CorrectionKey=NL-A;CA-A CorrectionKey=NL-C;CA … Documents/interview...Common Core Math Standards...

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Explore Understanding Discrete Exponential FunctionsRecall that a discrete function has a graph consisting of isolated points.

The table represents the cost of tickets to an annual event as a function of the number t of tickets purchased. Complete the table by adding 10 to each successive cost. Plot each ordered pair from the table.

Tickets t Cost ($) (t, f (t) )

1 10 (1, 10)

2 20 (2, 20)

3 (3, ) 4 (4, ) 5 (5, )

30 30

40 40

50 50

Module 15 693 Lesson 3

15 . 3 Constructing Exponential Functions

Essential Question: What are discrete exponential functions and how do you represent them?

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Common Core Math StandardsThe student is expected to:

F-LE.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Also F-IF.2, F-IF.7e

Mathematical Practices

MP.6 Precision

Language ObjectiveExplain to a partner what the graph of a discrete exponential function looks like.

HARDCOVER PAGES 545554

Turn to these pages to find this lesson in the hardcover student edition.

Constructing Exponential Functions

ENGAGE Essential Question: What are discrete exponential functions and how do you represent them?Possible answer: Discrete exponential functions are exponential functions whose domains are limited, such as the set of integers, making the graph appear as isolated points instead of a continuous curve.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo and why the population of invasive species might grow exponentially. Then preview the Lesson Performance Task.

693

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

F-LE.2 For the full text of these standards, see the table starting on page CA2. Also F-IF.2,

F-IF.7e

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Explore Understanding Discrete Exponential Functions

Recall that a discrete function has a graph consisting of isolated points.

The table represents the cost of tickets to an annual event as a function of the number t of

tickets purchased. Complete the table by adding 10 to each successive cost. Plot each ordered

pair from the table.

Tickets tCost ($)

(t, f (t) )

1

10(1, 10)

2

20(2, 20)

3

(3, )

4

(4, )

5

(5, )

3030

4040

5050

Module 15

693

Lesson 3

15 . 3 Constructing Exponential

Functions

Essential Question: What are discrete exponential functions and how do you represent them?

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3/24/14 11:20 AM

693 Lesson 15 . 3

L E S S O N 15 . 3

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B The number of people attending an event doubles each year. The table represents the total attendance at each annual event as a function of the event number n. Complete the table by multiplying each successive attendance by 2. Plot each ordered pair from the table.

Event Number n Attendance (n, g (n) )

1 20 (1, 20)

2 40 (2, 40)

3 (3, ) 4 (4, ) 5 (5, )

C Complete the table.

Function Linear? Discrete?

f (t) Yes

g (n)

Reflect

1. Communicate Mathematical Ideas What are the limitations on the domains of these functions? Why?

80 80

160 160

320 320

No

Yes

Yes

The domains include only whole numbers. You cannot buy a fraction of a ticket to an

event, nor stage a fraction of an event.


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EXPLORE Understanding Discrete Exponential Functions

INTEGRATE TECHNOLOGYHave students complete the Explore activity in either the book or online lesson.

QUESTIONING STRATEGIESHow are the tables for the functions different? They are different because

successive output values of the first function have a common difference, while successive output values of the second function have a common ratio.

PROFESSIONAL DEVELOPMENT

Learning ProgressionsIn this lesson, students learn how to identify and represent exponential functions. Some key understandings for students are as follows:• An exponential function can be represented by an equation of the form f (x) = ab x , where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1.

• For an exponential function ƒ (x) , if x is in the domain of the function, then the ratio of ƒ (x+1) to ƒ (x) is a constant ratio.

In the next module, students will learn more about exponential functions, including exponential growth and decay functions.

Constructing Exponential Functions 694


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Explain 1 Representing Discrete Exponential FunctionsAn exponential function is a function whose successive output values are related by a constant ratio. An exponential function can be represented by an equation of the form ƒ (x) = ab x , where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1. The constant ratio is the base b.

