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AlgebraGrades 9–124 min read

Function Transformations

Function transformations move, stretch, compress, and reflect a familiar graph without rebuilding it point by point.

Cheat sheet

Transformations move a parent graph in predictable ways. A faint parent parabola and a bold translated parabola share one plane. A horizontal arrow marks the displacement so corresponding features can be compared.

Transformations describe how the graph of $y=f(x)$ changes when numbers are added or multiplied inside and outside the function.

The transformation form

A useful general form is

y=a f\bigl(k(x-d)\bigr)+c.

Start with the parent graph $y=f(x)$. The parameters control four types of change:

$a$ changes vertical scale and may reflect across the $x$-axis;
$k$ changes horizontal scale and may reflect across the $y$-axis;
$d$ translates the graph horizontally;
$c$ translates the graph vertically.

The form is a compact instruction set, not a new function family.

Vertical changes happen outside

In $y=af(x)+c$, every output of $f$ is multiplied by $a$ and then increased by $c$.

If $|a|>1$, the graph stretches vertically. If $0<|a|<1$, it compresses vertically. A negative $a$ reflects outputs across the $x$-axis. Adding $c$ moves every point up by $c$; a negative $c$ moves it down.

A point $(x,y)$ on the parent becomes $(x,ay+c)$ under these outside changes.

Horizontal changes happen inside

In $f(k(x-d))$, inputs must compensate for what occurs inside the function. The horizontal scale factor is $1/|k|$, not $k$. If $|k|>1$, the graph compresses horizontally; if $0<|k|<1$, it stretches. A negative $k$ reflects across the $y$-axis.

The expression $x-d$ shifts the graph right by $d$. Thus $f(x-4)$ moves right $4$, while $f(x+4)=f(x-(-4))$ moves left $4$.

Point-mapping rule

If $(x,y)$ lies on $y=f(x)$, then the corresponding point on

$$ y=a f(k(x-d))+c $$

\left(\frac{x}{k}+d,\ ay+c\right).

This rule is especially useful when a parent graph is supplied as a table or sketch rather than a formula.

Worked example from a parent function

Order and structure

Read horizontal parameters from the inside but use the completed form $k(x-d)$. For example, $f(2x-6)$ must first be factored as $f(2(x-3))$, revealing $k=2$ and $d=3$.

For point mapping, horizontal coordinates divide by $k$ and then add $d$. Vertical coordinates multiply by $a$ and then add $c$. Keeping the horizontal and vertical rules separate reduces sign errors.

Domain, range, and key features

Transformations also move important features. Translate endpoints, intercepts, maxima, minima, and asymptotes using the same logic.

For $a\ne0$ and $k\ne0$, a horizontal transformation changes the domain through $x\mapsto x/k+d$, while a vertical transformation changes the range through $y\mapsto ay+c$. A horizontal asymptote $y=L$ becomes $y=aL+c$.

Building an equation from a description

Suppose $y=|x|$ is reflected across the $x$-axis, vertically stretched by $3$, shifted left $2$, and moved up $5$. Apply each instruction to the correct location:

$$ y=-3|x+2|+5. $$

The left shift appears inside with the opposite-looking sign. The vertical changes appear outside with direct signs.

Common mistakes

Using $k$ as the horizontal scale factor. The factor is $1/|k|$.

Reading $f(x+3)$ as right $3$. Rewrite it as $f(x-(-3))$: it moves left $3$.

Failing to factor the inside. $f(2x-6)=f(2(x-3))$ has shift $3$, not $6$.

Applying outside changes to $x$. Parameters $a$ and $c$ transform outputs.

Transforming only the picture. Domain, range, intercepts, and asymptotes must move too.

Quick self-check

Is the inside written as $k(x-d)$?
What is the horizontal factor $1/|k|$?
Are there reflections from negative $a$ or $k$?
Where do key points map under $(x,y)\mapsto(x/k+d,ay+c)$?
Do the new domain and range match the graph?

Explore the idea