Function Transformer
Every graph in Grade 11 is some parent shape that's been stretched, flipped and slid around. Learn that one idea here and the whole Functions course gets easier. Grab the graph and move it — works on any phone.
The one formula behind it all
Start with a simple parent function — say f(x) = x². Every related graph is built by wrapping four numbers around it:
- a — stretches it vertically (taller/shorter). Make it negative and the graph flips over the x-axis.
- b — stretches it horizontally (narrower/wider). Negative flips over the y-axis.
- h — slides it left or right.
- k — slides it up or down.
Move the graph 🟠
Drag the orange dot to slide the graph, use the sliders for the stretches, and switch the parent function with the buttons. The dashed grey curve is the untouched parent — watch how far your version has moved from it.
The thing that trips everyone up
Notice the split: a and k (the outside numbers) behave the friendly way — bigger a is taller, positive k goes up. But b and h live inside the brackets, next to the x — and inside, everything works backwards:
Outside the brackets — intuitive
Change a or k and the graph does exactly what you'd guess: it grows taller or moves up. These act on the output (the y-value), after the function runs.
Inside the brackets — reversed
x − 3 moves the graph right by 3 (not left!), and b = 2 makes it narrower, not wider. These act on the input before the function runs, so the effect is inverted.
To feel it: set the parent to x², then drag h using the dot. The equation shows (x − 3) when the graph moves right — the minus sign and the rightward move look contradictory until you see them happen together. That single "aha" is most of what this topic is testing.
Things worth trying
- Pick √x and set b negative — the curve flips to the left side. The domain flips with it.
- Pick 1/x and drag the dot — you're moving the asymptotes (the lines the curve never touches) to x = h and y = k.
- Pick sin x and grow b — the wave gets shorter front-to-back. That horizontal squeeze is exactly what "period" means.
- Turn off "Show parent" and try to read off a, b, h, k from a graph in your head, then turn it back on to check.