Function Transformer
Every graph in y = a·f(b(x − h)) + k is a parent graph that has been stretched, flipped and slid. Move the sliders, or drag the curve itself, and see which number does what.
Outside changes y, inside changes x
Numbers outside the function act on the outputs, so they move the graph vertically: a stretches it (and flips it if a < 0), k lifts it. Numbers inside act on the inputs, so they move it horizontally, and they work backwards: b = 2 squeezes the graph to half its width, and (x − 3) moves it 3 to the right.
Why "backwards"?
To get the same output as the parent gave at x = 0 you now need x − 3 = 0, which is x = 3. Every point slides 3 to the right. In the same way, f(2x) reaches each output when x is only half as big. Follow the highlighted point: the parent's point (x₀, y₀) lands at (x₀/b + h, a·y₀ + k).
Dragging the curve changes h and k. Dashed grey = parent graph.