Module 6: MORE ON FUNCTIONS
  Exploring Calculus

You will be working on the functions y1 = , y2 =  and y3 =  . You will determine the domain and range of these and other functions in this section and investigate the properties--translation and reflection.
 

    Display the graph for this module.  At first, select N when asked if you want to modify the original functions.

Critical Thinking Questions
 

1. What are the domain and range of each function? Click here to display the hint on domain and range.    Hint
 

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y1

y2

y3

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Domain

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Range

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2. Replace x in each function by x - 2 . What effect does this have on each of the graphs?
 

3. What do you think would happen if you had replaced x with x + 2 ?
 

4. Now add 2 to the right side of each function. What happens in each case?
 

5. Consider the function y + 2 = f (x) .

    a. Since you must enter functions into the computer in the form y = f (x) , you cannot key in this function. How do you fix this?

    b. Adding 2 to the left side of a function has what effect on the graph?
 

6. Write the equations for the graph y = f (x)

    a. shifted a units to the left ;

    b. shifted a units to the right ;

    c. shifted a units up ;

    d. shifted a units down ; and

    e. reflected about the x-axis.
 
 

Skill Exercises
 
 

1. Let f (x) = . Find f (0) and f (-5) .
 

2. Given

    y1 = 

    y2 = 

    y3 = 

fill in the chart below.
 

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y1

y2

y3

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Domain

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Range

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3. What are the domain and range of the following ?

    f (x) = 

    g (x) = 
 

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f (x)

g (x)

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Domain

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Range

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4. For a function f (x) of your choice

    a. Sketch the graph of f (x) and -f (x).

Specify the function you have chosen.

    b. Describe the relationship between the graphs of y = f (x) and y = -f (x) .
 

5. Without using the computer, determine how the graph of f (x) = | x + 2 |  -  4 is shifted with respect to f (x) = | x |  .
 

6. a. For f (x) =  , rewrite f (x) so that the graph of f (x) is reflected about the x-axis and shifted 4 units to the right.

    b. For g (x) = x2 , rewrite g (x) so that the graph of g (x) is shifted 1 unit to the left and 3 units up. What special point can you read when g (x) is written this way?
 
 

Exploring Calculus