Shifting and Reflecting

1. Six Basic Functions
   
   Below are six basic functions:
   
   

   1. 
      
      
      
      

   2. 
      
      
      

   3. 
      
      
      

   4. 
      
      
      

   5. 
      
      
      
      

   6. 
      
      
      Memorize the shapes of these functions.
      
      

2. Horizontal Shifting

    Consider the graphs
   

   y = 
   

   1. (x + 0)²
      

   2. (x + 1)²
      

   3. (x + 2)²
      

   4. (x + 3)³

   
   
   
   
   

   
   
   

   Exercise
   
   
   Use the list features of a calculator to sketch the graph of 
   
           y = 1/[x - {0,1,2,3}]
   

    

   Rule1:  f(x - a) = f(x) shifted a units to the right. Rule 2:  f(x + a) = f(x) shifted a units to the left

   
   

3. Vertical Shifting

   Consider the graphs
   

   y = 
   

   1. x³
      

   2. x³ + 1
      

   3. x³ + 2
      

   4. x³ + 3

   
   
   
   
   

   
   
   

   Exercise
   
   
   Use the list features of a calculator to sketch the graph of 
   
           y = x³  - {0,1,2,3}
   

    

   Rule 3:  f(x ) + a = f(x) shifted a units up. Rule 4:  f(x) - a = f(x) shifted a units down.

   
   

4. Reflecting About the x-axis

   Consider the graphs

    y = x²  and y = -x²  

   
   

   Rule 5:  -f(x ) = f(x)  reflected about the x-axis.

   
   
   

5. Reflecting About the y-axis.

   Exercise:
   
   Use the calculator to graph

            

   and
   
          
   

   Rule 6:  f(-x ) = f(x) reflected about the y-axis.

   
   

6. Stretching and Compressing

   Exercises
   
   Graph the following
   
   

   1. y = {1,2,3,4}x³
      

   2. y = {1/2,1/3,1/4,1/5}x³
      

   
   

   Rule 7:  cf(x ) = f(x) (for c > 1) stretched vertically. Rule 8:  cf(x ) = f(x) (c < 1) compressed vertically.

We will do some examples (including the graph of the winnings for the gambler and for the casino.  

Exercises:  Graph the following

1.  y = x² - 10
   

2. y = sqrt(x - 2)
   

3.  y = -|x - 5| + 3 
   
   

For an interactive investigation of the shifting rules go to 

http://mathcsjava.emporia.edu/~greenlar/Shifter/Shifter1.html

7. Increasing and Decreasing Functions

   A function is called increasing if as an object moves from left to right, it is moving upwards along the graph.  

           If 
   
           x < y 
   
   then 
   
           f(x) < f(y)

   A function is called decreasing if as an object moves from left to right, it is moving downwards along the graph.  

   If 
   
           x <  y 
   
   then 
   
           f(x) > f(y)

   Example:  

   The curve 
   
           y = x²  
   
   is increasing on (0, ) and decreasing on (-,0)

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