Transformations of Functions (Section 2.5) – Precalculus Study Notes

Transformations of Functions

Introduction to Function Transformations

Transformations are operations that alter the position, shape, or orientation of the graph of a function. Understanding these transformations is essential for analyzing and graphing functions in precalculus. The basic types include translations, reflections, stretchings, and shrinkings.

• Translation: Shifts the graph horizontally or vertically.

• Reflection: Flips the graph across an axis.

• Stretching/Shrinking: Changes the scale of the graph vertically or horizontally.

Common parent functions used as building blocks include: linear ($y = x$), squaring ($y = x^2$), square root ($y = \sqrt{x}$), cubing ($y = x^3$), reciprocal ($y = \frac{1}{x}$), and absolute value ($y = |x|$).

Types of Transformations

Vertical Translation

A vertical translation shifts the graph of a function up or down without changing its shape.

• Upward Shift: The graph of $y = f(x) + b$ is the graph of $y = f(x)$ shifted up $b$ units.

• Downward Shift: The graph of $y = f(x) - b$ is the graph of $y = f(x)$ shifted down $b$ units.

Example: The graph of $y = \frac{1}{5}x^4 + 5$ is the graph of $y = \frac{1}{5}x^4$ shifted up 5 units.

Horizontal Translation

A horizontal translation shifts the graph of a function left or right.

• Right Shift: The graph of $y = f(x - d)$ is the graph of $y = f(x)$ shifted right $d$ units.

• Left Shift: The graph of $y = f(x + d)$ is the graph of $y = f(x)$ shifted left $d$ units.

Example: The graph of $y = \frac{1}{5}(x + 7)^4$ is the graph of $y = \frac{1}{5}x^4$ shifted left 7 units.

Reflections

Reflections flip the graph of a function across the x-axis or y-axis.

• Reflection across the x-axis: The graph of $y = -f(x)$ is the reflection of $y = f(x)$ across the x-axis.

• Reflection across the y-axis: The graph of $y = f(-x)$ is the reflection of $y = f(x)$ across the y-axis.

• If $(x, y)$ is on the graph of $y = f(x)$, then $(x, -y)$ is on $y = -f(x)$, and $(-x, y)$ is on $y = f(-x)$.

Example: For $f(x) = x^3 - 4x^2$:

• $g(x) = (-x)^3 - 4(-x)^2 = -x^3 - 4x^2 = g(x)$ is a reflection across the y-axis.

• $h(x) = 4x^2 - x^3$ is a reflection across the x-axis.

If $(2, -8)$ is on $f$, then $(-2, -8)$ is on $g$ (y-axis reflection), and $(2, 8)$ is on $h$ (x-axis reflection).

Vertical Stretching and Shrinking

Vertical stretching and shrinking change the steepness of the graph by multiplying the output by a constant.

• Vertical Stretch: The graph of $y = a f(x)$ is stretched vertically if $|a| > 1$.

• Vertical Shrink: The graph of $y = a f(x)$ is shrunk vertically if $0 < |a| < 1$.

• If $a < 0$, the graph is also reflected across the x-axis.

• Each y-coordinate is multiplied by $a$.

Example: $y = 2f(x)$ is a vertical stretch by a factor of 2; $y = \frac{1}{2}f(x)$ is a vertical shrink by a factor of 1/2.

Horizontal Stretching and Shrinking

Horizontal stretching and shrinking change the width of the graph by multiplying the input by a constant.

• Horizontal Shrink: The graph of $y = f(cx)$ is shrunk horizontally if $|c| > 1$.

• Horizontal Stretch: The graph of $y = f(cx)$ is stretched horizontally if $0 < |c| < 1$.

• If $c < 0$, the graph is also reflected across the y-axis.

• Each x-coordinate is divided by $c$.

Example: $y = f(2x)$ is a horizontal shrink by a factor of 2; $y = f(\frac{1}{2}x)$ is a horizontal stretch by a factor of 2.

Summary Table: Types of Function Transformations

Transformation	Equation Form	Effect on Graph
Vertical Translation	$y = f(x) + b$	Shift up $b$ units
Vertical Translation	$y = f(x) - b$	Shift down $b$ units
Horizontal Translation	$y = f(x - d)$	Shift right $d$ units
Horizontal Translation	$y = f(x + d)$	Shift left $d$ units
Reflection (x-axis)	$y = -f(x)$	Flip over x-axis
Reflection (y-axis)	$y = f(-x)$	Flip over y-axis
Vertical Stretch	$y = a f(x)$, $|a| > 1$	Stretches vertically
Vertical Shrink	$y = a f(x)$, $0 < |a| < 1$	Shrinks vertically
Horizontal Stretch	$y = f(cx)$, $0 < |c| < 1$	Stretches horizontally
Horizontal Shrink	$y = f(cx)$, $|c| > 1$	Shrinks horizontally

Key Takeaways

• Transformations allow you to graph complex functions by modifying basic parent functions.

• Always apply transformations in the correct order: stretches/shrinks, reflections, then translations.

• Understanding transformations is essential for analyzing function behavior and solving equations graphically.