Function Transformations

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

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.

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

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?