How to Put Absolute Value Function on Desmos Grpahing Calculator

Absolute value functions are fundamental in mathematics and appear in many real-world applications. Desmos Graphing Calculator provides an excellent platform for visualizing these functions. This guide will walk you through the process of graphing absolute value functions in Desmos, from basic examples to more complex transformations.

Introduction

The absolute value of a number is its distance from zero on the number line, regardless of direction. Mathematically, the absolute value of x is denoted as |x| and is defined as:

|x| = x if x ≥ 0 -x if x < 0

Desmos Graphing Calculator allows you to graph absolute value functions with ease. The calculator supports LaTeX input for mathematical expressions, making it straightforward to input absolute value functions.

Basic Absolute Value Function

The simplest absolute value function is y = |x|. This function creates a V-shape with its vertex at the origin (0,0). Here's how to graph it in Desmos:

Open the Desmos Graphing Calculator at https://www.desmos.com/calculator.
In the left panel, type: y = abs(x)
Press Enter to graph the function.

The graph will show a V-shaped line with the vertex at (0,0). The function is symmetric about the y-axis, meaning |x| = |-x| for all x.

Note: Desmos uses the abs() function for absolute value. This is equivalent to the mathematical notation |x|.

Transformed Absolute Value Functions

Absolute value functions can be transformed using vertical and horizontal shifts, stretches, and reflections. Here are some common transformations:

Vertical Shift

To shift the graph vertically, add or subtract a constant to the function. For example:

y = |x| + 2 (shift up by 2 units)
y = |x| - 3 (shift down by 3 units)

Horizontal Shift

To shift the graph horizontally, add or subtract a constant inside the absolute value function. For example:

y = |x - 2| (shift right by 2 units)
y = |x + 3| (shift left by 3 units)

Vertical Stretch

To stretch the graph vertically, multiply the function by a constant greater than 1. For example:

y = 2|x| (stretch vertically by factor of 2)

Horizontal Stretch

To stretch the graph horizontally, divide the x variable by a constant greater than 1. For example:

y = |x/2| (stretch horizontally by factor of 2)

Absolute Value Equations

Absolute value equations can be solved graphically in Desmos. For example, to solve |x| = 3:

Graph y = abs(x)
Graph y = 3
The points of intersection will show the solutions: x = 3 and x = -3

For more complex equations like |x - 2| = 5:

Graph y = abs(x - 2)
Graph y = 5
The solutions will be x = 7 and x = -3

Tip: Use Desmos' sliders to explore how changing parameters affects the graph of absolute value functions.

FAQ

Can I graph piecewise absolute value functions in Desmos?: Yes, Desmos supports piecewise functions. You can define different expressions for different intervals using the piecewise function syntax.
How do I graph the absolute value of a quadratic function?: To graph |x² - 4|, simply type y = abs(x^2 - 4) in Desmos. The graph will show the absolute value of the quadratic function.
Can I use absolute value functions with inequalities in Desmos?: Yes, you can graph inequalities involving absolute value functions. For example, to graph |x| < 2, you can type y = abs(x) and then use the inequality tool to shade the region between y = -2 and y = 2.
How do I find the vertex of an absolute value function in Desmos?: The vertex of a basic absolute value function y = |x| is at (0,0). For transformed functions like y = |x - h| + k, the vertex is at (h, k). You can find the vertex by looking for the point where the graph changes direction.
Can I export my absolute value graph from Desmos?: Yes, Desmos allows you to export graphs as images or share them via link. Look for the "Download" or "Share" options in the top-right corner of the calculator.