Transforming Graphs

BEWARE! THIS IS HARD!!

NOTE: f(x) just means an equation like y = something x. It’s not something to be scared by!

There are a few ways you might be asked to transform graphs.

y Shift: y = f(x) + a

Adding or subtracting a number to the end of the equation just moves the graph up or down.

So below:

  • Red line is y = sin x
  • Blue line is y = (sin x) + 3
  • Green line is y  = (sin x) – 1

y shift

x Shift: y = f(x + a)

This is where you replace x everywhere in the equation with (x + a)

Anything you do to x you must now do to (+ a). So if before you had xyou would now replace it with (x + a)²

If a is positive you shift the graph to the left. If a is negative you shift it to the right. BE CAREFUL because this is probably the opposite of what you would expect instinctively.

So below:

  • Red line is y = 
  • Yellow line is y = (x + 2)²

x shift

y Stretch: y = kf(x)

This involves multiplying the whole function by a number.

If k is greater than one then it stretches the graph parallel to the y axis. If it is less than 1 is squashes the graph parallel to the y axis.

So below:

Yellow line is y = 
Red line is y = 2

y stretch

x Stretch: y = f(kx)

For this one you replace all the values of x with x multiplied or divide by k.

If k is a multiplier it scrunches up the graph parallel to the x axis.

If k is a divider it stretches the graph.

Beware this is again probably counter intuitive!

So below:

  • Red line is y = sin(x)
  • Blue line is y = sin(2x)

x stretch

Reflections: y = -f(x) and y = f(-x)

These are quite easy….just remember:

  • To get y = -f(x) you just flip y = f(x) in the x axis
  • To get y = f(-x) you just flip y = f(x) in the y axis