When evaluating exponential functions, you will need to use the properties of exponents, including zero and negative exponents.

Recall that, for any nonzero number c:

c 0 = 1, c ≠ 0

c -n = 1 _ c n , c ≠ 0.

Example 1 Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table.

ƒ (x) = 3 ⋅ ( 1 _ 2 ) x with a domain of ⎧ ⎨ ⎩ -1, 0, 1, 2, 3, 4 ⎫

⎬ ⎭

ƒ (-1) = 3 ⋅ ( 1 _ 2 ) -1

= 3 ⋅ 1 _ ( 1 _ 2 )

= 3 ⋅ 2 = 6

x f (x) (x, f (x) )

-1

0

1

2

3

4

6 (-1, 6)

3 (0, 3)

1 1 _ 2

(1, 1 1 _ 2

)

3 _ 4

(2,

3 _ 4

)

3 _ 8

(3,

3 _ 8

)

3 _ 16

(4,

3 _ 16

)


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COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave students work in pairs. Instruct one student in each pair to construct a table of values and graph for the function defined by h (x) = 3x with the domain ⎧ ⎨ ⎩ -2, -1, 0, 2 ⎫

⎬ ⎭ . Instruct the other student to do the same for the function defined

by g (x) = 3 x with the same domain. Have the students compare and contrast the tables of values and the graphs.

EXPLAIN 1 Representing Discrete Exponential Functions

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Point out that the constant b in a function of the form f (x) = ab x , where a and b are real numbers, a ≠ 0, b > 0, and b ≠ 1, corresponds to the common ratio, r, in a geometric sequence. Explain to students that if the domain of the function is the set of positive integers, the points of the graph of a discrete exponential function represent a geometric sequence. The x-values are the term numbers of the sequence and the y-values are the corresponding terms.

QUESTIONING STRATEGIES

How do you evaluate a complex fraction, such

as 1 __ 1 __ 2

? Multiply the numerator by the

reciprocal of the denominator: 1∙ 2 __ 1 = 2 __ 1 = 2.

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B ƒ (x) = 3 ( 4 _ 3 ) x ; domain = ⎧ ⎨ ⎩ -2, -1, 0, 1, 2, 3 ⎫

⎬ ⎭

ƒ (-2) = 3 ( 4 _ 3 ) -2

= 3 ⋅ 1 _

( 2

_

2

) = 3 ⋅

2

_

2

= 3 ⋅ _ = _ = 1 _

x f (x) (x, f (x) )

-2 (-2, )

-1 (-1, )

0 (0, )

1 (1, )

2 (2, )

3 7 1 _ 9

(3, 7 1 _ 9

)

Reflect

2. What If What would happen to the function ƒ (x) = a b x if a were 0? What if b were 1?

3. Discussion Why is a geometric sequence a discrete exponential function?

1 11 __ 16

2 1 _ 4

1 11 __ 16

2 1 _ 4

3 3

4 4

5 1 _ 3 5 1 _ 3

4

3 9

3

4 16

27 11

16 16

If a = 0 then ƒ (x) = 0 for all x, and if b = 1, then ƒ (x) = a for all x. Neither of these

constant functions are exponential functions.

A sequence is a function. When the input values increase by 1, the successive output

values are related by a constant ratio, the common ratio. The general recursive rule for a

geometric sequence is the form of an exponential function.


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DIFFERENTIATE INSTRUCTION

Cognitive StrategiesEncourage students to isolate the necessary information from any problem scenario they are given. It may help to circle or underline values or to write the important information on a separate sheet of paper.



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Your Turn

Make a table for the function using the given domain. Then graph the function using the ordered pairs from the table.

4. (x) = 4 ( 3 _ 2 ) x ; domain = ⎧ ⎨ ⎩ -3, -2, -1, 0, 1, 2 ⎫

⎬ ⎭

Explain 2 Constructing Exponential Functions from Verbal Descriptions

You can write an equation for an exponential function f (x) = ab x by finding or calculating the values of a and b.

