McGrawHill Grade 11 Functions Unit 3

147

graph an exponential relation, given its equation in the form y = ax (a > 0, a ≠ 1), define this relation as the function f (x) = ax, and explain why it is a function

determine the value of a power with a rational exponent

simplify algebraic expressions containing integer and rational exponents and evaluate numerical expressions containing integer and rational exponents and rational bases

determine and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes for exponential functions represented in a variety of ways

distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways

determine and describe the roles of the parameters a, k, d, and c in functions of the form y = af [k(x — d )] + c in terms of transformations on the graph of f (x) = ax (a > 0, a ≠ 1)

sketch graphs of y = af [k(x — d )] + c by applying one or more transformations to the graph of f (x) = ax (a > 0, a ≠ 1), and state the domain and range of the transformed functions

determine that the equation of a given exponential function can be expressed using different bases

represent an exponential function with an equation, given its graph or its properties

collect data that can be modelled as an exponential function, from primary sources, using a variety of tools, or from secondary sources, and graph the data

identify exponential functions, including those that arise from real-world applications involving growth and decay, given various representations, and explain any restrictions that the context places on the domain and range

solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications by interpreting the graphs or by substituting values for the exponent into the equations

Exponential

Functions

CHAPTER

3

By the end of this chapter, you will

Refer to the Prerequisite Skills Appendix on pages 478 to 495 for examples of the topics and further practice.

Exponent Rules

1.

Rulea) xa xb

b) xa

xb

c) xa b

Method

A xa b

B xa b

C xa b

2.

3.

a) x x b) y y y

c) m m d) hh

e) a a b b f) x yx y

g) ab c h) uv

i) ab j) w

r

4.

a) b)

c) d)

e) f)

g) h)

Zero and Negative Exponents

5. y x

a)

x y

4 24 = 16

3 23 = 8

2 22 =

1 21 =

0 20 =

b) y

c)

x y

4 24 = 16

3 23 = 8

2 22 =

1 21 =

0 20 =

—1 2—1 = 1

2 = 1

21

—2 2—2 = 1

4 =

—3 2—3 =

—n 2—n =

6.

a) b)

c) d)

e) f)

g) h)

7.

a) x x b) y

c) u v u v

d) a b

148 MHR • Functions 11 • Chapter 3

Prerequisite Skills

Graph Functions

8.

i)

ii)

iii) x y

a)

2

y

x42 860—4 —2—8 —6

4

b)

2

—4

y

x42 860—4 —2—6

4

—2

—8

9.

i)

ii)

iii) x y

a) y x b) y x

Transformations of Functions

10. y x

2

—4

—2

x420—4 —2

4

y

11. y x

a) y x b) y x

c) y x d) y x

e) y x f) y x

Chapter Problem

Prerequisite Skills • MHR 149

Investigate

How can you discover the nature of exponential growth?

Pattern 3

Pattern 1 Pattern 2

1.

2.

a)

b)

3. a)

b)

The Nature of Exponential Growth


3.1

Tools

• coloured tiles or linking cubes

• grid paper

or

• graphing calculator

or

• computer with The Geometer’s Sketchpad®

4. a)

Pattern 1

Term number,

n

Number of Squares,

t

1 2

2 4

3 6

4

5

b)

c)

d) tn

e)

5.

6. Reflect

a)

b)

c)

Example 1

Model Exponential Growth

a)

b)

c)

d)

i)

ii)

e)

3.1 The Nature of Exponential Growth • MHR 151

First Differences

2

2

Second Differences

0


Solution

a)

Day Population

0 100

1 100 3 = 300

2 300 3 = 900

3 900 3 = 2700

4 2700 3 = 8100

b)

Day Population

0 100

1 100 31 = 300

2 100 32 = 900

3 100 33 = 2700

4 100 34 = 8100

n 100 3n

pn p n n

c) p n n

d) n

p n n

i) n

p

ii) n

p

The initial population is 100.

The population triples each day.

After 1 day, the initial population triples.

After 2 days, the initial population triples again.

After n days, the initial population has tripled n times.

5400

7200

p

3600

1800

n20 4

9000

p(n) = 100 3n

e)

Day Population

0 100

1 300

2 900

3 2700

4 8100

exponential growth

Example 2

Apply a Zero Exponent in a Model

Solution

Day Population

4 100 34 = 8100

3 100 33 = 2700

2 100 32 = 900

1 100 31 = 300

0 100 30 = 100

y x y

x y

Technology TipYou can use a graphing calculator to calculate finite differences.

Refer to the Technology Appendix on pages 496 to 516.

exponential growth pattern of growth in

which each term is multiplied by a constant amount (greater than one) to produce the next term

produces a graph that increases at a constantly increasing rate

has a repeating pattern of finite differences: the ratio of successive finite differences is constant

y

x0

Technology TipYou can use a graphing calculator to view and analyse this function. To find the y-intercept:

Press 2nd [CALC].

