Mathematics · Functions

Transformations of Functions

Reference entry · last updated September 20, 2026

A transformation of a function is a change to its graph produced by altering the input or the output of the rule. Every transformation of a parent function \(f\) can be written in the form below, where each parameter has a fixed geometric effect.

1. First principles: the general form

\( g(x) = a \, f\big(b(x - h)\big) + k \)

The parent function is \(f\). The parameters act independently: \(a\) scales the output, \(b\) scales the input, \(h\) shifts horizontally, and \(k\) shifts vertically [1, 2]. A point \((x, y)\) on the parent graph maps to the point \(\left(\dfrac{x}{b} + h, \; a y + k\right)\) on the transformed graph.

2. Translations

Vertical shift by \(k\): \( g(x) = f(x) + k \). Positive \(k\) moves the graph up.
Horizontal shift by \(h\): \( g(x) = f(x - h) \). Positive \(h\) moves the graph right. The direction is opposite the sign in the expression because the input must change to compensate.

3. Reflections

Across the \(x\)-axis: \( g(x) = -f(x) \). Outputs change sign, so the graph flips vertically.
Across the \(y\)-axis: \( g(x) = f(-x) \). Inputs change sign, so the graph flips horizontally.

4. Stretches and compressions

Parameter	Effect	Rule
\( a > 1 \)	Vertical stretch	Outputs multiplied, graph taller
\( 0 < a < 1 \)	Vertical compression	Outputs reduced, graph shorter
\( b > 1 \)	Horizontal compression	Inputs divided, graph narrower
\( 0 < b < 1 \)	Horizontal stretch	Inputs divided by a fraction, graph wider

Horizontal scaling works with the reciprocal of \(b\), which reverses the intuition from vertical scaling. A factor of \(2\) inside the function doubles the horizontal rate and halves the width.

5. Order of operations

Apply the inside changes first: horizontal shift and horizontal scaling act on the input. Then apply the outside changes: vertical scaling, reflection, and vertical shift. In the form \(a f(b(x - h)) + k\), the input is first shifted by \(h\), then scaled by \(b\).

6. Even and odd symmetry

Even function: \( f(-x) = f(x) \). The graph is symmetric about the \(y\)-axis. Example: \( f(x) = x^2 \).
Odd function: \( f(-x) = -f(x) \). The graph has rotational symmetry about the origin. Example: \( f(x) = x^3 \).

References

^OpenStax, College Algebra 2e, Rice University (Transformation of functions).
^Ron Larson, Precalculus with Limits, 4th ed., Cengage Learning, 2018.