Transformations

General Form

For a parent function $f(x)$, the transformed function is:

$g(x) = a \cdot f(b(x - h)) + k$

Parameter	Effect
$a$	Vertical stretch/compression and reflection
$b$	Horizontal stretch/compression and reflection
$h$	Horizontal shift
$k$	Vertical shift

Vertical Transformations

Vertical Shift

$y = f(x) + k$

• $k > 0$: Shift up $k$ units
• $k < 0$: Shift down $|k|$ units

Vertical Stretch/Compression

$y = a \cdot f(x)$

• $|a| > 1$: Vertical stretch by factor of $|a|$
• $0 < |a| < 1$: Vertical compression by factor of $|a|$
• $a < 0$: Reflection across x-axis

Horizontal Transformations

Horizontal Shift

$y = f(x - h)$

• $h > 0$: Shift right $h$ units
• $h < 0$: Shift left $|h|$ units

Note: The shift is opposite to the sign in the equation!

Horizontal Stretch/Compression

$y = f(bx)$

• $|b| > 1$: Horizontal compression by factor of $\frac{1}{|b|}$
• $0 < |b| < 1$: Horizontal stretch by factor of $\frac{1}{|b|}$
• $b < 0$: Reflection across y-axis

Note: The effect is the reciprocal of $b$!

Reflections

Transformation	Effect
$y = -f(x)$	Reflect across x-axis
$y = f(-x)$	Reflect across y-axis
$y = -f(-x)$	Reflect across origin

Order of Transformations

When applying multiple transformations, follow this order:

1. Horizontal stretch/compression (inside function)
2. Horizontal shift (inside function)
3. Reflection (if applicable)
4. Vertical stretch/compression (outside function)
5. Vertical shift (outside function)

Memory aid: Work from inside out, horizontal before vertical.

Common Transformations Summary

Original	Transformed	Description
$f(x)$	$f(x) + 3$	Up 3
$f(x)$	$f(x) - 2$	Down 2
$f(x)$	$f(x - 4)$	Right 4
$f(x)$	$f(x + 1)$	Left 1
$f(x)$	$2f(x)$	Vertical stretch $\times 2$
$f(x)$	$\frac{1}{2}f(x)$	Vertical compression $\times \frac{1}{2}$
$f(x)$	$f(2x)$	Horizontal compression $\times \frac{1}{2}$
$f(x)$	$f(\frac{1}{2}x)$	Horizontal stretch $\times 2$
$f(x)$	$-f(x)$	Reflect over x-axis
$f(x)$	$f(-x)$	Reflect over y-axis

Example: Quadratic Transformations

Starting with $f(x) = x^2$:

Function	Transformation
$x^2 + 3$	Up 3
$(x - 2)^2$	Right 2
$(x + 1)^2 - 4$	Left 1, Down 4
$2(x - 3)^2 + 1$	Stretch $\times 2$, Right 3, Up 1
$-x^2$	Reflect over x-axis
$(-x)^2 = x^2$	No change (even function)

Finding Transformations from Equations

Given: $g(x) = -2(x + 3)^2 - 5$

1. Parent: $f(x) = x^2$
2. Horizontal shift: Left 3 (from $x + 3$)
3. Vertical stretch: $\times 2$ (from coefficient 2)
4. Reflection: Over x-axis (from negative)
5. Vertical shift: Down 5 (from $-5$)

Vertex: $(-3, -5)$