Function Transformations

move and flip Just like Transformations in Geometry, we can move and resize the graphs of functions

Let us start with a function, in this case it is f(x) = x2, but it could be anything:

Square function

f(x) = x2

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:

Translation

g(x) = x2 + C

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:

Translation

g(x) = (x+C)2

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 would get it when you are 21. Adding 4 made it happen earlier.

  • C > 0 moves it left
  • C < 0 moves it right

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

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

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)

 

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

Scaling

g(x) = 0.35(x2)

  • 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.

Scaling

g(x) = (2x)2

  • 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:

Scaling

g(x) = −(x2)

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:

Scaling

g(x) = (−x)2

It really does flip it left and right! But you can't see it, because x2 is symmetrical about the y-axis. So here is another example using √(x):

Scaling

g(x) = √(−x)

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

Summary

y = f(x) + C
  • C > 0 moves it up
  • C < 0 moves it down
y = f(x + C)
  • C > 0 moves it left
  • C < 0 moves it right
y = Cf(x)
  • C > 1 stretches it in the y-direction
  • 0 < C < 1 compresses it
y = f(Cx)
  • C > 1 compresses it in the x-direction
  • 0 < C < 1 stretches it
y = −f(x)
  • Reflects it about x-axis
y = f(−x)
  • Reflects it about y-axis

 

Examples

Example: the function g(x) = 1/x

Here are some things we can do:

Move 2 spaces up:   h(x) = 1/x + 2
Move 3 spaces down:   h(x) = 1/x − 3
Move 4 spaces to the right:   h(x) = 1/(x−4) graph
Move 5 spaces to the left:   h(x) = 1/(x+5)
Stretch it by 2 in the y-direction:   h(x) = 2/x
Compress it by 3 in the x-direction:   h(x) = 1/(3x)
Flip it upside down:   h(x) = −1/x

Example: the function v(x) = x3 − 4x

Here are some things we can do:

Move 2 spaces up:   w(x) = x3 − 4x + 2
Move 3 spaces down:   w(x) = x3 − 4x − 3
Move 4 spaces to the right:   w(x) = (x−4)3 − 4(x−4)
Move 5 spaces to the left:   w(x) = (x+5)3 − 4(x+5)  graph
Stretch it by 2 in the y-direction:   w(x) = 2(x3 − 4x) = 2x3 − 8x
Compress it by 3 in the x-direction:   w(x) = (3x)3 − 4(3x) = 27x3 − 12x
Flip it upside down:   w(x) = −x3 + 4x

All In One ... !

We can do all transformation in one go using this:

af( b(x + c) ) + d

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

 

Example: 2√(x+1)+1

a=2, c=1, d=1

So it takes the square root function, and then

  • Stretches it by 2 in the y-direction
  • Shifts it left 1, and
  • Shifts it up 1

Play with this graph