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.

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.

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