Shift, stretch, flip and invert

Move a graph around with y = a·f(x − h) + k, and mirror it to see its inverse.

Interactive

Start from

a: stretch (negative flips)1h: shift right0k: shift up0Show the inverse

The math, with your numbers

y = x^2

f(x) = x^2 \quad\longleftrightarrow\quad f^{-1}(x) = \sqrt{x}

Subtracting h inside moves the graph right by h: the input has to be h bigger to give the same output.

Try this

→x²: move the lowest point to (2, −1). Which sliders did you use?
→Turn on the inverse for 2ˣ. Which function's graph do you see?
→x²: why does its mirror image fail to be a function? Look at x = 4.

Want to know why it behaves like this?

The lesson “Functions & Inverses: Running Rules Backwards” explains it step by step: How do you turn a formula around to answer the opposite question?

Read the lesson

Shift, stretch, flip and invert

Move a graph around with y = a·f(x − h) + k, and mirror it to see its inverse.

Shift, stretch, flip and invert

Interactive

Start from

a: stretch (negative flips)1h: shift right0k: shift up0Show the inverse

The math, with your numbers

y = x^2

f(x) = x^2 \quad\longleftrightarrow\quad f^{-1}(x) = \sqrt{x}

Subtracting h inside moves the graph right by h: the input has to be h bigger to give the same output.

Try this

→x²: move the lowest point to (2, −1). Which sliders did you use?
→Turn on the inverse for 2ˣ. Which function's graph do you see?
→x²: why does its mirror image fail to be a function? Look at x = 4.

Want to know why it behaves like this?

The lesson “Functions & Inverses: Running Rules Backwards” explains it step by step: How do you turn a formula around to answer the opposite question?

Read the lesson