Exponential Functions and Graphs ACTIVITY Sizing Up the ......Apr 04, 2012 · Exponential...

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Unit 2 • Series, Exponential and Logarithmic Functions 97

My Notes

ACTIVITY

2.3Exponential Functions and GraphsSizing Up the SituationSUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/Retell, Create Representations, Look for a Pattern, Quickwrite

Ramon Hall, a graphic artist, needs to make several diff erent-sized draft copies of an original design. His original graphic design sketch is contained within a rectangle with a width of 4 cm and a length of 6 cm. Using the offi ce copy machine, he magnifi es the original 4 cm × 6 cm design to 120% of the original design size, and calls this his fi rst draft . Ramon’s second draft results from magnifying the fi rst draft to 120% of its new size. Each new draft is 120% of the previous draft .

1. Complete the table with the dimensions of Ramon’s fi rst fi ve draft versions, showing all decimal places.

Number of Magnifi cations Width (cm) Length (cm)

0 4 612345

2. Th e resulting draft for each number of 120% magnifi cations has a unique width and a unique length. Th us, there is a functional rela-tionship between the number of magnifi cations n and the resulting width W. Th ere is also a functional relationship between the number of magnifi cations n and the resulting length L. Make a graph using the grid in the My Notes space and the data from the table in Item 1.

a. Plot the ordered pairs (n, W).

b. Use a diff erent color or symbol to plot the ordered pairs (n, L).

3. Examine the data and graphs in Items 1 and 2. Do these functions appear to be linear? Explain why or why not.

Magnifying a design creates similar fi gures. The ratio between corresponding lengths of similar fi gures is called the constant of proportionality, or the scale factor. For a magnifi cation of 120%, the scale factor is 1.2.

97-108_SB_A2_2-3_SE_Act.indd 9797-108_SB_A2_2-3_SE_Act.indd 97 1/14/10 9:00:31 PM1/14/10 9:00:31 PM

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My Notes

98 SpringBoard® Mathematics with Meaning™ Algebra 1

ACTIVITY 2.3continued

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations

4. Explain why each table contains data that can be represented by a linear function. Write an equation to show the linear relationship between x and y.

a.

y =

b.

y =

5. Consider the data in the table below.

x 0 1 2 3 4y 24 12 6 3 1.5

a. Can the data in the table be represented by a linear function? Explain why or why not.

b. Describe any patterns that you see in the consecutive y-values.

x −3 −1 1 3 5y 8 5 2 −1 −4x −3 −1 1 3 5y 8 5 2 −1 −4

x 2 5 11 17 26y 3 7 15 23 35x 2 5 11 17 26y 3 7 15 23 35

Linear functions have the prop-erty that the rate of change of the output variable y with respect to the input variable x is constant,

i.e., the ratio Δy

___ Δ x is constant for

linear functions.

Exponential Functions and Graphs Sizing Up the SituationSizing Up the Situation


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SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Close Reading, Vocabulary Organizer

6. Consider the data in the table created in Item 1.

a. Copy the width values into the second row of the table below.

x 0 1 2 3 4 5y

b. How does the relationship of the data in this table compare to the relationship of the data in the table in Item 5?

Th e data in the tables in Items 5 and 6 were generated by functions that are exponential. In the special case when the change in the input variable x is constant, the output variable y of an exponential function changes by a multiplicative constant. For example, in the table in Item 5, the increase in the consecutive x-values results from repeatedly adding 1, while the decrease in y-values results from repeatedly multiplying by the constant 1 __ 2 , known as the exponential decay factor.

7. In the table in Item 6, the increase in the x-values results from repeatedly adding 1.

a. What is the exponential growth factor?

b. What is the growth rate and how is it related to the exponential growth factor?

To compare change in size, you could also use the growth rate, or percent increase. This is the percent that is equal to the ratio of the increase amount to the original amount.

ACADEMIC VOCABULARY

An exponential function is a function of the form f(x) = a · bx, where a and b are constants, x is the domain, f(x) is the range, and a ≠ 0, b > 0, b ≠ 1.

In an exponential function, the constant multiplier, or scale factor, is called an exponential decay factor when the constant is less than 1. When the constant is greater than 1, it is called an exponential growth factor.

