top of page
This site was designed with the
.com
website builder. Create your website today.Start Now
f(x)

Building Functions

Read about Building Functions.  Then, test your knowledge!

Click To Watch Your Functions Come To Life!

Building Functions:  Transformations

​

We can move and resize, TRANSFORM, the graphs of functions.

​

Let us start with a function, in this case it is                , but it could be anything:

Here are some simple things we can do to move or scale it on the graph:

​

We can move it up or down by adding a constant to the y-value:

Note: to move the line down, we use a negative value for C.

​

  • C > 0 moves it up

​​

  • C < 0 moves it down

​

We can move it left or right by adding a constant to the x-value:

Adding C moves the function to the left (the negative direction).

​

Why? Well imagine you will inherit a fortune when your age=25. If you change that to (age+4) = 25 then you will get it when you are 21. Adding 4 made it happen earlier.

​

  • C > 0 moves it left

​​

  • C < 0 moves it right

 

BUT we must add C wherever x appears in the function (we are substituting x+C for x).

 

Example: the function v(x) = x3 - x2 + 4x

​

To move C spaces to the left, add C to x wherever x appears:

​

w(x) = (x + C)3 − (x + C)2 + 4(x + C)

 

An easy way to remember what happens to the graph when we add a constant:

 

add to y to go high
add to x to go left

 

We can stretch or compress it in the y-direction by multiplying the whole function by a constant.

  • C > 1 stretches it

​​

  • 0 < C < 1 compresses it

 

We can stretch or compress it in the x-direction by multiplying x by a constant.

  • C > 1 compresses it

​​

  • 0 < C < 1 stretches it

​​

Note that (unlike for the y-direction), bigger values cause more compression.

 

We can flip it upside down by multiplying the whole function by −1:

This is also called reflection about the x-axis (the axis where y=0)

​

We can combine a negative value with a scaling:

​

Example: multiplying by −2 will flip it upside down AND stretch it in the y-direction.

 

We can flip it left-right by multiplying the x-value by −1:

It really does flip it left and right! But you can't see it, because        is symmetrical about the y-axis. So here is another example using √(x):

This is also called reflection about the y-axis (the axis where x=0)

​

​

Summary

This is also called reflection about the y-axis (the axis where x=0)

​

​

Examples

All In One!

​

We can do all transformation in one go using this:

a is vertical stretch/compression

  • |a| > 1 stretches 

  • |a| < 1 compresses 

  • a < 0 flips the graph upside down

 

b is horizontal stretch/compression

  • |b| > 1 compresses 

  • |b| < 1 stretches 

  • b < 0 flips the graph left-right

 

c is horizontal shift

  • c < 0 shifts to the right 

  • c > 0 shifts to the left

 

d is vertical shift

  • d > 0 shifts upward 

  • d < 0 shifts downward

bottom of page