equation
P 10.1(1.0125)x, where x is the
number of years since 1991.
Determine the population in
each year.
a) 2000
b) 1991
c) 1988
( 101 )
6
5. a) Graph the equation y 400(0.7)n
b)
for n 1, 2, 3, and 4.
What could this equation represent?
Pose a problem that could be solved
by the equation and solve it.
4.5 13. a) Use a table of values to graph
y 5x, for x from 2 to 2.
b) Describe the graph in as much
detail as you can.
6. A ball is dropped from a height of
300 cm. On each bounce, the ball
rises to 50% of its previous height.
a) Model the decay with a table of
values, a graph, and an equation.
b) What is the height of the ball, to the
nearest centimetre, after the 5th
bounce?
4.3
14. a) Graph y 2x and y 1 x on the
(2)
same set of axes for x from 3 to 3.
b) Compare the two graphs. How are
they similar? How are they different?
15. Which of the following relations are
exponential? How do you know?
a) y 7(3)x
b) y 7 x 3
c) y 3 x7
d) y 2(0.3)x
7. Simplify using the exponent rules.
731 712
c) 514 59
a)
b)
d)
(67)10
32 37
34
8. When simplifying expressions with
powers, when do you multiply exponents
and when do you add them?
Explain using examples.
200
CHAPTER 4: Exponential Relations
240
b) 218 222
12. The population, P million, of Ontario
4. Evaluate without a calculator.
a)
175
(172)3
4.6
16. The equation V 15 000(1.08)t models
the value, V dollars, of an antique car t
years after it was purchased.
a) Graph the equation. Use values of t
from 0 to 10 in steps of 2 years.
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Use the graph to estimate the number
of years it takes for the value of the
car to double.
4.8 19. The temperature of a bowl of soup
4.9
changes with time.
Cooling Curve of
Bowl of Soup
17. The graph shows the population of
100
Temperature (°C)
Smalltown during the 1990s.
Population of Smalltown
Population (thousands)
10
8
6
80
60
40
20
0
10 20 30 40
Time (min)
4
2
How much does the temperature
drop in the first 10 min?
How much does in drop in the next
10 min?
b) How would the graph change if the
soup had been placed in an insulated
storage container?
a)
1988
1992
1996
Year
2000
Estimate the population in 1990 and
in 1994. By what factor did the
population increase in this time?
b) Repeat part a for two other dates
separated by 4 years.
c) Estimate the doubling time of the
population. Explain your answer.
a)
20. Classify each relation as linear, quadratic,
or exponential.
a) y 5x2
b) y 5x
y 5x
c)
4.10 21. The graph shows how the population
of Alberta has changed over time.
Describe the trends in the graph.
4.7 18. The table shows how the number of
frogs in a conservation area decreased
over time.
Population of Alberta
Year
0
1
2
3
4
Number of Frogs
60
48
38
31
25
Describe the trends in the data that
suggest the relation is exponential.
b) Determine the decay factor
c) Estimate the time it takes for the
number of frogs to be half the original
number of frogs.
a)
3000
Population (thousands)
Chapter 04
2500
2000
1500
1000
500
1920
1940
1960 1980
Year
Chapter Review
2000
201
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Practice Test
Multiple Choice: Choose the correct answers for questions 1 and 2. Justify each choice.
1. Substituting n 1, 2, 3, and 4 in the expression 3n gives:
A.
1, 2, 3, 4
B.
3, 6, 9, 12
C.
3, 9, 27, 81
D.
2. Which type of relation best fits the graph?
Exponential
C. Linear
A.
What tools can
you select?
1 1 1 1
, , ,
3 9 27 81
16
Quadratic
D. None of the above
y
B.
12
8
Show your work for questions 3 to 8.
4
x
3. Knowledge and Understanding Write each expression as a whole
number or a fraction.
a) 5 23
b) 43
c)
( 25 )4
d)
0
1
2
3
523 519
(513)3
4. Communication British Columbia’s population, P million, can be modelled
by the equation P 2.9(1.0257)t, where t is the number of years after 1983.
a) Determine the population of British Columbia in 1980, 1983, and 1986.
b) Use your answers to explain the meaning of an exponent
that is zero or negative.
Intensity of Light
through Tinted Glass
5. Application Tinted glass is often used in shower stalls.
The graph at the right shows the percent, P, of light that passes
through tinted glass of varying thickness.
a) About what percent of the light passes through tinted
glass 3 cm thick?
b) Which thickness of glass allows 50% of the light to
pass through?
100
Light passing
through glass (%)
Chapter 04
80
60
40
20
6. Thinking The speed of a model rocket increases
exponentially with time.
At 0 s, the speed of the rocket is 2.0 m/s.
The growth factor for each 1-s interval is 1.07.
a) Write an equation that models the rocket’s speed s at time t.
b) Draw a graph to show the rocket’s speed every second from 0 s to 5 s.
c) Explain how the graph in part b would change in each scenario.
i) The speed at the beginning of the second stage is 5 m/s.
ii) The growth factor for each 1-s interval is 1.17.
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CHAPTER 4: Exponential Relations
0
1
2
3
4
5
Glass thickness (cm)