MATH TERMS


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My Notes

Exponential Functions and Graphs ACTIVITY 2.3continued Sizing Up the SituationSizing Up the Situation

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations

8. You can write an equation for the exponential function relating W and n.

a. Complete the table below to show the calculations used to fi nd the width of each magnifi cation.

Number of Magnifi cations Calculation to Find Width (cm)

0 41 4(1.2)2 4(1.2)(1.2)3 4(1.2)(1.2)(1.2)45

10n

b. Write a function that expresses the resulting width W aft er n magnifi cations of 120%.

c. Use the function in Part b to fi nd the width of the 11th magnifi cation.

Confi rm the reasonableness of your function in Item 8b by using a graphing calcula-tor to make a scatter plot of the data in the table in Item 6. Then graph the function to see how it compares to the scatter plot.

TECHNOLOGY


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SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Quickwrite, Create Representations, Think/Pair/Share

Th e general form of an exponential function is f (x) = a(bx), where a and b are constants, and a ≠ 0, b > 0, b ≠ 1.

9. For the exponential function written in Item 8b, identify the value of the parameters a and b. Th en explain their meaning in terms of the problem situation.

10. Starting with Ramon’s original 4 cm × 6 cm rectangle containing his graphic design, write an exponential function that expresses the resulting length L aft er n magnifi cations of 120%.

Ramon decides to print fi ve diff erent reduced draft copies of his original design rectangle. Each one will be reduced to 90% of the previous size.

11. Complete the table below to show the dimensions of the fi rst fi ve draft versions. Include all decimal places.

Number of Reductions Width (cm) Length (cm)

0 4 612345

12. Write the exponential decay factor and the decay rate for the data in the table in Item 11.

To compare change in size, you could also use the decay rate, or percent decrease. This is the percent that is equal to the ratio of the decrease amount to the original amount.


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SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share, Activating Prior Knowledge, Vocabulary Organizer

13. Use the data in the table in Item 11.

a. Write an exponential function that expresses the width w of a reduction in terms of n, the number of reductions performed.

b. Write an exponential function that expresses the length l of a reduction in terms of n, the number of reductions performed.

c. Use the functions to fi nd the dimensions of the design if the original design undergoes ten reductions.

14. Graph the functions y = 6(1.2)x and y = 6(0.9)x on a graphing calculator or other graphing utility. Sketch the results on the grid in the My Notes space.

15. Determine the domain and range for each function.

Domain Range

a. y = 6(1.2)x

b. y = 6(0.9)x

A function is said to increase if the y-values increase as the x-values increase. A function is said to decrease if the y-values decrease as the x-values increase.

16. Describe each function as increasing or decreasing.

a. y = 6(1.2)x

b. y = 6(0.9)x

CONNECT TO APAP

Not all functions increase or decrease over the entire domain of the function. Functions may increase, decrease, or remain constant over various intervals of the domain. Functions that either increase or decrease over the entire domain are called strictly monotonic.

ACADEMIC VOCABULARY

increasing functiondecreasing function


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SUGGESTED LEARNING STRATEGIES: Close Reading, Vocabulary Organizer, Create Representations, Debriefing, Think/Pair/Share, Group Presentation

Th e end behavior of a graph describes the y-values of the function as x increases without bound and as x decreases without bound. If the end behavior approaches some constant a, then the graph of the function has a horizontal asymptote at y = a.

When x increases without bound, the values of x approach positive infi nity, ∞. When x decreases without bound, the values of x approach negative infi nity, −∞.

17. Describe the end behavior of each function as x approaches ∞. Write the equation for any horizontal asymptotes.

a. y = 6(1.2)x

b. y = 6(0.9)x

18. Describe the end behavior of each function as x approaches −∞. Write the equation for any horizontal asymptotes.

a. y = 6(1.2)x

b. y = 6(0.9)x

19. Identify any x- or y-intercepts of each function.

a. y = 6(1.2)x

b. y = 6(0.9)x

20. Consider how the parameters a and b aff ect the graph of the general exponential function f (x) = a(b)x. Use a graphing calculator to graph f for various values of a and b.

a. When does the function increase?

b. When does the function decrease?

c. What determines the y-intercept of the function?

d. State any horizontal asymptotes of the function.

If the graph of a relation gets closer and closer to a line, the line is called an asymptote of the graph.

