Transformations of Functions: Shifts, Stretches and Reflections

One parent, many graphs

Start with a function and call it $f$. Small changes to the formula move, stretch or flip its graph without changing its basic shape.

That is worth knowing because it turns a library of graphs into one graph plus a set of moves. Every parabola is $y = x^2$ transformed.

Change	Effect on the graph
$f(x) + k$	up $k$
$f(x) - k$	down $k$
$f(x + k)$	left $k$
$f(x - k)$	right $k$
$k \cdot f(x)$, $k > 1$	vertical stretch
$k \cdot f(x)$, $0 < k < 1$	vertical squeeze
$f(kx)$, $k > 1$	horizontal squeeze
$-f(x)$	reflect over the $x$-axis
$f(-x)$	reflect over the $y$-axis

Outside the parentheses acts on the output, so it behaves as written. Inside acts on the input, so it behaves backwards. That single sentence covers most of the table.

Vertical shifts read as written

$$g(x) = f(x) + 3$$

Every output is $3$ larger, so every point moves up $3$. Nothing about the horizontal position changes.

• y = x^2
• y = x^2 + 3

The two curves are the same width and the same shape. Only their height differs.

Why horizontal shifts go the other way

$$g(x) = f(x + 3)$$

This one surprises people, so it is worth slowing down.

$g$ at $x = -3$ computes $f(-3 + 3) = f(0)$. So whatever $f$ did at $0$, $g$ does at $-3$ — three units to the left.

$x$	$g(x) = f(x+3)$ evaluates	Which $f$ did at
$-3$	$f(0)$	$0$
$-2$	$f(1)$	$1$
$0$	$f(3)$	$3$

Every landmark of $f$ appears $3$ earlier in $g$. Adding inside makes the input arrive sooner, which slides the picture left.

• y = x^2
• y = (x + 3)^2

This is the same fact as vertex form. In $y = (x - h)^2 + k$, the minus inside is why the vertex is at $+h$.

Stretches

$$g(x) = k \cdot f(x)$$

Multiplying the output scales every height by $k$.

$k$	Effect
$k > 1$	taller — a vertical stretch
$0 < k < 1$	flatter — a vertical squeeze
$k < 0$	scaled and flipped over the $x$-axis

Points already on the $x$-axis do not move, because $k \cdot 0 = 0$. That makes the $x$-intercepts the fixed points of a vertical stretch — a useful check.

Multiplying the input squeezes horizontally, and again the direction is backwards:

$$g(x) = f(2x)$$

$g$ at $x = 3$ gives $f(6)$, so $g$ reaches at $3$ what $f$ reached at $6$. Everything arrives at half the $x$-value, so the graph is squeezed toward the $y$-axis by a factor of $2$.

Reflections

Form	Reflects over	Because
$-f(x)$	the $x$-axis	every output flips sign
$f(-x)$	the $y$-axis	every input flips sign

Once more the pattern holds: the minus outside affects heights, the minus inside affects positions along the $x$-axis.

For $f(x) = x^2$ the second does nothing visible, since $(-x)^2 = x^2$. A parabola is already symmetric about the $y$-axis, so reflecting it there lands it on itself.

Reading k off a picture

Given the original and the moved graph, compare one landmark on each.

A parabola with vertex $(0, 0)$ becomes one with vertex $(2, -5)$.

The vertex went right $2$ and down $5$:

$$g(x) = f(x - 2) - 5$$

Right $2$ is a minus inside; down $5$ is a minus outside.

Take the transformations in order: horizontal first, then vertical. Stated the other way, the shift amounts stay the same but any stretch has to be applied before the vertical shift, or the shift gets stretched too.

Worked examples

Common mistakes

Practice problems