How To Graph Absolute Value On Ti-84

How to Graph Absolute Value on TI-84: A Comprehensive Guide

Graphing absolute value functions on your TI-84 calculator can seem daunting at first, but with a little understanding of the absolute value function and the calculator's capabilities, it becomes straightforward. This comprehensive guide will walk you through the process, covering various scenarios and providing tips for accurate and efficient graphing.

Understanding the Absolute Value Function

Before diving into the graphing process, let's refresh our understanding of the absolute value function. The absolute value of a number is its distance from zero, always resulting in a non-negative value. Mathematically, it's represented as |x|.

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

This piecewise definition is crucial for understanding how the graph will appear. The graph will always be V-shaped, with the vertex at the point where the expression inside the absolute value becomes zero.

Graphing Basic Absolute Value Functions on the TI-84

Let's start with the simplest case: graphing y = |x|.

Step 1: Accessing the Y= Editor

Press the Y= button on your TI-84 calculator. This will take you to the equation editor where you'll enter your function.

Step 2: Entering the Absolute Value Function

The absolute value function isn't directly available as a single button. You'll need to use the MATH menu.

1. Press MATH.
2. Use the right arrow key to navigate to the NUM menu.
3. Select option 1:abs(. This will paste "abs(" into your Y= editor.

Now, enter the expression you want to take the absolute value of within the parentheses. For y = |x|, you'll type: abs(X) (Note: X is accessed using the ALPHA button followed by the X,T,θ,n button).

Step 3: Setting the Window

Before graphing, it's essential to set an appropriate viewing window. The default window might not always show the complete graph, especially for functions with transformations. Press WINDOW to access the window settings. Adjust the following parameters as needed:

• Xmin: The minimum x-value displayed.
• Xmax: The maximum x-value displayed.
• Xscl: The scale for the x-axis (distance between tick marks).
• Ymin: The minimum y-value displayed.
• Ymax: The maximum y-value displayed.
• Yscl: The scale for the y-axis.
• Xres: Leave this at 1 unless you need to improve the speed of graphing complex functions.

For y = |x|, a standard window like Xmin = -10, Xmax = 10, Ymin = -10, Ymax = 10 will suffice.

Step 4: Graphing the Function

Press GRAPH to view the graph of y = |x|. You should see a V-shaped graph with its vertex at the origin (0,0).

Graphing More Complex Absolute Value Functions

Now let's explore graphing more intricate absolute value functions. These will involve transformations such as horizontal shifts, vertical shifts, stretches, and compressions.

Example: y = |x - 2| + 1

This function represents a horizontal shift of 2 units to the right and a vertical shift of 1 unit upwards.

1. Enter abs(X-2)+1 in the Y= editor.
2. Adjust the window if necessary to see the entire graph. Because of the shift, you might want to adjust Xmin and Xmax accordingly.
3. Press GRAPH.

You'll observe that the vertex of the V-shaped graph is now located at (2,1).

Example: y = 2|x + 3| - 4

This function includes a vertical stretch by a factor of 2, a horizontal shift of 3 units to the left, and a vertical shift of 4 units downwards.

1. Enter 2abs(X+3)-4 in the Y= editor.
2. Adjust the window as needed.
3. Press GRAPH.

Notice the steeper slope due to the stretch, and the vertex at (-3,-4).

Using the Table Feature for Verification

The TI-84's table feature is a powerful tool for verifying your graph.

1. Press 2nd then GRAPH (TABLE) to access the table.
2. You can scroll through the table to see corresponding x and y values. This helps confirm the vertex and the general shape of the graph.
3. You can also adjust the table setup (2nd then WINDOW) to change the starting x-value and the increment between x-values.

Advanced Techniques: Piecewise Functions

Absolute value functions can be represented as piecewise functions. For example, y = |x| can be defined as:

• y = x, if x ≥ 0
• y = -x, if x < 0

The TI-84 allows you to graph piecewise functions using the test operators. However, this method is more involved and generally not necessary for simple absolute value functions. The abs() function is usually the most efficient approach.

Troubleshooting Common Issues

• Incorrect Graph: Double-check that you've entered the function correctly in the Y= editor. Pay close attention to parentheses and signs.
• Graph Not Visible: Adjust the window settings (WINDOW) to encompass the relevant portion of the graph.
• Unexpected Results: Review the order of operations to ensure that the calculator interprets your function correctly.

Beyond Basic Graphing: Exploring Features

• Trace Function: Use the TRACE function (press TRACE) to move a cursor along the graph and see the coordinates of points.
• Calculate Features: The CALC menu (2nd then TRACE) provides tools like finding the minimum, maximum, and zeros of the function. This can be especially useful for determining the vertex of the absolute value graph.
• Zoom Features: The ZOOM menu allows you to zoom in or out to examine specific regions of the graph in greater detail.

This comprehensive guide provides a thorough understanding of how to graph absolute value functions on your TI-84 calculator. By mastering these techniques, you can effectively visualize and analyze absolute value functions, enhancing your mathematical understanding and problem-solving skills. Remember to practice with various examples to solidify your understanding and build confidence in using this powerful graphing tool. Experiment with different transformations and observe how they affect the graph's shape and position. This hands-on experience is crucial for mastering the art of graphing absolute value functions on the TI-84.