The value of a is the value of the function when x = 0. The value of b is the common ratio of successive function

values, b = ƒ (x + 1)

_______ ƒ (x) . For discrete functions with integer or whole number domains, these will be successive values

of the function.

Example 2 Write an equation for the function.

When a piece of paper is folded in half, the total thickness doubles. Suppose an unfolded piece of paper is 0.1 millimeter thick. The total thickness t (n) of the paper is an exponential function of the number of folds n.

The value of a is the original thickness of the paper before any folds are made, or 0.1 millimeter.

Because the thickness doubles with each fold, the value of b (the constant ratio) is 2.

The equation for the function is t (n) = 0.1 (2) n .

x f (x) (x, f (x) )

-3 ( 32 _ 27

) = 1 5 _ 27

(-3, 1 5 _ 27

)

-2 ( 16 _ 9

) = 1 7 _ 9

(-2, 1 7 _ 9

)

-1 ( 8 _ 3

) = 2 2 _ 3

(-1, 2 2 _ 3

)

0 4 (0, 4)

1 6 (1, 6)

2 9 (2, 9)




LANGUAGE SUPPORT

Connect VocabularyIn this lesson there are several key words that begin with the prefix in-. Learning the meaning of prefixes helps all students of math and science, since many key terms in both disciplines originated in Latin or have Latin prefixes. In this lesson, the words increase and input both start with the prefix in-, which means in, into, on, near, or towards. The prefix in- can also mean not, as in the word inverse. Another key prefix in this lesson is non-, which means not. The term nonzero, for example, means not zero.

EXPLAIN 2 Constructing Exponential Functions from Verbal Descriptions

MODELINGHave students fold a piece of paper in half repeatedly. Have them make a table representing the number of layers of paper after 0 folds, 1 fold, 2 folds, and so on.

QUESTIONING STRATEGIESHow do you find the value of b in order to write an exponential function in the form

ƒ (x) = ab x ? Choose a number x in the domain of the function. Then divide f(x + 1) by f(x) to get b.

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A savings account with an initial balance of $1000 earns 1% interest per month. That means that the account balance grows by a factor of 1.01 each month if no deposits or withdrawals are made. The account balance in dollars B (t) is an exponential function of the time t in months after the initial deposit.

Let B represent the balance in dollars as a function of time t in months.

The value of a is the original balance, .

The value of b is the factor by which the balance changes every month, .

The equation for the function is B(t) = .

Reflect

5. Why is the exponential function in the paper-folding example discrete?

Your Turn

6. A piece of paper that is 0.2 millimeters thick is folded. Write an equation for the thickness t of the paper in millimeters as a function of the number n of folds.

Explain 3 Constructing Exponential Functions from Input-Output Pairs

You can use given two successive values of a discrete exponential function to write an equation for the function.

Example 3 Write an equation for the function that includes the points.

(3, 12) and (4, 24)

Find b by dividing the function value of the second pair by the function value of the first: b = 24 _ 12 = 2.

Evaluate the function for x = 3 and solve for a.

Write the general form. ƒ (x) = a b x

Substitute the value for b. ƒ (x) = a ⋅ 2 x

Substitute a pair of input-output values. 12 = a ⋅ 2 3

Simplify. 12 = a ⋅ 8

Solve for a. a = 3 _ 2

Use a and b to write an equation for ƒ (x) = 3 _ 2 ⋅ 2 x the function.

1000

1000 (1.01) t

1.01

The paper can only be folded a whole number of times.

Initial thickness (n = 0) a = 0.2

Change per fold: double b = 2

Equation: t (n) = 0.2 ⋅ 2 n





EXPLAIN 3 Constructing Exponential Functions from Input-Output Pairs

AVOID COMMON ERRORSWhen calculating the value of b, students may divide in the wrong order. Remind them that they should always divide an output value by the preceding output value. For example, if they are using (4, 27) and (5, 9), they should divide the function value of the second pair by the function value of the first pair:

b = 9 ___ 27 = 1 __ 3 .