Select 1:value and enter 0 to identify the y-intercept.

For this pattern to be extended, 30 must equal 1, because 100 1 = 100.


First Differences

300 — 100 = 200

900 — 300 = 600

2700 — 900 = 1800

8100 — 2700 = 5400

Second Differences

600 — 200 = 400

1800 — 600 = 1200

5400 — 1800 = 3600


Key Concepts

b b b

y

x0

Communicate Your Understanding

C1

a)

A p n n

B p n n

C p n n

b)

C2

y x y x y x

a)

b)

C3

y x y x y x

C4

a)

b)

c)

C5

A Practise

For help with question 1, refer to Example 1.

1.

a)

Day Population

0 20

1 80

2

3

4

5

b)

c)

d)

e)

For help with questions 2 to 4, refer to Example 2.

2.

3. a) aa

b)

c) aa

d)

4.

a) b) c) d) x

B Connect and Apply 5. a)

t n n nt n

b) t

c)

i) ii)

d)

e)

6. Use Technology

a)y x

i)

ii)

iii)

b)

c) y x


Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating

First

Differences Second

Differences


7.

a)

i) ii)

b)

c)

8.

9.

a)

p t

b)

c)

d)

10.

A

A P i n Pi

n

a)

b)

Number of

Compounding

Periods

(years) Amount ($)

0 100

1 105

2

3

4

c)

d)

e)

f)

Connecting

Problem Solving


Reflecting


Communicating

A = 100(1 + 0.05)1

= 100(1.05)1

= 105

11.

a) b)

12.

a)

b)

I Prt IP r

t

C Extend 13.

14.

a)

b)

15.

a)

b)

16. Math Contest

A B C D

17. Math Contest

A B C D

18. Math Contestx y

z

x zy

x y z

A B C D

19. Math Contest

20. Math Contest

n n


Use Technology


Use the Lists and Trace Features on a TI-NspireTM CAS Graphing

Calculator

A: Calculate First and Second Differences for the Population Data in Example 1 on pages 151 to 153

1. Lists & Spreadsheet

2.

day ·

pop ·

3.

first_diff

k

List ·

Operations·

Difference List·

pop·

Tools

• TI-NspireTM CAS graphing calculator

B: Calculate the Population Data in Example 2 on page 153

1. Lists & Spreadsheet

2.

day

pop

·

C: Trace the Population Function in Example 2

1. Graphs & Geometry

2.

f1 x ·

b 4:Window 1:Window Settings

x y

b 5:Trace 1:Graph Trace

Use Technology: Use the Lists and Trace Features on a TI-Nspire™ CAS Graphing Calculator • MHR 159


Exponential Decay: Connecting to Negative Exponents

Investigate

How can you explore exponential decay?

half-life

1.

Time

(years)

Number of Half-Life Periods

(2 years)

Amount of U-239 Remaining

(mg)

0 0 1000

2 1 500

4 2

6 3

8 4

2. a)

b)

•

3.2

Tools


half-life length of time for an

unstable element to spontaneously decay to one half its original amount

Connections

You will learn more about nuclear fission if you study physics in high school or university.Visit the Functions 11 page on the McGraw-Hill Ryerson Web site and follow the links to Chapter 3 to find out more about CANDU reactors.

3. a)

i) ii)

b)

4. a)

• STAT EDIT

• L1 L2

• 2nd Plot1

•

• ZOOM 9:ZoomStat

b) Reflect

5. a)

b)

6. a)

• STAT

CALC

• 0:ExpReg

• 2nd , 2nd , VARS

Y-VARS 1:Function

• 1:Y1 ENTER

b) a by abx

c) Reflect a b

7. a) GRAPH

b)

8. Reflect Zoom Trace

3.2 Exponential Decay: Connecting to Negative Exponents • MHR 161

Technology Tip

Press 2nd MODE for [QUIT] to return to the home screen from other screens.


exponential decay

Example 1

Model Exponential Decay

Solution

Method 1: Use a Table or a Spreadsheet

Enter

exponential decay a pattern of decay in

which each term is multiplied by a constant fraction between zero and one to produce the next term

produces a graph that decreases at a constantly decreasing rate

has a repeating exponential pattern of finite differences: the ratio of successive finite differences is constant

y

x0


Method 2: Use Systematic Trial

A nn

n A

A n

A nn

n

n

n

n 0.5n Mathematical Reasoning

5 0.03125 Try 5 half-lives. This gives a value much lower than 0.1. Try a

shorter period of time.

2 0.25 Too high. The correct value is between 2 and 5. Try 3 next.

