Learn on PengienVision, Algebra 1Chapter 10: Working With Functions

Lesson 5: Compressions and Stretches of Functions

In this Grade 11 enVision Algebra 1 lesson from Chapter 10, students learn how multiplying a function's output or input by a constant produces vertical stretches, vertical compressions, horizontal stretches, and horizontal compressions of its graph. Students explore how the constant k in g(x) = kf(x) and g(x) = f(kx) determines whether a graph is stretched or compressed, and in which direction, using quadratic, square root, and absolute value functions as examples. The lesson also covers reflections across the x-axis as a special case of output multiplication by −1.

Section 1

Reflections Across the X-Axis

Property

When a function is multiplied by $-1$ , the graph reflects across the x-axis:

g(x) = -f(x)

Every point $(x, y)$ on the original graph becomes $(x, -y)$ on the reflected graph.

Section 2

General Function Transformations

Property

For any function $f(x)$ , transformations follow these patterns:

Vertical: $g(x) = af(x)$ where $|a| > 1$ stretches, $0 < |a| < 1$ compresses
Horizontal: $g(x) = f(bx)$ where $|b| > 1$ compresses, $0 < |b| < 1$ stretches
Reflection: $g(x) = -f(x)$ reflects across x-axis

Examples

Section 3

Identifying Vertical vs Horizontal Transformations

Property

Vertical transformations modify the output: $g(x) = k \cdot f(x)$

Horizontal transformations modify the input: $g(x) = f(k \cdot x)$

Section 4

Graph Quadratic Functions of the Form f(x) = ax^2

Property

The coefficient $a$ in the function $f(x) = ax^2$ affects the graph of $f(x) = x^2$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax^2$ will be “wider” than the graph of $f(x) = x^2$ .
If $|a| > 1$ , the graph of $f(x) = ax^2$ will be “skinnier” than the graph of $f(x) = x^2$ .

Examples

The graph of $f(x) = 4x^2$ is skinnier than $f(x) = x^2$ . For any given x-value, the y-value is multiplied by 4, stretching the parabola vertically.

The graph of $f(x) = \frac{1}{3}x^2$ is wider than $f(x) = x^2$ . The y-values are compressed to one-third of their original height, making the parabola open more broadly.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Reflections Across the X-Axis

Property

When a function is multiplied by $-1$ , the graph reflects across the x-axis:

g(x) = -f(x)

Every point $(x, y)$ on the original graph becomes $(x, -y)$ on the reflected graph.

Section 2

General Function Transformations

Property

For any function $f(x)$ , transformations follow these patterns:

Vertical: $g(x) = af(x)$ where $|a| > 1$ stretches, $0 < |a| < 1$ compresses
Horizontal: $g(x) = f(bx)$ where $|b| > 1$ compresses, $0 < |b| < 1$ stretches
Reflection: $g(x) = -f(x)$ reflects across x-axis

Examples

Section 3

Identifying Vertical vs Horizontal Transformations

Property

Vertical transformations modify the output: $g(x) = k \cdot f(x)$

Horizontal transformations modify the input: $g(x) = f(k \cdot x)$

Section 4

Graph Quadratic Functions of the Form f(x) = ax^2

Property

The coefficient $a$ in the function $f(x) = ax^2$ affects the graph of $f(x) = x^2$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax^2$ will be “wider” than the graph of $f(x) = x^2$ .
If $|a| > 1$ , the graph of $f(x) = ax^2$ will be “skinnier” than the graph of $f(x) = x^2$ .

Examples

The graph of $f(x) = 4x^2$ is skinnier than $f(x) = x^2$ . For any given x-value, the y-value is multiplied by 4, stretching the parabola vertically.

The graph of $f(x) = \frac{1}{3}x^2$ is wider than $f(x) = x^2$ . The y-values are compressed to one-third of their original height, making the parabola open more broadly.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Reflections Across the X-Axis

Property

When a function is multiplied by $-1$ , the graph reflects across the x-axis:

g(x) = -f(x)

Every point $(x, y)$ on the original graph becomes $(x, -y)$ on the reflected graph.

Section 2

General Function Transformations

Property

For any function $f(x)$ , transformations follow these patterns:

Vertical: $g(x) = af(x)$ where $|a| > 1$ stretches, $0 < |a| < 1$ compresses
Horizontal: $g(x) = f(bx)$ where $|b| > 1$ compresses, $0 < |b| < 1$ stretches
Reflection: $g(x) = -f(x)$ reflects across x-axis

Examples

Section 3

Identifying Vertical vs Horizontal Transformations

Property

Vertical transformations modify the output: $g(x) = k \cdot f(x)$

Horizontal transformations modify the input: $g(x) = f(k \cdot x)$

Section 4

Graph Quadratic Functions of the Form f(x) = ax^2

Property

The coefficient $a$ in the function $f(x) = ax^2$ affects the graph of $f(x) = x^2$ by stretching or compressing it.

If $0 < |a| < 1$ , the graph of $f(x) = ax^2$ will be “wider” than the graph of $f(x) = x^2$ .
If $|a| > 1$ , the graph of $f(x) = ax^2$ will be “skinnier” than the graph of $f(x) = x^2$ .

Examples

The graph of $f(x) = 4x^2$ is skinnier than $f(x) = x^2$ . For any given x-value, the y-value is multiplied by 4, stretching the parabola vertically.

The graph of $f(x) = \frac{1}{3}x^2$ is wider than $f(x) = x^2$ . The y-values are compressed to one-third of their original height, making the parabola open more broadly.