2 · IB Maths
Function transformations
Build y = a·f(b(x − c)) + d from a base function. Each parameter stretches, shifts or reflects the graph in its own way; track one point to see exactly where it goes.
Predict first
Start from the default settings (f(x) = x², a = 1, b = 1, c = 0, d = 0, Tracked point u = 1.0). Set b to 2.0. Where is the image of the tracked point (1, 1)?
The image of (u, f(u)) is (u ÷ b + c, a f(u) + d). With b = 2 the x-coordinate is 1 ÷ 2 = 0.5 while y stays 1, because b squashes the graph horizontally towards the y-axis.
Check it in the model: Move the b slider to 2.0 and read Its image in the readout; watch the gold dot move to (0.5, 1).
Grey: y = f(x). Dark: the transformed graph. Open dot: tracked point on f. Gold dot: its image.
(u, f(u)) → (u ÷ b + c, a f(u) + d)
b, c act inside the bracket (horizontal)
a, d act outside (vertical)
Try this
- Press "Squash: b = 2" and follow the tracked point. Does the image move towards the y-axis or away from it? Explain using u ÷ b.
- Set c = 2 and d = 1 using the sliders, then swap to f(x) = |x|. Find the new vertex and check it against (c, d).
- For f(x) = sin(x), make a = −1 and b = −1 together. Compare the result with a = −1 alone and with b = −1 alone, and say what each reflection does.
Common error: shifting the wrong way inside the bracket. In f(b(x − c)) the graph moves c units to the right, and a bigger b squashes the graph horizontally by a factor of b rather than stretching it.