3 0.125 Close, but high.

3.3 0.1015… This is very close to 0.1. This will give a reasonable approximation.

Method 3: Use a Graphical Model

A nn

Trace

Divide both sides by 50.

Connections

You will learn how to find exact solutions to equations such as

5 = 50 1

2 n

when you

study logarithms in grade 12.


A nn

n

b n

bn b b n

Example 2

Evaluate Expressions Involving Negative Exponents

a) b) c) d)

Solution

a)

b)

c)

d)

Apply the product rule. Add the exponents.

Apply the power of a power rule first. Multiply exponents.

Apply the quotient rule. Subtract exponents.

Example 3

Simplify Expressions Involving Negative Exponents

a) x x x b)a ba b

c) u v

Solution

a) x x x x x –

x

b) a ba b

a b

a b

ab

c) u v u v

u v

vu

Example 4

Evaluate Powers Involving Fractional Bases

a) b)

Solution

a)

b)

Apply the product rule.

Apply the quotient rule.

Apply the power of a power rule.

Write with a positive exponent.

Multiply by the reciprocal.

Apply the property of negative exponents.


Connections

The result in Example 4 part a) can be generalized as follows.

a b

n

b a

n

for

a, b , a, b 0, and n .


Key Concepts

b n

bn b b n

a b

n b a n

a b a b n

y

x0


C1

C2 A nn

A

n

A n n

a)

b)

C3

C4

A Practise


1.

a) b) x c) y

d) ab e) x f) x

2.

a) b) k

3.

a) b) c)

d) e) f)

4.

a) b)

c) d)

e) f)

g) h)


5.

a) m m b) v v

c) p p d) ww

e) k f) ab

For help with questions 6 and 7, refer to Example 4.

6.

a) b) c)

d) e) f)

7.

a)ab

b) u

c)gw

d) ab

e) ab

f) x y

B Connect and Apply 8.

a)

b) f x abx

c)

d)

e)

f)

9. a)

i) ii)

b)

c)

d)

e)

10.

11.


Connecting

Problem Solving


Reflecting


Communicating


12.

v tv t t

a)

b)

i) ii)

c)

d)

13.

a)

b)

i)

ii)

14.

a b

n b a n

a b

a b n

15. A P i n

P A i n

Achievement Check

16. P n A i n

P n

A

in

a)

i)

ii)

b)

c) A

17.

18. Chapter Problem

Connecting

Problem Solving


Reflecting


Communicating

F GMmr

GM m

r

Mm

F

C Extend

19.

a) F GMmr

b)

c)

20.

a)

b) r

21. Math Contest f x x

f x f x

A B C D

22. Math Contest y

yx

A B C D

23. Math Contest

x xx xx xx xx x x

x x

A B C D


Career Connection


3.3

Rational Exponents

Investigate

How can you find the meaning of a rational exponent?

1.

2.

a) P

b)

i)

ii)

c)

3. a)

b) t P t

1200

1600

2000

P

800

400

t20 4

P(t) = 100(4)t

Technology TipIf you use a graphing calculator, you can use the Zoom and Trace operations to find the coordinates of points with greater accuracy.

4. a)

b)

c) Reflect

5. a)

b) Reflect

6. a) Reflect

b)

c)

b b b b

Example 1

Powers With Rational Exponents of the Form 1

n

a) b) c)

d) e)

Solution

a)

3.3 Rational Exponents • MHR 171

The cube of 8 1

3 is equal to 8, so 8 1

3 is equal to the cube root of 8.

Connections

3

__

8 is read as “the cube root of eight.”

4

___

16 is read as “the fourth root of sixteen.”

The terminology used for this type of expression is as follows:

4

___

16 is called a radical. 16 is the radicand. 4 is the index.


b)

c)

d)

e)

Example 2

Powers With Rational Exponents of the Form m

n

a) b) c)

Solution

a)

b)

Think: What number raised to the exponent 5 produces —32?

5

____

—32 = —2, because (—2)(—2)(—2)(—2)(—2) = —32.

4

____

16

81 = 2

3 , because 2

3 4

= 2

3 2

3 2

3 2

3

= 2 2 2 2

3 3 3 3

= 16

81

Rewrite the power with a positive exponent.

Evaluate the cube root in the denominator.

Express the exponent as a product: 2

3 = 1

3 2

Use the power of a power rule.

Write 8 1

3 as a cube root.

Evaluate the fourth root of 81.

Connections

The result in Example 1a) can be generalized as follows.

b 1

n = n

__

b for any n . If n is even, b must be greater than or equal to 0.


c)

bm n

n b

m

b m

n b n

Example 3

Apply Exponent Rules

a) x x

xb) y y c) x x

Solution

a) x x

x

x

x

x

x

x

b) y y y y

y y

y

y

y

c) x x x x

x

Rewrite the power with a positive exponent.