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 After students have found an equation for an exponential function, have them check both pairs in the equation to verify that the equation is correct.



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B (1, 3) and (2, 9 _ 4 ) Find b by dividing the function value of the second pair by the first: b = 9 _ 4 ÷ 3 = .

Write the general form. ƒ (x) =

Substitute the value for b. ƒ (x) = a ⋅

x

Substitute a pair of input-output values. = a ⋅ ( 3 _ 4 ) Simplify. 3 = a ⋅

Solve for a. a =

Use a and b to write an equation for ƒ (x) = the function.

Your Turn

Write an equation for the function that includes the points.

7. (-2, 2 _ 5 ) and (-1, 2)

Elaborate 8. Explain why the following statement is true: For 0 < b < 1 and a > 0, the function ƒ (x) = ab x decreases

as x increases.

9. Explain why the following statement is true: For b > 1 and a > 0, the function ƒ (x) = ab x increases as x increases.

10. Essential Question Check-In What property do all pairs of adjacent points of a discrete exponential function share?

a b x

( 3 _ 4 )

3

3 _ 4

4

4 ( 3 _ 4 ) x

1

3 _ 4

When you multiply a number a by a fraction b between 0 and 1, the product is less than

the original number a. When you multiply by this fraction b repeatedly, the results

decrease further.

b = 2 ÷ 2 _ 5

= 5

2 = a ⋅ 5 -1 ; 2 = a _ 5

; a = 10

f (x) = 10 ⋅ 5 x

When you multiply a number a by a number b greater than 1, the product is greater

than the original number a. When you multiply by this number b repeatedly, the results

increase further.

The ratio of function values is the same for any two pairs.




QUESTIONING STRATEGIESIf an exponential function includes the points (4, 64) and (5, 32), what is the common ratio?

Explain how you found it. The common ratio is 0.5. I found it by dividing 32 by 64: 64 ___ 32 = 1 __ 2 = 0.5 .

ELABORATE CRITICAL THINKINGHave students give an example of an increasing exponential function and of a decreasing exponential function.

SUMMARIZE THE LESSONWhat are discrete exponential functions and how can you represent them? A discrete

exponential function is a function of the form f (x) = ab x , with a ≠ 0, b > 1, and b ≠ 1, such that the graph of the function is a set of isolated points. For example, the domain of the function could be the set of whole numbers. Such a function can be represented by a set of input and output values in a table or by a graph of ordered pairs.

699 Lesson 15 . 3


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• Online Homework• Hints and Help• Extra Practice

Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table.

1. ƒ (x) = 1 _ 2 ⋅ 4 x ; domain = ⎧ ⎨ ⎩ -2, -1, 0, 1, 2 ⎫

⎬ ⎭ .

2. ƒ (x) = 9 ( 1 _ 3 ) x ; domain = ⎧ ⎨ ⎩ 0, 1, 2, 3, 4, 5 ⎫

⎬ ⎭ .

3. ƒ (x) = 6 ( 2 _ 3 ) x ; domain = ⎧ ⎨ ⎩ -1, 0, 1, 2, 3, 4 ⎫

⎬ ⎭ .

