Graph Transformations

Shifts, reflections, and stretches read off a parent graph.

The idea

A graph transformation rewrites a function's formula to slide, reflect, or stretch its graph. A handful of parent curves — $x^{2}$, $|x|$, $\sqrt{x}$, $\sin x$ — underlie most graphs you meet, so recognising the parent graph and the moves applied to it is faster than plotting points.

Proposition (Transformation rules).

Let $y = f(x)$ be a parent graph, and let $k$, $h$ be constants and $a, c > 0$. The graph of $y = f(x) + k$ is the parent shifted up by $k$; of $y = f(x - h)$, shifted right by $h$; of $y = -f(x)$, reflected over the $x$-axis; of $y = a\,f(x)$, stretched vertically by the factor $a$; of $y = f(-x)$, reflected over the $y$-axis; and of $y = f(cx)$ with $c > 1$, compressed horizontally by the factor $c$.

A change written outside $f$ happens after the function acts, so it alters the output and moves the graph vertically, in the direction the sign suggests: $f(x) + k$ raises every height by $k$, $a\,f(x)$ multiplies every height by $a$, and $-f(x)$ reflects the curve over the $x$-axis.

A change written inside $f$ happens before the function acts, so it alters the input, and the effect is horizontal and opposite to the sign. The graph of $y = f(x - h)$ shows at position $x$ whatever the parent shows at $x - h$, so every feature of the parent reappears $h$ units to the right: subtracting inside shifts the graph right. For the same reason $f(-x)$ reflects the graph over the $y$-axis, and $f(cx)$ with $c > 1$ compresses it horizontally.

Applied one at a time, the moves rebuild the graph from its parent and carry the parent's key points — a vertex, an intercept, a peak — along with them.

Ways to work on it

• Walkthrough. Shifts, reflections, and stretches of a parent graph.
• Practice. Identify the effect of a single transformation.
• Hardest. Combine an inside scaling with a shift and locate the new endpoint.

Not sure where to start? Take the ten-question placement test.