Apply the quotient rule.


Write the exponents with common denominators.


Express with a positive exponent.

Connections

Note that 1

y

1 6 can also

be expressed as 1

6

__ y .




Example 4

Solve a Problem Involving a Rational Exponent

V S

V Sπ

S

Solution

S V

Vπ

π

V

Key Concepts

nn

b n bn n b n b

mn n m

b bm n

n b

m

m n n b


C1 a)

b)

C2 a)

i)

ii)

b)

C3

Step Explanation

x x x x

x

x

x

x

x

A Practise


1.

a) b)

c) d)

2.

a) b)

c) d)


3.

a) b)

c) d)

4.

a) b) c)

d) e) f)


5.

a) x x b) m m c) w

w

d) ab

a be) y f) u v

6.

a) k k b) p

p

c) y d) w

e) x x f) y




7. SV

S V π V


a) A

b) A

c)

i) ii) iii)

d)S

e)

i) ii) iii)

9.

a) V

b) V

c)

i) ii) iii)

10.

a)

A S V B S V

C V S D V S

b)

c)

11. a)

i) ii)

b)

c)

i) ii)

d)

12. Chapter Problem

Tr

Connecting

Problem Solving


Reflecting


Communicating

Connections

The square-cube law is important in many areas of science. It can be used to explain why elephants have a maximum size and why giant insects cannot possibly exist, except in science fiction! Why do you think this is so? Go to the Functions 11 page on the McGraw-Hill Ryerson Web site and follow the links to Chapter 3 to find out more about the square-cube law and its impact on humans and other creatures.

T kr k

T

a) k

b)

c)

13.

a)

T kr r T

b)

Achievement Check

14. h m r m

hr

m

a)

i)

ii)

iii)

b)

c) B m

B

C Extend 15. a)

b)

16.

a)

b)

c)

d)

17. PV

P V

a) V

b)

c)

d)

18. Math Contesta b b a

A B

C D

19. Math Contest


Connections

Johannes Kepler lived in the late 16th and early 17th centuries in Europe. He made some of science’s most important discoveries about planetary motion.

Connecting

Problem Solving


Reflecting


Communicating


Properties of Exponential Functions

f x abx ab b a b

Investigate

How can you discover the characteristics of the graph of an exponential function?

A: The Effect of b on the Graph of y = abx

f x x a

1. a)

b)

2.

a)

b) y

c) x

d) interval

i)

ii)

3. b

a) y x

b) b

c) b

Tools

• computer with The Geometer’s Sketchpad®

or

• grid paper

or


Connections

It is important to be careful around discarded electrical equipment, such as television sets. Even if the device is not connected to a power source, stored electrical energy may be present in the capacitors.

3.4

interval an unbroken part of the

real number line

is either all of or has one of the following forms: x < a, x > a, x ≤ a, x ≥ a, a < x < b, a ≤ x ≤ b, a < x ≤ b, a ≤ x < b, where a, b , and a < b

3.4 Properties of Exponential Functions • MHR 179

The Geometer’s Sketchpad b

Graph New Parameter bOK

Graph Plot New Functionb x OK

b

bb

b Edit Parameter

b Animate ParameterMotion Controller b

4. b

a)

b) b

c) b

5. b

6. Reflect bf x bx

B: The Effect of a on the Graph of y = abx

f x a x ba

1. a f x x

a) a

b) a

c) a

2. Reflect af x a x

x ya

Connections

In Section 3.3, you saw how to deal with rational exponents. The definition of an exponent can be extended to include real numbers, and so the domain of a function like f (x) = 2x is all real numbers. Try to evaluate 2π using a calculator.

Technology TipWith a graphing calculator, you can vary the line style to see multiple graphs simultaneously. Refer to the Technology Appendix on pages 496 to 516.


f x x

xyx x

Example 1

Analyse the Graph of an Exponential Function

x y

a) yx

b) y x

Solution

a) yx

Method 1: Use a Table of Values

xy

x y

—2 16

—1 8

0 4

1 2

2 1

3 1

2

4 1

4

Connections

You encountered horizontal and vertical asymptotes in Chapter 1 Functions.

4 1

2 —2

= 4 2

1 2

= 16

x

12

16

20

—4

y

8

4

2—2 0 4

y = 4( )x12


Method 2: Use a Graphing Calculator

xx

y yy y

xx

y y

yx

x yx x y

b) y x

x

y y

x

y

x

x y

Connections

Could you use transformations to quickly sketch the graph of y = —3—x ? You will explore this option in Section 3.5.