Evaluate: Homework and Practice

x f (x) (x, f (x) )

-2 1 __ 32 (-2, 1 __ 32 )

-1 1 _ 8 (-1, 1 _ 8 )

0 1 _ 2 (0, 1 _ 2 )

1 2 (1, 2)

2 8 (2, 8)

x f (x) (x, f (x) )

0 9 (0, 9)

1 3 (1, 3)

2 1 (2, 1)

3 1 _ 3 (3, 1 _ 3 )

4 1 _ 9 (4, 1 _ 9 )

5 1 __ 27 (5, 1 __ 27 )

x f (x) (x, f (x) )

-1 9 (-1, 9)

0 6 (0, 6)

1 4 (1, 4)

2 2 2 _ 3 (2, 2 2 _ 3 )

3 1 7 _ 9 (3, 1 7 _ 9 )

4 1 5 __ 27 (4, 1 5 __ 27 )




A1_MNLESE368187_U6M15L3.indd 700 3/24/14 11:15 AMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–4 1 Recall of Information MP.6 Precision

5–8 2 Skills/Concepts MP.4 Modeling

9–21 2 Skills/Concepts MP.2 Reasoning

22 1 Recall of Information MP.2 Reasoning

23–24 3 Strategic Thinking MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreUnderstanding Discrete Exponential Functions

Exercise 19

Example 1Representing Discrete Exponential Functions

Exercises 1–4, 22–23

Example 2Constructing Exponential Functions from Verbal Descriptions

Exercises 5–8, 20–21

Example 3Constructing Exponential Functions from Input-Output Pairs

Exercises 9–18, 24



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4. ƒ (x) = 6 ( 4 _ 3 ) x ; domain = ⎧ ⎨ ⎩ -3, -2, -1, 0, 1 ⎫

⎬ ⎭

Write an equation for each function.

5. Business A recent trend in advertising is viral marketing. The goal is to convince viewers to share an amusing advertisement by e-mail or social networking. Imagine that the video is sent to 100 people on day 1. Each person agrees to send the video to 5 people the next day, and to request that each of those people send it to 5 people the day after they receive it. The number of viewers v (n) is an exponential function of the number n of days since the video was first shown.

6. A pharmaceutical company is testing a new antibiotic. The number of bacteria present in a sample when the antibiotic is applied is 100,000. Each hour, the number of bacteria present decreases by half. The number of bacteria remaining r (n) is an exponential function of the number n of hours since the antibiotic was applied.

7. Optics A laser beam with an output of 5 milliwatts is directed into a series of mirrors. The laser beam loses 1% of its power every time it reflects off of a mirror. The power p (n) is a function of the number n of reflections.

x f (x) (x, f (x) )

-3 2 17 __ 32 (-3, 2 17 __ 32 )

-2 3 3 _ 8 (-2, 3 3 _ 8 )

-1 4 1 _ 2 (-1, 4 1 _ 2 )

0 6 (0, 6)

1 8 (1, 8)

a = 100

b = 5

v (n) = 100 ⋅ 5 n

a = 100,000

b = 1 _ 2

r (n) = 100,000 ( 1 _ 2 ) n

a = 5

b = 0.99, (power remaining after 1% lost)

p (n) = 5 (0.99) n


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A1_MNLESE368187_U6M15L3 701 6/8/15 12:35 PM

QUESTIONING STRATEGIESWhy is the restriction b ≠ 1 in the definition of an exponential function necessary? What

would happen if b = 1? 1 raised to any power is 1, so the function would always have the same value. It would not increase or decrease.

701 Lesson 15 . 3


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8. The NCAA basketball tournament begins with 64 teams, and after each round, half the teams are eliminated. The number of remaining teams t (n) is an exponential function of the number n of rounds already played.

Write an equation for the function that includes the points.

9. (2, 100) and (3, 1000) 10. (-2, 4) and (-1, 8)

11. (1, 4 _ 5 ) and (2, 2 _ 3 ) 12. (-3, 1 _ 16 ) and (-2, 3 _ 8 )

Use two points to write an equation for the function.

