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Transformation of Functions Exploration Lesson

Module by: Debbie Trahan. E-mail the author

Summary: In this module the students will explore transformations of a square root function algebraically, graphically, and using a table of values.

Transformation of Functions Exploration

In this lesson you will be investigating the transformation of a function by examining the notations, the tables of values and the graphs of the function and its transformations. After working through problems 1 - 11, you will be able to explain the effects of the following transformations on the graph of the function f⁢x f x if c>0 c 0 .

f⁢x+c f x c ; f⁢x−c f x c ; f⁢x+c f x c ; f⁢x−c f x c ; c⁢f⁢x c f x if 0<c<1 0 c 1 ; c⁢f⁢x c f x if c>1 c 1 ; f⁢c⁢x f c x if 0<c<1 0 c 1 ; f⁢c⁢x f c x if c>1 c 1 ; −f⁢x f x ; f⁢−x f x .

In problems 1 - 11, you will examine the function f⁢x=x f x x and its transformations.

Exercise 1

Given the function f⁢x=x f x x and the table of values for f⁢x f x , graph y=f⁢x y f x on the grid (figure 1).

Figure 1
Figure 1 (prob1.png)

Solution

Figure 2
Figure 2 (ans1.png)

Exercise 2

The function y=x+2 y x 2 is a transformation of f⁢x f x . The notation f⁢x+2 f x 2 can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 3) . Explain how the transformation changed the graph of f⁢x f x .

Figure 3
Figure 3 (proba.png)

Solution

The graph of f f has been translated 2 units to the left. (figure 4)

Figure 4
Figure 4 (ansa.png)

Exercise 3

The function y=x−1 y x 1 is a transformation of f⁢x f x . The notation f⁢x−1 f x 1 can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 5) . Explain how the transformation changed the graph of f⁢x f x .

Figure 5
Figure 5 (probb.png)

Solution

The graph of f f has been translated 1 unit to the right. (figure 6)

Figure 6
Figure 6 (ansb.png)

Exercise 4

The function y=-3+x y -3 x is a transformation of f⁢x f x . The notation f⁢x−3 f x 3 can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 7) . Explain how the transformation changed the graph of f⁢x f x .

Figure 7
Figure 7 (probc.png)

Solution

The graph of f f has been translated 3 units down. (figure 8)

Figure 8
Figure 8 (ansc.png)

Exercise 5

The function y=1+x y 1 x is a transformation of f⁢x f x . The notation f⁢x+1 f x 1 can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 9) . Explain how the transformation changed the graph of f⁢x f x .

Figure 9
Figure 9 (probd.png)

Solution

The graph of f f has been translated 1 unit up. (figure 10)

Figure 10
Figure 10 (ansd.png)

Exercise 6

The function y=2⁢x y 2 x is a transformation of f⁢x f x . The notation 2⁢f⁢x 2 f x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 11) . Explain how the transformation changed the graph of f⁢x f x .

Figure 11
Figure 11 (probe.png)

Solution

The graph of f f has been vertically stretched by a scale factor of 2. (figure 12)

Figure 12
Figure 12 (anse.png)

Exercise 7

The function y=.5⁢x y .5 x is a transformation of f⁢x f x . The notation .5⁢f⁢x .5 f x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 13) . Explain how the transformation changed the graph of f⁢x f x .

Figure 13
Figure 13 (probf.png)

Solution

The graph of f f has been vertically compressed by a scale factor of 0.5 . (figure 14)

Figure 14
Figure 14 (ansf.png)

Exercise 8

The function y=2⁢x y 2 x is a transformation of f⁢x f x . The notation f⁢2⁢x f 2 x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 15) . Explain how the transformation changed the graph of f⁢x f x .

Figure 15
Figure 15 (probg.png)

Solution

The graph of f f has been horizontally compressed by a scale factor of 0.5 . (figure 16)

Figure 16
Figure 16 (ansg.png)

Exercise 9

The function y=.5⁢x y .5 x is a transformation of f⁢x f x . The notation f⁢.5⁢x f .5 x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 17) . Explain how the transformation changed the graph of f⁢x f x .

Figure 17
Figure 17 (probh.png)

Solution

The graph of f f has been horizontally stretched by a scale factor of 2. (figure 18)

Figure 18
Figure 18 (ansh.png)

Exercise 10

The function y=−x y x is a transformation of f⁢x f x . The notation −f⁢x f x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 19) . Explain how the transformation changed the graph of f⁢x f x .

Figure 19
Figure 19 (probi.png)

Solution

The graph of f f has been reflected over the x - axis. (figure 20)

Figure 20
Figure 20 (ansi.png)

Exercise 11

The function y=−x y x is a transformation of f⁢x f x . The notation f⁢−x f x can be used to show this transformation. Graph f⁢x f x in red. Complete the table of values then use the table of values to help you graph the transformation (figure 21) . Explain how the transformation changed the graph of f⁢x f x .

Figure 21
Figure 21 (probj.png)

Solution

The graph of f f has been reflected over the y - axis. (figure 22)

Figure 22
Figure 22 (ansj.png)

Answer questions 12 - 21 to test your knowledge of functions and their transformations.

Exercise 12

Given the function g⁢x g x , c c is a real number , and c>0 c 0 explain how the transformation g⁢x+c g x c changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be translated c c units up.

Exercise 13

Given the function g⁢x g x , c c is a real number , and c>0 c 0 explain how the transformation g⁢x−c g x c changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be translated c c units down.

Exercise 14

Given the function g⁢x g x , c c is a real number , and c>0 c 0 explain how the transformation g⁢x+c g x c changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be translated c c units to the left.

Exercise 15

Given the function g⁢x g x , c c is a real number , and c>0 c 0 explain how the transformation g⁢x−c g x c changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be translated c c units to the right.

Exercise 16

Given the function g⁢x g x , c c is a real number , and c>1 c 1 explain how the transformation g⁢c⁢x g c x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be horizontally compressed by a scale factor of 1c 1 c .

Exercise 17

Given the function g⁢x g x , c c is a real number , and 0<c<1 0 c 1 explain how the transformation g⁢c⁢x g c x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be horizontally stretched by a scale factor of c c .

Exercise 18

Given the function g⁢x g x , c c is a real number , and c>1 c 1 explain how the transformation c⁢g⁢x c g x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be vertically stretched by a scale factor of c c .

Exercise 19

Given the function g⁢x g x , c c is a real number , and 0<c<1 0 c 1 explain how the transformation c⁢g⁢x c g x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be vertically compressed by a scale factor of 1c 1 c .

Exercise 20

Given the function g⁢x g x explain how the transformation −g⁢x g x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be reflected over the x - axis.

Exercise 21

Given the function g⁢x g x explain how the transformation g⁢−x g x changes the graph of g⁢x g x .

Solution

The graph of g⁢x g x will be reflected over the y - axis.

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 (Graphically) Module

Go to Transformation of Functions (Verbally) Module

Go to Transformation of Functions using a Graphing Calculator Module

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