Example 2

Write an Exponential Equation Given Its Graph

y abx

Solution

x y

x y

0 2

1 6

2 18

3 54

by abx

b a

x y b

y abx

a

a

a

y x

x

18

24

30

36

42

48

54y

12

6

2—2 0 4

x

18

24

30

36

42

48

54y

12

6

2—2 0 4

(3, 54)

(2, 18)

(1, 6)

(0, 2)

Change in y

3

3

3


Example 3

Write an Exponential Function Given Its Properties

a)

b)

Solution

a)

A x Ax

x A

Ax

Start with 200 mg. After every half-life, the amount is reduced by half.

A A xx

A x

A t x t

The half-life of this material is 3 days. Therefore, the number of elapsed half-lives at any given point is the number of days divided by 3.

A tt

At

b)

t

t

A tt

t t

t

120

160

280

80

40

20 4

200

240

A(t) = 200( )12

t3

A

120

160

280

A

80

40

t20 4

200

240



C1 a)

b) y abx x

C2 f xx

g x x

a)

i) y y ii) y y

b)

i) ii)

C3

Key Concepts

y abx

y abx a b

x y y

y y a

y abx a b

x y y

y y a

y

0 x

x

y

0 x

y

0 x

y

0

ab

ab

ab

ab


A Practise


1.

a)

x

4

—8

—4

420—4 —2

8

y

b)

2

—4

—2

x42—4

4

y

0—2

c)

x

4

—8

—4

42—4

8

y

0—2

d)

x

4

—8

—4

42—4

8

y

0—2

A y x

B yx

C y x

D y x

2. a)

x

y y

y

b)

3. a)

x

y y

y

b)


4.

x

12

42—4

16

4 (0, 4)

(1, 8)

(2, 16)

8

y

0—2


5.


6.

a)t

A

A At

B At

C At

D At

b)


i)

ii)

iii) x y

iv)

v)

a) f xx

b) y x

c) yx

8. a)

i) f x x

ii) r x x

b)

c)

9. a)

i) g xx

ii) r x x

b)

c)

10. a)

i) f x x ii) g xx

b)

c)

11.

t

a)

b)

c)

d)

e)

24

—8

y

16

8

2—2 0 4

32

48

56

40

x

(0, 24)

(1, 12)(2, 6)

Connecting

Problem Solving


Reflecting


Communicating

2

3

V

1

t30 21Time (milliseconds)

Vo

lta

ge

Dro

p (

V)


12.

R

tR t t

a)

b)

c)

d)

i) ii)

C Extend 13. Use Technology

V V b t

RC V

V tR

C

R C

a) b

b)

c)

d)

e)

14.

a)

b)

c)

15.

a)

b)

c)

d)

16. Math Contestx x

17. Math Contest

yx

y b

x a a b

A B C D

18. Math Contest

A B

C D

Connecting

Problem Solving


Reflecting


Communicating


Transformations of Exponential Functions

f x f x c f x f x d

f x af x f x f kx

Investigate

How can you transform an exponential function?

1. a)y x y x

b)

c)

i) y x y x

ii)

d)

2. a) y x y x

b)

c)

3. a) y x

y x

b)

Tools

• computer with The Geometer’s Sketchpad

or


3.5

3.5 Transformations of Exponential Functions • MHR 189

4. a) y x y x

b)

c)

5.

6. Reflect

7. a) Reflect y x

b)

Example 1

Horizontal and Vertical Translations

y x

a) y x

b) y x

c) y x

Solution

a) y x

y x

y x

x

y y y

2

—2

x42—4

4

y

0—2

—4

y = 3x

y = 3x — 4


b) y x

y x

y x

y

x

y y

c) y x

y x

y

x

y y

Example 2

Stretches, Compressions, and Reflections

y x

a) y x

b) y x

c) y x

2

—2

x42—4

4

y

0—2

—4

y = 3xy = 3x — 2

6

2

4

x42—4

8

y

0—2

y = 3x

y = 3x + 1 + 3

(0, 1)

(—1, 4)


Solution

a) y x

y x

b) y x

y x

y x

c) y x

y x

y x

x

12

4

8

x42—4

16

y

0—2

y = 4x

y = 2(4x)

8

—16

—8

x

y = 2(4x)

y = —2(4x)

42—4

16

y

0—2

y x

x

y

12

4

8

x

y = 4x

42—4

16

y

0—2

y = 412 x

12

4

8

x42—4

16

y

0—2

y = 412 xy = 4

12 x—

y


Example 3

Graph y = abk(x — d) + c

y x

Solution

y x

y x

y x

y x

Example 4

Circuit Analysis

I tI t t

a)

b) i) ii)

c)

x

Connections

You explored compound transformations of functions in Chapter 2.

