how to graph absolute value equations

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25-03-2025

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The absolute value of a number is its distance from zero. This means it's always non-negative. Graphing absolute value equations involves understanding this fundamental concept and applying some simple techniques. This guide will walk you through the process step-by-step.

Understanding Absolute Value

Before diving into graphing, let's solidify our understanding of absolute value. The absolute value of a number x, denoted as |x|, is defined as:

• |x| = x, if x ≥ 0
• |x| = -x, if x < 0

In simpler terms:

• The absolute value of a positive number is the number itself. For example, |5| = 5.
• The absolute value of a negative number is its positive counterpart. For example, |-5| = 5.
• The absolute value of zero is zero: |0| = 0.

Graphing Basic Absolute Value Equations

The simplest absolute value equation is y = |x|. Let's explore how to graph it:

1. Create a table of values: Choose several x-values (both positive and negative), calculate the corresponding y-values, and record them in a table.

2. Plot the points: Plot the (x, y) pairs from your table on a coordinate plane.

3. Connect the points: You'll notice the points form a V-shape. Connect them with two straight lines, forming the characteristic V-shape of an absolute value graph. The vertex of the V is at (0,0).

Graphing Transformations of Absolute Value Equations

More complex absolute value equations involve transformations of the basic y = |x| graph. These transformations include shifts (translations), reflections, and stretches/compressions.

Translations (Shifts)

The general form of a translated absolute value equation is:

y = a|x - h| + k

Where:

• a affects the vertical stretch or compression and reflection.
• h represents a horizontal shift (right if positive, left if negative).
• k represents a vertical shift (up if positive, down if negative).

Example: Graph y = |x - 2| + 1

This graph is a shift of the basic y = |x| graph:

• 2 units to the right (because of the -2 inside the absolute value).
• 1 unit upward (because of the +1 outside the absolute value).

The vertex of this graph is (2, 1).

Reflections

A negative sign in front of the absolute value reflects the graph across the x-axis. A negative sign in front of the x inside the absolute value reflects across the y-axis.

Example: Graph y = -|x|

This graph is a reflection of y = |x| across the x-axis. The V-shape now opens downwards.

Stretches and Compressions

The value of a in the general equation y = a|x - h| + k determines the vertical stretch or compression:

• |a| > 1: Vertical stretch (makes the V narrower).
• 0 < |a| < 1: Vertical compression (makes the V wider).

Example: Graph y = 2|x|

This graph is a vertical stretch of y = |x| by a factor of 2. The V-shape is narrower.

Step-by-Step Guide: Graphing a More Complex Equation

Let's graph y = -2|x + 1| - 3.

1. Identify the transformations: This equation involves:

   • A vertical stretch by a factor of 2.
   • A reflection across the x-axis (due to the negative sign).
   • A horizontal shift 1 unit to the left (due to the +1 inside the absolute value).
   • A vertical shift 3 units down (due to the -3 outside the absolute value).

2. Find the vertex: The vertex of the transformed graph is (-1, -3). The horizontal shift affects the x-coordinate, and the vertical shift affects the y-coordinate.

3. Plot the vertex and a few other points: Start by plotting the vertex. Then, choose a few x-values on either side of the vertex, calculate the corresponding y-values, and plot those points.

4. Sketch the graph: Connect the points to create the V-shape. Remember the graph opens downwards due to the reflection.

Solving Absolute Value Inequalities Graphically

You can also use graphs to solve absolute value inequalities. For example, to solve |x| < 2, look at the graph of y = |x| and find the x-values where the graph is below the line y = 2. This will give you the solution -2 < x < 2.

Remember to practice! The more you graph absolute value equations, the better you'll understand the transformations and be able to quickly sketch the graphs. Use online graphing calculators to verify your work and explore different equations.