Graphing Techniques and Transformations in College Algebra

Section 1.5 – Graphing Techniques; Transformations

Learning Objectives

• Graph functions using vertical and horizontal shifts.

• Graph functions using compressions and stretches.

• Graph functions using reflections about the axes.

• Graph functions using combinations of transformations.

• Find the equation of a function from the transformations.

Graphing Using Vertical and Horizontal Shifts

Introduction

Transformations allow us to modify the graph of a function by shifting, stretching, shrinking, or reflecting it. Understanding these transformations is essential for graphing functions efficiently and interpreting their behavior.

Vertical Shifts

A vertical shift moves the graph of a function up or down without changing its shape. This is accomplished by adding or subtracting a constant outside the function.

• If c is added: $y = f(x) + c$ shifts the graph up by c units.

• If c is subtracted: $y = f(x) - c$ shifts the graph down by c units.

• The sign of c determines the direction of the shift.

Vertical Shifts	Effect
$y = f(x) + d$	Shift up d units
$y = f(x) - d$	Shift down d units

Example: $y = x^2 + 2$ is the graph of $y = x^2$ shifted up 2 units.

Horizontal Shifts

A horizontal shift moves the graph left or right. This is done by adding or subtracting a constant inside the function's argument.

• If c is added: $y = f(x + c)$ shifts the graph left by c units.

• If c is subtracted: $y = f(x - c)$ shifts the graph right by c units.

• The sign is opposite of what you might expect: plus means left, minus means right.

Horizontal Shifts	Effect
$y = f(x + c)$	Shift left c units
$y = f(x - c)$	Shift right c units

Example: $y = (x - 4)^2 + 2$ is the graph of $y = x^2$ shifted right 4 units and up 2 units.

Reflections

Reflections over the x-axis

A reflection over the x-axis flips the graph upside down. This is achieved by multiplying the entire function by -1.

• $y = -f(x)$ reflects the graph across the x-axis.

• The negative sign is outside the function.

Reflections	Effect
$y = -f(x)$	Reflect across the x-axis

Example: $y = -x^2$ is the reflection of $y = x^2$ over the x-axis.

Reflections over the y-axis

A reflection over the y-axis flips the graph left to right. This is done by replacing x with -x inside the function.

• $y = f(-x)$ reflects the graph across the y-axis.

• The negative sign is inside the function.

Reflections	Effect
$y = f(-x)$	Reflect across the y-axis

Example: $y = (-x)^3$ is the reflection of $y = x^3$ over the y-axis.

Vertical Stretching and Shrinking

Introduction

Vertical stretching and shrinking change the steepness of a graph. This is done by multiplying the function by a constant a.

• If $a > 1$, $y = a f(x)$ is a vertical stretch (graph becomes steeper).

• If $0 < a < 1$, $y = a f(x)$ is a vertical shrink (graph becomes flatter).

Example: $y = 2x^2$ is a vertical stretch of $y = x^2$ by a factor of 2.

Horizontal Stretching and Shrinking

Introduction

Horizontal stretching and shrinking affect the width of the graph. This is done by multiplying the input variable x by a constant b.

• If $b > 1$, $y = f(bx)$ is a horizontal shrink (graph becomes narrower).

• If $0 < b < 1$, $y = f(bx)$ is a horizontal stretch (graph becomes wider).

Example: $y = f(2x)$ is a horizontal shrink of $y = f(x)$ by a factor of 2.

Combinations of Transformations

Introduction

Multiple transformations can be applied to a function in sequence. The order of operations is important: typically, perform horizontal shifts and stretches first, then reflections, and finally vertical shifts and stretches.

• Each transformation affects the graph in a predictable way.

• Labeling key points after each transformation helps track changes.

Example: Given $f(x) = x^2$, the function $g(x) = -2(x-3)^2 + 4$ is:

• Shifted right 3 units

• Stretched vertically by 2

• Reflected over the x-axis

• Shifted up 4 units

Summary Table: Types of Transformations

Transformation	Equation	Effect
Vertical Shift	$y = f(x) + d$	Up d units
Vertical Shift	$y = f(x) - d$	Down d units
Horizontal Shift	$y = f(x + c)$	Left c units
Horizontal Shift	$y = f(x - c)$	Right c units
Vertical Stretch	$y = a f(x)$, $a > 1$	Steeper (stretched vertically)
Vertical Shrink	$y = a f(x)$, $0 < a < 1$	Flatter (shrunk vertically)
Horizontal Stretch	$y = f(bx)$, $0 < b < 1$	Wider (stretched horizontally)
Horizontal Shrink	$y = f(bx)$, $b > 1$	Narrower (shrunk horizontally)
Reflection over x-axis	$y = -f(x)$	Flips graph upside down
Reflection over y-axis	$y = f(-x)$	Flips graph left to right

Practice Examples

• Example 1: $f(x) = x^2$; $g(x) = (x-4)^2 + 2$ Transformation: Shift right 4 units, up 2 units.

• Example 2: $f(x) = x^3$; $g(x) = -x^3 + 2$ Transformation: Reflect over x-axis, shift up 2 units.

• Example 3: $f(x) = \sqrt{x}$; $g(x) = -\sqrt{x-1} - 2$ Transformation: Shift right 1 unit, reflect over x-axis, shift down 2 units.

• Example 4: $f(x) = x^2$; $g(x) = 2x^2$ Transformation: Vertical stretch by factor of 2.

• Example 5: $f(x) = x^2$; $g(x) = (2x)^2$ Transformation: Horizontal shrink by factor of 2.

Additional info: The notes provide a comprehensive overview of function transformations, including definitions, formulas, and examples relevant to College Algebra. The tables and examples are reconstructed for clarity and completeness.