12

4

8

x42

16

y

0—2

y = 22x

y = 2x

8

—16

—8

x

y = 22x

y = —22x

42

16

y

0—2

xy x

8

—16

—8

x

y = —22x

y = —22(x — 3) + 5

42

16

y

0—2


Solution


a)

t t

b) i)

2nd 1:value

ENTER

t

ii)

t

c) t t

Method 2: Use The Geometer’s Sketchpad®

a) GraphPlot New Function

OK

GraphGrid FormRectangular Grid

t t


b) i) It

Construct Point On Function Plot

Measure Coordinates

tI

t

ii)

t

c)

t t

Key Concepts

y abk x d cy bx

Horizontal and Vertical Translations

d dd

c cc

Vertical Stretches, Compressions, and Reflections

a a

a a

a xx

yy = abx, a = 1

a = —1

0 < a < 1

—1 < a < 00

a > 1

a < —1

0 x

y

y = bx + c, c = 0

c < 0

c > 0

x

y

d > 0

d < 0

y = bx — d, d = 0

0


Key Concepts

Horizontal Stretches, Compressions, and Reflections

kk

kk

k y

y x y x

x

yy = bkx, k = 1k = —1

0 < k < 1—1 < k < 0

k > 1k < —1

0


C1 y x

Transformation

a)

b)

c)

d)

e)

f)

g) x

C2

C3 a) y x

b) y x

C4 a k c d y bx

y abk x d c

C5

a)

b)

A y = x

B y = x +

C y = – x

D y = x

E y = x

F y = x

G y = x


A Practise


1. y x

a) y x b) y x

c) y x d) y x

2. y x

3.

y x

a)

b)

c)

d)


4. y x

a) y x b) y x

c) y x d) y x

5. y x

6.

y x

a) x

b)

c)

d) y

For help with question 7 and 8, refer to Example 3.

7. y x

y x

8. y x

y x

B Connect and Apply


9. Tt

T t t

a)

b)

c)

10. a) f x x

b)

i)

ii)

iii)

11. a) y x

b) y x

y x

c) y x

y x

d)

12. a) y x

b)

Connecting

Problem Solving


Reflecting


Communicating

13. a)

y y

b)

14. a)

b)

c)

15.

a)

y bx

b)

c)

d)

e)

f)

16.

Term 1 Term 2 Term 3

a)

b)

c)s

n

d)

i)

ii)

17.

a)n

t

b)

c)

d) tn

e) tn

f)

18. Use Technologya

a

a)

b)a

c)


Connecting

Problem Solving


Reflecting


Communicating


19. Use Technology

d

a)

b)d

c)

Achievement Check

20. a) y x

yx

b) yx

c)

i)

ii)

iii)

iv)

C Extend 21.

a)

b)

c)

d)

22. Use Technology

yx x

a)

b)

c)

d)

e)

f)

23. Math Contest f x x

f x f x f x kf xk

24. Math Contest y

y

A B C D

25. Math Contest

yx

26. Math Contest

A B C D

A B C D

3.6 Making Connections: Tools and Strategies for Applying Exponential Models • MHR 199

3.6

Making Connections: Tools and Strategies for Applying Exponential Models

Example 1

Construct a Model of Exponential Growth

a)

b)

c)

Solution


a)

STAT 1:Edit

L1 L2

Y=

Plot1

2nd Plot1

Year Earnings ($)

2002 679

2003 688

2004 702

2005 725

2006 747


ZOOM 9:ZoomStat

2nd

STAT CALC0:ExpReg

2nd , 2nd ,

VARS

Y-VARS 1:Function

1:Y1 ENTER

a by abx

y x

x yn E

nE n n

GRAPH

b) E

n

2nd

1:value

c)

y


2nd

5:intersect

Method 2: Use Fathom™

a)

y abx

a b

a

b

b


Graph Plot Function

OK

n

E

n

E n n

b) n

E n n

E

c)

Graph Moveable Line

Technology TipYou can find parameters a and b under the Global Values heading and Year under the Attributes heading.

Double-click on these headings to show or hide the list of values.

Double-click on the value to enter it into the equation.


Example 2

Choose a Model of Depreciation

n

a)

b)

c)

Solution

a) L1 L2 Table Editor

L3 2nd OPS 7: List( 2nd

) ENTER

Number of Years, n Value ($)

1 1200

2 960

3 768

4 614

5 492

6 393


b) Linear Model

Y1

v n n v

n

Plot1 Y1

ZOOM 9:ZoomStat

Quadratic Model

Y2

v n n n v

n

Y1Plot1 Y2 ZOOM

9:ZoomStat

ZOOM 3:Zoom Out ENTER

Technology Tip To perform a regression, press

STAT , cursor over to CALC, select the desired type of regression, and enter the appropriate data lists.

