absolute value functions and graphs worksheet

3 min read 22-11-2024

absolute value functions and graphs worksheet

Meta Description: Master absolute value functions and their graphs! This comprehensive guide provides a worksheet with examples, explanations, and practice problems to help you understand absolute value, its graph transformations, and how to solve absolute value equations and inequalities. Perfect for students and anyone wanting to improve their algebra skills.

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. It's always non-negative. We denote the absolute value of a number x as |x|.

|x| = x if x ≥ 0 (The number is already positive or zero)
|x| = -x if x < 0 (We take the opposite of the negative number to make it positive)

For example:

|5| = 5
|-5| = 5
|0| = 0

The Graph of a Basic Absolute Value Function

The simplest absolute value function is f(x) = |x|. Its graph is a V-shape with the vertex at (0,0). The left branch has a slope of -1, and the right branch has a slope of 1.

[Insert image here: A graph of f(x) = |x|, clearly showing the vertex at (0,0) and the slopes of the branches.] Alt text: Graph of the absolute value function f(x) = |x|.

Transformations of Absolute Value Functions

We can transform the basic absolute value function using various techniques, affecting its position, shape, and orientation.

Vertical Shifts

Adding or subtracting a constant k outside the absolute value shifts the graph vertically.

f(x) = |x| + k: Shifts the graph k units upwards (positive k) or downwards (negative k).

Horizontal Shifts

Adding or subtracting a constant h inside the absolute value shifts the graph horizontally.

f(x) = |x - h|: Shifts the graph h units to the right (positive h) or to the left (negative h).

Vertical Stretches and Compressions

Multiplying the absolute value function by a constant a results in a vertical stretch or compression.

f(x) = a|x|: Stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a is negative, it also reflects the graph across the x-axis.

Combining Transformations

We can combine multiple transformations. For example, f(x) = 2|x - 3| + 1 involves a vertical stretch by a factor of 2, a horizontal shift 3 units to the right, and a vertical shift 1 unit upwards.

[Insert image here: A graph showing f(x) = 2|x - 3| + 1, clearly indicating the transformations.] Alt text: Graph of the transformed absolute value function f(x) = 2|x - 3| + 1.

Solving Absolute Value Equations and Inequalities

Solving equations and inequalities involving absolute values requires careful consideration of the definition of absolute value.

Absolute Value Equations

To solve an equation like |x| = 5, we consider two cases:

x = 5 (The expression inside the absolute value is already positive)
x = -5 (The expression inside the absolute value is negative, so we take the opposite)

Therefore, the solutions are x = 5 and x = -5.

Absolute Value Inequalities

Solving inequalities like |x| < 5 or |x| > 5 involves similar casework, but with different solution sets.

|x| < 5: This means -5 < x < 5 (x is between -5 and 5).
|x| > 5: This means x < -5 or x > 5 (x is less than -5 or greater than 5).

Worksheet: Absolute Value Functions and Graphs

(Note: This is a sample worksheet. Expand this section with more diverse problems.)

Part 1: Graphing

Graph f(x) = |x - 2| + 1. Identify the vertex.
Graph f(x) = -3|x + 1|. Describe the transformations.
Graph f(x) = ½|x| - 4. What is the y-intercept?

Part 2: Solving Equations and Inequalities

Solve |2x - 1| = 7.
Solve |x + 3| < 4.
Solve |x - 5| ≥ 2.
Solve |3x + 2| = |x - 4|.

Part 3: Applications

The distance between two points x and a can be represented as |x - a|. If the distance between x and 3 is less than or equal to 2, what are the possible values of x?
A manufacturing process requires the thickness of a metal sheet to be within 0.01 cm of 2 cm. Write an absolute value inequality that describes acceptable thicknesses.

This worksheet and guide should provide a solid foundation in understanding and working with absolute value functions and their graphs. Remember to practice regularly to improve your skills! For further help, explore resources like [link to Khan Academy's algebra section] and [link to another reputable math resource].