More Parabolas

 

Horizontal Shifting

 Consider the graphs

     y = 

  1. (x + 0)2

  2. (x + 1)2

  3. (x + 2)2

  4. (x + 3)3

 

We see that adding a number inside the parenthesis shifts the graph left or right.  The rules below state this precisely.

 


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

 

 

Vertical Shifting

Consider the graphs

     y  = 

  1. x3

  2. x3 + 1

  3. x3 + 2

  4. x3 + 3

We see that adding a number outside the parenthesis shifts the graph up or down.  The rules below state this precisely.

 


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

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

 

 



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.


Reflecting About the x-axis

Consider the graphs

 y = x2  and y = -x2  

       

We can see that one is a reflection of the other.  This leads us to the fifth rule.

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



Reflecting About the y-axis.

Exercise:

The graphs of 

          

and

       

are the same except they are reflected about the x axis.

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


For an interactive investigation of the shifting rules go to 

Shifting

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