To store the function, press VARS , select Y-VARS, and select the desired function location.

Remember to use commas to separate data lists and function variables.

Connections

You may see values of r and/or r2 after performing a regression. These values are related to how well a regression line or curve fits the data. The closer r is to 1 or —1, and the closer r2 is to 1, the better the fit. To enable the viewing of these parameters, press 2nd STAT 0 for [CATALOG] and then scroll down and select Diagnostics On. You will learn more about these parameters if you take Mathematics of Data Management in grade 12.


Exponential Model

Y3

v n n v

n

Y1 Y2

Plot1 Y3 ZOOM

9:ZoomStat

ZOOM

3:Zoom Out ENTER

c) n

Method 1: Use the Graph

2nd

1:value

ENTER

Method 2: Use the Equation

v n n

v


Key Concepts

Fathom


C1

a)

x

y

0

b)

x

y

0

c)

x

y

0

d)

x

y

0

C2 a)

b)

C3 a)

i)

ii)

b)


A Practise


1.

a)

12

16

20

—4

y

8

4

x2—2 0 4

b)

12

16

20

—4

y

8

4

x2—2 0 4

c)

12

16

20

—4

y

8

4

x2—2 0 4

2.

3.

95100105110115120

130135

125

V

t30

(0, 100)

(1, 110)

(2, 121)

(3, 133.1)

21Time (years)

Va

lue

($

)

a)

b) a b

V n a bn

c)

d)

e)

A y x

B y x

C y x

D y x

E y x

F y x

Connecting

Problem Solving


Reflecting


Communicating

Connections

Data from investment earnings can look linear, but often involve an exponential relationship. You will learn more about investment and loan calculations in Chapter 7 Financial Applications.



4.

a)

b)

c)

d)


a)

b)

6.

a)

b)

c)

d)

7.

8.

a)

b)

c)

d)

9.

Year CPI ($)

2002 100

2003 102.8

2004 104.7

2005 107

2006 109.1

2007 111.8

2008 115.6

Connecting

Problem Solving


Reflecting


Communicating

Year Population

0 800

1 830

2 870

3 900

4 940

5 970


a)

b)

10.

a)

b)

c)

d)

11.

a)

b)

12.

a)

t

b)

c)

d)

e)

f)

13.

a)

b)

C Extend 14.

15. Math Contest

Connecting

Problem Solving


Reflecting


Communicating

A

B C

X

Y

Z

3.1 The Nature of Exponential Growth, pages 150 to 157

1.

A P n B P n

C P n D P n

2. a)

b)

c)

d)

e)

3. a)

b)

3.2 Exponential Decay: Connecting to Negative Exponents, pages 160 to 169

4.

a)

b)

c)

5.

a)

b)

6.

a) b) c)

d) e) f)

7.

a) x x x b) km k m

c) w w d) u vu v

e) z f) ab

3.3 Rational Exponents, pages 170 to 177

8.

a) b) c)

d) e) f)

g) h) i)

9. xk

U

x Uk

a)

b)

c)

3.4 Properties of Exponential Functions, pages 178 to 187

10. a) yx

b)

i) ii)

iii) x y

iv)

v)

Chapter 3 Review


11.

3.5 Transformations of Exponential Functions, pages 188 to 198

12. a) y x

b)

i)

ii)

iii)

13.

y x

a) y x b) y x

c) y x d) y x

3.6 Making Connections: Tools and Strategies for Applying Exponential Models, pages 199 to 209

14. hn

Number of Bounces, n Height, h (cm)

0 100

1 76

2 57

3 43

4 32

5 24

a)

b)

c)

d)

i)

ii)

e)

120

160

y

80

40

x2—4 —2 0 4(0, 10)

(1, 40)

(2, 160)

Chapter Problem WRAP-UP

a)

b)

Chapter 3 Review • MHR 211


For questions 1 to 4, select the best answer.

1. One day, 5 friends started a rumour. They agreed that they would each tell the rumour to two different friends the next day. On each day that followed, every person who just heard the rumour would tell another two people who had not heard the rumour. Which equation describes the relation between the number of days that have elapsed, d, and the number of people, P, who hear the rumour on that day?

A P 5 2 5d B P 5 5 2d

C P 5 5 ( 1 _ 2 ) d D P 5 2 ( 1 _

5 ) d

2. What is the value of 4 1 _ 2

?

A 2 B 1 _ 2 C

1 _ 2 D1 _ 16

3. Which is the correct equation when y 5 5x is translated 3 units down and 4 units left?

A y 5 5x 4 3 B y 5 5x 4 3

C y 5 5x 3 4 D y 5 5x 3 4

4. Which is correct about exponential functions?

A The ratio of successive first differences is constant.

B The first differences are constant.

