Function Transformations Explorer

Parent Function

Select f(x)

Transformed Function

y = f(x)

Transformation Parameters

Vertical

Vertical stretch (|a|>1) / compress (|a|<1)

Vertical shift up (k>0) / down (k<0)

Horizontal

Horizontal stretch (|b|<1) / compress (|b|>1)

Horizontal shift right (h>0) / left (h<0)

Quick Examples

Transformation Steps

Parent f(x)

Transformed y

The Master Transformation Formula

y = a · f(b(x − h)) + k

a — Vertical stretch/compress and reflection. If |a| > 1 the graph stretches away from the x-axis; if 0 < |a| < 1 it compresses toward the x-axis. Negative a reflects over the x-axis.

b — Horizontal stretch/compress and reflection. If |b| > 1 the graph compresses toward the y-axis (period gets shorter); if 0 < |b| < 1 it stretches away. Negative b reflects over the y-axis.

h — Horizontal shift. Replace x with (x − h): positive h shifts right, negative h shifts left.

k — Vertical shift. Added outside f(...): positive k shifts up, negative k shifts down.

Order of Operations on Transformations

When applying multiple transformations, the inside (horizontal) transformations happen before the outside (vertical) ones — but always read the formula from the inside out:

Horizontal shift: replace x with (x − h) inside f.
Horizontal stretch/reflect: replace x with b·(x − h).
Vertical stretch/reflect: multiply the whole function by a.
Vertical shift: add k to the entire output.
Order matters! h and b are counter-intuitive — always check with a test point.

For sin/cos: period = 2π / |b|, amplitude = |a|. Adjust b to change how many cycles fit in 2π.

Function Transformations Explorer

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