Graph Transformations

Transformations allow you to build new functions from familiar ones by shifting, stretching, compressing, or reflecting their graphs.

original: f(x) = sin  transformed: y = 2.0·f(x)

k = 2.0

■ original f(x)■ transformed

Vertical Transformations

⚡Theorem — Vertical Scaling and Reflection

Given $y = f(x)$ , the graph of $y = kf(x)$ :

Example: $y = 3x^2$ stretches $y = x^2$ vertically by 3. $y = -x^2$ reflects it over the $x$ -axis.

⚡Theorem — Horizontal Scaling

The graph of $y = f(kx)$ :

$k > 1$ : horizontal compression by factor $1/k$ (squeezes toward $y$ -axis)
$0 < k < 1$ : horizontal stretch by factor $1/k$ (pulls away from $y$ -axis)

Example: $y = (2x)^2 = 4x^2$ compresses $y = x^2$ horizontally by $1/2$ .

Note the counterintuitive direction: multiplying the input by $k > 1$ makes the graph narrower, not wider.

Transformation	Effect	Example with $f(x) = x^2$
$y = f(x) + b$	Shift up by $b$ (down if $b < 0$ )	$y = x^2 + 3$ : shift up 3
$y = f(x + a)$	Shift left by $a$ (right if $a < 0$ )	$y = (x - 2)^2$ : shift right 2

⚡Theorem — Translations

$y = f(x) + b$ shifts the graph vertically by $b$ units.
$y = f(x + a)$ shifts the graph horizontally by $-a$ units (left if $a > 0$ , right if $a < 0$ ).

Transformation	Effect	Example with $f(x) = x^2 - 4$
$y = \\|f(x)\\|$	Reflect all parts below the $x$ -axis upward	Negative portion of $x^2 - 4$ flips up
$y = f(\\|x\\|)$	Keep the graph for $x \geq 0$ , mirror it to the left	Right half of $x^2 - 4$ is mirrored

ℹNote — Order Matters

When multiple transformations are combined, the order of operations matters. In general:

$f(2x + 1) \neq 2f(x) + 1$

Apply transformations in this order: (1) horizontal scaling, (2) horizontal shift, (3) vertical scaling, (4) vertical shift.

✏Exercise

Describe how to obtain $y = 2(x - 3)^2 + 1$ from $y = x^2$ .

Hint: Identify the horizontal shift, vertical stretch, and vertical shift, then state the order in which they are applied.