C The first differences are zero.

D The second differences are constant.

5. Evaluate. Express your answers as integers or fractions in lowest terms.

a) 49 1 _ 2

b) 53 c) (4)0

d) 16 1 _ 4

e) (8) 5 _ 3

f) ( 3 _ 4 ) 4

g) ( 27 _ 64

) 1 _

3

h) ( 8 _ 125

) 4 _

3

6. Simplify. Express your answers using only positive exponents.

a) (x2)(x3)(x4) b)p3

_ p2

c) (2k4)1 d) ( a 1 _ 2

)( a 2 _ 3

)

e) ( y 2 _ 3

)6 f) ( u 1 _ 2

v3)2

7. Build or sketch a model that shows an exponential growing pattern that is doubling from one term to the next.

a) Sketch the first three terms of the model.

b) Make a table of values that relates the number of objects to the term number.

c) Graph the relation.

d) Find the first and second differences.

e) Write an equation for this model.

f) Give at least three reasons that confirm that the pattern is exponential in nature.

8. Match each graph with its corresponding equation.

a)

6

x42—4

8

2

4

y

0—2

b)

6

x42—4

8

2

4

y

0—2

c)

6

x42—4

8

2

4

y

0—2

Chapter 3 Practice Test

d)

6

x42—4

8

2

4

y

—2 0

A y 5 x2

B y 5 2x

C y 5 2x

D y 5 ( 1 _ 2 )

x

9. a) Sketch the graph of the function y 5 2x – 5 3.

b) Identify the

i) domain

ii) range

iii) equation of the asymptote

10. Describe the transformation(s) that map the base function y 5 8x onto each function.

a) y 5 1 _ 3

(8x) b) y 5 84x

c) y 5 –8–x d) y 5 8–3x – 6

11. A radioactive substance with an initial mass of 80 mg has a half-life of 2.5 days.

a) Write an equation to relate the mass remaining to time.

b) Graph the function. Describe the shape of the curve.

c) Limit the domain so that the model accurately describes the situation.

d) Find the amount remaining after

i) 10 days

ii) 15 days

e) How long will it take for the sample to decay to 5% of its initial mass?

12. a) Sketch the function y 5 ( 1 _ 2 ) 2x 3 1

by applying transformations to the graph of the base function y 5 2x.

b) For the transformed function, find the

i) domain

ii) range

iii) equation of the asymptote

13. The height, h, of a square-based pyramid is related to its volume, V, and base side length, b, by the equation h 5 3Vb2. A square-based pyramid has a volume of 6250 m3 and a base side length of 25 m. Find its height.

14. The population of Pebble Valley over a period of 5 years is shown.

Year Population

0 2000

1 2150

2 2300

3 2500

4 2700

5 2950

a) Make a scatter plot for this relationship. Do the data appear to be exponential in nature? Explain your reasoning.

b) Find the equation of the curve of best fit.

c) Limit the domain of the function so that it accurately models this situation.

d) Predict the population of Pebble Valley 7 years after the first table entry. State any assumptions you must make.

e) How long will it take for the town to double in size? State any assumptions you must make.

Chapter 3 Practice Test • MHR 213


Radioactive Isotopes

Radioactive isotopes are used in nuclear medicine to allow physicians to explore bodily structures in patients. Very small quantities of these isotopes, such as iodine-131 (I-131; half-life 8.065 days), are injected into patients. The isotopes are traced through the body so a medical diagnosis can be made. These elements have a very short half-life, so the label on the bottle will state the radioactivity of the element (the amount of the element that remains) at a particular moment in time.

a) A bottle of I-131 is delivered to a hospital on September 5. It states on the label that the radioactivity at 12:00 p.m. on September 10 will be 380 megabecquerels (MBq).

i) Write a defining equation for the relationship between the amount of radioactivity and the time since the bottle was delivered.

ii) What is the radioactive activity at

• 12:00 p.m. on the delivery day?

• 6:00 a.m. on September 25?

iii)Sketch the graph of this relationship.

b) Effective half-life is the time required for a radioactive isotope contained in a body to reduce its radioactivity by half, due to a combination of radioactive decay and the natural elimination of the isotope from the body. In patients with Graves disease, the effective half-life of I-131 is 62.5% of the normal half-life. In patients with a disease called toxic nodular goitre, the effective half-life is 75% of the normal half-life. Compare these results to the standard rate of decay for I-131 numerically, graphically, and algebraically.

c) Research one of the topics below and write a short report on your findings.

• other radioactive isotopes that are used in medical diagnoses or treatments

• diseases that can be treated with radioactive isotopes• the production and transport of radioactive isotopes• the training required for medical personnel to use radioactive isotopes

Task

Documents

McGrawHill Grade 11 Functions Unit 3