Transformations of Functions: Precalculus Study Guide

Functions and Graphs

Transformations of Functions

Transformations are operations that alter the appearance of a function's graph without changing its fundamental nature. Understanding these transformations is essential for graphing and analyzing functions in Precalculus.

• Rigid Transformations: These include vertical shifts, horizontal shifts, and reflections. They do not change the shape of the graph, only its position or orientation.

• Non-rigid Transformations: These include vertical stretching/shrinking and horizontal stretching/shrinking. They alter the shape of the graph by changing its size.

Graphs of Common Functions

Common functions such as linear, quadratic, absolute value, and piecewise functions serve as the basis for understanding transformations. Their graphs are often used as reference points for applying transformations.

Example: The absolute value function $y = |x|$ is a V-shaped graph centered at the origin.

Vertical Shifts

A vertical shift moves the graph of a function up or down. If $f(x)$ is a function, then $f(x) + k$ shifts the graph upward by $k$ units, and $f(x) - k$ shifts it downward by $k$ units.

• Formula: $y = f(x) + k$

• Example: Shifting $y = f(x)$ up by 2 units gives $y = f(x) + 2$.

Horizontal Shifts

A horizontal shift moves the graph left or right. If $f(x)$ is a function, then $f(x + h)$ shifts the graph left by $h$ units, and $f(x - h)$ shifts it right by $h$ units.

• Formula: $y = f(x - h)$

• Example: Shifting $y = f(x)$ right by 3 units gives $y = f(x - 3)$.

Reflections of Graphs

Reflections flip the graph over a specified axis.

• Reflection about the x-axis: $y = -f(x)$ reflects the graph over the x-axis.

• Reflection about the y-axis: $y = f(-x)$ reflects the graph over the y-axis.

Example: Reflecting $y = |x|$ over the x-axis gives $y = -|x|$; reflecting over the y-axis gives $y = | -x | = |x|$ (for absolute value).

Vertically Stretching and Shrinking Graphs

Vertical stretching and shrinking change the height of the graph. If $c > 1$, $y = c f(x)$ stretches the graph vertically; if $0 < c < 1$, it shrinks the graph vertically.

• Formula: $y = c f(x)$

• Example: $y = 2 f(x)$ doubles the y-values; $y = 0.5 f(x)$ halves them.

Horizontally Stretching and Shrinking Graphs

Horizontal stretching and shrinking change the width of the graph. If $c > 1$, $y = f(c x)$ shrinks the graph horizontally; if $0 < c < 1$, it stretches the graph horizontally.

• Formula: $y = f(c x)$

• Example: $y = f(2x)$ compresses the graph; $y = f(0.5x)$ expands it.

Graphing with Multiple Transformations

When a function involves several transformations, the order in which they are applied is important. The recommended sequence is:

1. Horizontal shifting

2. Stretching or shrinking

3. Reflecting

4. Vertical shifting

Example: To graph $g(x) = 2 f(\frac{1}{2} x)$, first horizontally stretch by a factor of 2, then vertically stretch by a factor of 2.

Sequence of Transformations: Example

Applying a sequence of transformations to a function can be visualized step-by-step. For instance, starting with $y = f(x)$, you might shift, stretch, reflect, and then shift again.

Practice: Fill in the Blanks and Graphing

Practice problems often ask you to apply transformations to a given function and graph the result. Use the properties of stretching, shrinking, shifting, and reflecting to complete these exercises.

Summary Table: Types of Transformations

Transformation	Formula	Effect
Vertical Shift	$y = f(x) + k$	Up/down by $k$ units
Horizontal Shift	$y = f(x - h)$	Right/left by $h$ units
Reflection (x-axis)	$y = -f(x)$	Flip over x-axis
Reflection (y-axis)	$y = f(-x)$	Flip over y-axis
Vertical Stretch/Shrink	$y = c f(x)$	Stretch if $c > 1$, shrink if $0 < c < 1$
Horizontal Stretch/Shrink	$y = f(c x)$	Shrink if $c > 1$, stretch if $0 < c < 1$