13.

x f (x)

1 2

2 2 _ 7

3 2 _ 49

4 2 _ 343

a = 64

b = 1 _ 2

t (n) = 64 ( 1 _ 2 ) n

b = 1000 ____ 100

= 10

100 = a ⋅ 1 0 2

100 = 100a

a = 1

f (x) = 1 0 x

b = 8 _ 4

= 2

8 = a ⋅ 2 -1

8 = a _ 2

a = 16

f (x) = 16 ⋅ 2 x

b = ( 2 _ 3 )

___ ( 4 _ 5 )

= 2 _ 3 ⋅ 5 _ 4

= 5 _ 6

4 _ 5 = a ⋅ ( 5 _ 6 ) 1

4 _ 5 = a ⋅ 5 _ 6

a = 4 _ 5 ⋅ 6 _ 5

= 24 __ 25

f (x) = 24 __ 25 ⋅ ( 5 _ 6 ) x

b = ( 3 _ 8 )

___ ( 1 __ 16 )

= 3 _ 8 ⋅ 16 __ 1

= 6

1 __ 16 = a ⋅ 6 -3

1 __ 16 = a ⋅ 1 ___ 216

a = 1 __ 16 ⋅ 216

= 13.5

f (x) = 13.5 ⋅ 6 x

b = ( 2 _ 7 )

___ 2

= 1 _ 7

2 = a ⋅ ( 1 _ 7 ) 1

2 = a ⋅ 1 _ 7

a = 14

f (x) = 14 ⋅ ( 1 _ 7 ) x





AVOID COMMON ERRORSStudents may forget to find the value of both a and b when writing a function rule for an exponential function of the form ƒ (x) = ab x . Remind students that they need to determine the common ratio and the initial value of an exponential function in order to write the function rule.



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14.

x f (x)

-4 0.53

-3 5.3

-2 53

-1 530

15.

16.

17. The height h (n) of a bouncing ball is an exponential function of the number n of bounces. One ball is dropped and on the first bounce reaches a height of 6 feet. On the second bounce it reaches a height of 4 feet.

Use points (-2, 53) and (-1, 530)

b = 530 ___ 53

= 10

530 = a ⋅ (10) -1

530 = a ⋅ 1 __ 10

a = 5300

f (x) = 5300 ⋅ (10) x

Use points (0, 3) and (1, 6)

b = 6 _ 3

= 2

a = 3

f (x) = 3 ⋅ 2 x

Use points (1, 8) and (2, 4)

b = 4 _ 8

= 1 _ 2

8 = a ( 1 _ 2 ) 1

8 = a _ 2

a = 16

f (x) = 16 ⋅ ( 1 _ 2 ) x

b = 4 _ 6

= 2 _ 3

6 = a ( 2 _ 3 ) 1

6 = a ⋅ 2 _ 3

a = 6 ⋅ 3 _ 2

= 9

h (n) = 9 ⋅ ( 2 _ 3 ) n




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18. A child starts a playground swing from standing and doesn’t use her legs to keep swinging. On the first swing she swings forward by 18 degrees, and on the second swing she only comes 13.5 degrees forward. The measure in degrees of the angle m (n) is an exponential function of the number n of swings.

19. Make a Prediction A town’s population has been declining in recent years. The table shows the population since 1980. Is this data consistent with an exponential function? Explain. If so, predict the population for 2010 assuming the trend holds.

Year Population

1980 5000

1990 4000

2000 3200

2010

20. A piece of paper has a thickness of 0.15 millimeters. Write an equation to describe the thickness t(n) of the paper when it is repeatedly folded in thirds, where n is the number of foldings.

b = 13.5 ___ 18

= 3 _ 4

18 = a ( 3 _ 4 ) 1

18 = a ⋅ 3 _ 4

a = 18 ⋅ 4 _ 3

= 24

m (n) = 24 ( 3 _ 4 ) n

2560

To check for exponential behavior, find the ratios between successive

values of the function.

4000 ____ 5000 = 4 _ 5 , 3200 ____ 4000 = 4 _ 5

The ratio between successive function values is 4 _ 5 , so this is consistent with

an exponential function.

3200 ⋅ 4 _ 5 = 2560

b = 3

a = 0.15

t (n) = 0.15 ⋅ 3 n




A1_MNLESE368187_U6M15L3 704 6/12/15 9:53 AM

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Point out to students that, unlike linear functions, the letter b in an exponential function of the form ƒ (x) = ab x does not represent the y-intercept.



