Combined transformations: y = a·f(b(x − h)) + k

Year 11 Mathematics Advanced · axes fixed −8 to 8 · anchor point tracked through every stage · mapping x → x/b + h, y → ay + k

Base f(x)

base f(x) transformed anchor image

Drag the anchor point along the dashed base curve to test the mapping numerically.

a vertical 1

b horizontal 1

h across 0

k up 0

a and b skip zero. b = 2 divides every x-coordinate by 2, so it compresses horizontally.

anchor on base (1, 1)

image (1, 1)

check 1/1 + 0, 1×1 + 0

factored y = f(x)

same curve y = f(x)

substituted y = √x

Both lines are the same object: the bracketed form shows the true translation, the expanded form hides it.

Load one, then the other with ghosting on.

Dilate horizontally by 1/b, translate by h, dilate vertically by a, translate by k.

A hidden transformation of √x in gold — match it with the four sliders.