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Transformation of Functions (Verbally)

Module by: Debbie Trahan

Summary: The objective of this module is for students to relate the transformations of functions graphically to real-world applications.

The objective of this module is to relate the transformation of a function to a real-world application.

Denise's Bicycle Trip Problem

Denise took a bicycle ride away from her home today. She left home at 12 a.m. and arrived back at home at 8 pm. The graph shown in figure 1 represents Denise's distance from home during her ride.

Figure 1
Figure 1 (vprob.gif)

Let the function dt d t represent Denise's distance in miles from home during her trip in terms of the time in hours.

Example 1

The graph shown in figure 2 is a transformation of the function dt d t shown in figure 1.

Figure 2
Figure 2 (vprob1.gif)

Example a : Explain the real-world meaning of this transformation.

Answer a: The graph has been translated 3 units to the right. Denise started her trip 3 hours later (3 p.m.) and arrived home 3 hours later (12 p.m.).

Example b: Use function notation to represent this transformation.

Answer b: dt-3 d t 3

Transfer the graph of the function dt d t ,shown in figure 1, to your paper. Use the graph to help you answer parts a and b for each of the transformations of the function dt d t graphed in problems 1 - 5.

Exercise 1

Figure 3
Figure 3 (vprob2.gif)

a. Explain the real-world meaning of this transformation.

b. Use function notation to represent this transformation.

Solution 1

Answer a: The graph has been translated up 5 units. Denise started her ride 5 miles from home and ended her ride 5 miles from home.

Answer b: dt+5 d t 5

Exercise 2

Figure 4
Figure 4 (vprob3.gif)

a. Explain the real-world meaning of this transformation.

b. Use function notation to represent this transformation.

Solution 2

Answer a: The graph has been horizontally compressed by a scale factor of 1/2. Denise rode the same distance in 1/2 of the time (4.5 hours).

Answer b: d2t d 2 t

Exercise 3

Figure 5
Figure 5 (vprob4a.gif)

a. Explain the real-world meaning of this transformation.

b. Use function notation to represent this transformation.

Solution 3

Answer a: The graph has been horizontally stretched by a scale factor of 2. Denise rode the same distance in twice the time (18 hours).

Answer b: d12t d 1 2 t

Exercise 4

Figure 6
Figure 6 (vprob5.gif)

a. Explain the real-world meaning of this transformation.

b. Use function notation to represent this transformation.

Solution 4

Answer a: The graph has been vertically stretched by a scale factor of 2. Denise rode twice the distance in the same amount of time.

Answer b: 2dt 2 d t

Exercise 5

Figure 7
Figure 7 (vprob6.gif)

a. Explain the real-world meaning of this transformation.

b. Use function notation to represent this transformation.

Solution 5

Answer a: The graph has been vertically compressed by a scale factor of 1/4. Denise rode 1/4 the distance in the same amount of time.

Answer b: 14dt 1 4 d t

Exercise 6

Figure 8
Figure 8 (vprob.gif)

Create your own transformation of the graph of Denise's bicycle trip. Explain the real-world meaning of your transformation and use function notation to represent your transformation.

Solution 6

Answers will vary. Exchange transformations with another student and check each other's transformation.

Fish in a Lake Problem

Daniel has a lake in his backyard. He plans to raise fish in his lake. He released 100 fish in his lake in 1996. The graph in figure 9 represents the number of fish in Daniel's lake in terms of the number of years since 1996.

Figure 9
Figure 9 (fprob.gif)

Let the function ft f t represent the number of fish in Daniel's lake in terms of the number of years since 1996.

Transfer the graph of the function ft f t to your paper. Use this graph to help you answer problems 7 - 13.

Exercise 7

Figure 10
Figure 10 (fprob1.gif)

Explain the meaning of the transformation of the function ft f t shown in figure 10 in the context of this problem then use function notation to represent this transformation.

Solution 7

Daniel released the fish in 1998.

ft-2 f t 2

Exercise 8

Figure 11
Figure 11 (fprob2.gif)

Explain the meaning of the transformation of the function ft f t shown in figure 11 in the context of this problem then use function notation to represent this transformation.

Solution 8

Daniel released the fish in 1992.

ft+4 f t 4

Exercise 9

Figure 12
Figure 12 (fprob3.gif)

Explain the meaning of the transformation of the function ft f t shown in figure 12 in the context of this problem then use function notation to represent this transformation.

Solution 9

Daniel released 100 more fish into the pond in 1996 which increased the number of fish in the pond each year by 100 more fish.

ft+1 f t 1

Exercise 10

Figure 13
Figure 13 (fprob4.gif)

Explain the meaning of the transformation of the function ft f t shown in figure 13 in the context of this problem then use function notation to represent this transformation.

Solution 10

The fish population increased the same amount as the population in 3.5 years which is 1/2 the time as the function shown in figure 9.

f2t f 2 t

Exercise 11

Figure 14
Figure 14 (fprob5.gif)

Explain the meaning of the transformation of the function ft f t shown in figure 14 in the context of this problem then use function notation to represent this transformation.

Solution 11

The fish population increased by 1/2 as much as the fish population shown in figure 9.

12ft 1 2 f t

Exercise 12

Figure 15
Figure 15 (fprob6.gif)

Use function notation to represent the transformation of the function ft f t shown in figure 15.

Solution 12

f-t f t

Exercise 13

Figure 16
Figure 16 (fprob7.gif)

Use function notation to represent the transformation(s) of the function ft f t shown in figure 16. (Hint: You may need to use more than one transformation.)

Solution 13

-ft+13 f t 13

Transformations of Functions Exploration Modules

After you have completed this lesson you can continue to explore transformations of functions by choosing a module listed below.

Go to Transformation of Functions Exploration Module

Go to Transformation of Functions (Graphically) Module

Go to Transformation of Functions using a Graphing Calculator Module

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