Function Transformations

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 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.

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 right:h(x) = 1/(x−4) graph

Move 5 spaces 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 right:w(x) = (x−4)³ − 4(x−4)

Move 5 spaces 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 transformations on f() 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

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