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21. Probability The probability of getting heads on a single coin flip is 1 __ 2 . The probability of getting nothing but heads on a series of coin flips decreases by 1 __ 2 for each additional coin flip. Write an exponential function for the probability p(n) of getting all heads in a series of n coin flips.

22. Multipart Classification Determine whether each of the functions is exponential or not.a. ƒ (x) = x 2 Exponential Not exponentialb. ƒ (x) = 3 ⋅ 2 x Exponential Not exponentialc. ƒ (x) = 3 ⋅ 1 _

2 x Exponential Not exponential

d. ƒ (x) = 1.00 1 x Exponential Not exponentiale. ƒ (x) = 2 ⋅ x 3 Exponential Not exponentialf. ƒ (x) = 1 _

10 ⋅ 5 x Exponential Not exponential

H.O.T. Focus on Higher Order Thinking

23. Explain the Error Biff observes that in every math test he has taken this year, he has scored 2 points higher than the previous test. His score on the first test was 56. He models his test scores with the exponential function s (n) = 28 · 2 n where s (n) is the score on his nth test. Is this a reasonable model? Explain.

24. Find the Error Kaylee needed to write the equation of an exponential function from points on the graph of the function. To determine the value of b, Kaylee chose the ordered pairs (1, 6) and (3, 54) and divided 54 by 6. She determined that the value of b was 9. What error did Kaylee make?

b = 1 _ 2

1 _ 2

= a ( 1 _ 2 ) 1

1 _ 2

= a ⋅ 1 _ 2

a = 1

p (n) = ( 1 _ 2

) n

No; Biff’s test scores do not increase by a constant ratio. Instead they go up by 2 points each test. His scores are not consistent with an exponential function.

She should use ordered pairs with x-values that differ by 1. The value Kaylee found is not the constant ratio of successive values.




JOURNALHave students describe a specific discrete exponential function in at least two different ways. They can describe it with words, a function rule, a table, and/or a graph.

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Lesson Performance Task

In ecology, an invasive species is a plant or animal species newly introduced to an ecosystem, often by human activity. Because invasive species often lack predators in their new habitat, their populations typically experience exponential growth. A small initial population grows to a large population that drastically alters an ecosystem. Feral rabbits that populate Australia and zebra mussels in the Great Lakes are two examples of problematic invasive species that grew exponentially from a small initial population.

An ecologist monitoring a local stream has been collecting samples of an unfamiliar fish species over the past four years and has summarized the data in the table.

Here are the results so far:

Year Average Population Per Mile

2009 32

2010 48

2011 72

2012 108

2013

2014

a. Look at the data in the table and confirm that the growth pattern is exponential.

b. Write the equation that represents the average population per square meter as a function of years since 2009.

c. Predict the average populations expected for 2013 and 2014.

d. Graph the population versus time since 2009 and include the predicted values.

The function is f (t) = 32 ( 3 _ 2 ) t

.

48 _ 32

= 1.5 72 _ 48

= 1.5 108 _ 72

= 1.5

Since there is a common ratio, r = 1.5, the growth pattern is exponential.

2013: 162 individuals per mile

2014: 243 individuals per mile




EXTENSION ACTIVITY

Have students research the feral rabbit population in Australia or the zebra mussel population in the Great Lakes. Have students find out how the population was introduced into the new habitat and describe efforts to control the population.

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Have students explain why they would choose to use the fraction 3 __ 2 or its decimal equivalent, 1.5, when solving Part C of the Lesson Performance Task.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 After students have completed the graph of the populations versus time, discuss how the value of b, when a is positive, causes the graph to increase over time. Then discuss what happens to the graph when students let b = 2 __ 3 in f (t) = 32 ( 2 __ 3 ) t .

Scoring Rubric

2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



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