MATH TERMS


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SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Quickwrite

You can use transformations of the graph of the function f (x) = bx to graph functions of the form g(x) = a(b)x − c + d, where a and b are constants, and a ≠ 0, b > 0, b ≠ 1. Rather than having a single parent graph for all exponential functions, there is a diff erent parent graph for each base b.

21. Graph the parent graph f and the function g by applying the correct vertical stretch, shrink, and/or refl ection over the x-axis. Write a description of each transformation.

a. f (x) = ( 1 __ 2 ) x g(x) = 4 ( 1 __ 2 ) x

b. f (x) = 3x

g(x) = − 1 __ 2 (3)x

You can draw a quick sketch of each parent graph by plotting the

points ( −1, 1 __ b ) , (0, 1) and (1, b).

123456

–1

789

10111213

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x

123456

–4–3–2

–7–6–5

–1

7

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x

CONNECT TO APAP

Exponential functions play an important part in the study of calculus.


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SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite

22. Sketch the parent graph f and the graph of g by applying the correct horizontal or vertical translation. Write a description of each transformation and give the equations of any asymptotes.

a. f(x) = 2x

g(x) = 2(x − 3)

b. f (x) = ( 1 __ 3 ) x g(x) = ( 1 __ 3 ) x − 2

You can use a graphing calculator to approximate the range values when thex-coordinates are notintegers. For f (x) = 2x, use a calculator to fi nd

f ( 1 __ 2 ) and f ( √__

3 ).

2 1 __ 2 ≈ 1.414

2 √__

3 ≈ 3.322

Then use a graphing calculator to verify that the

points ( 1 __ 2 , 2 1 __ 2 ) and ( √

__ 3 , 2 √

__ 3 ) lie

on the graph of f (x) = 2x.

TECHNOLOGY

123456

–1

789

10111213

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x

123456

–4–3

–7–6–5

–1

7

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x

–2


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23. Describe how each function results from transforming a parent graph of the form f(x) = bx. Th en sketch the parent graph and the given function on the same axes. State the domain and range of each function. Give the equations of any asymptotes.

a. g(x) = 3x + 4 + 1

b. g(x) = 2 ( 1 __ 3 ) x - 4

123456

–2–1

7

1 2 3–1–2–3–4–5–6–7

y

x

123456

–4–3–2

–7–6–5

–1

7

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x


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23. (continued)

c. g(x) = 1 __ 2 (4)x - 4 - 2

d. g(x) = -3(2)x - 6 + 5

123456

–3–2–1

7

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

x

123456

–4–3–2

–7–6–5

–1

7

1 2 3 4 5 6 7–1–2–3–4–5–6–7

y

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SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation

24. Explain how the parameters a, c, and d transform the parent graph f (x) = bx to produce a graph of the function g(x) = a(b)x − c + d.

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper or grid paper. Show your work.

1. Ida paints violets onto porcelain plates. She paints a spiral that is a sequence of violets, the size of each consecutive violet being a fraction of the size of the preceding violet. Th e table below shows the width of the fi rst three violets in the continuing pattern.

Violet Number 1 2 3Width (in cm) 4 3.2 2.56

a. Is Ida’s shrinking violet pattern an example of an exponential function? Explain.

b. Find the width of the fourth and fi ft h violets in the sequence.

c. Write an equation to express the size of the smallest violet in terms of the number of violets on the plate.

d. If a plate has a total of 10 violets, explain two diff erent ways to determine the size of the smallest violet.

2. For each exponential function, state the domain and range, whether the function increases or decreases, and the y-intercept.

a. y = 2(4)x b. y = 3 ( 1 __ 2 ) x c. y = −(0.3)x d. y = −3(5.2)x

3. Describe how each function results from transforming a parent graph of the formf(x) = bx. Th en sketch the parent graph and the given function on the same axes. State the domain and range of each function and give the equations of any asymptotes.

a. g(x) = 2x + 3 - 4

b. g(x) = -3 ( 1 __ 2 ) x + 2

c. g(x) = 1 __ 2 (3)x + 3 - 4

4. MATHEMATICAL R E F L E C T I O N

Which function “grows” faster, a linear function or

an exponential function? Explain your reasoning.


Exponential Functions and Graphs ACTIVITY Sizing Up the ......Apr 04, 2012 · Exponential...

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