Problem 39 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(g(x)=|x|-5\)

Step-by-Step Solution

Answer

The graph of the function \(g(x) = |x| - 5\) is a 'V' shaped curve that pivots at the point (0,-5)

Step 1: Understanding the function

The function given is an absolute value function subtracted by a constant. The graph of this function is a V-shaped graph which pivots at the point (0,-5). The 'V' shape comes from the absolute value function, and the '-5' shifts this 'V' downwards by 5 units.

Step 2: Setting the viewing window

In the graphing utility, the viewing window should be set considering that we want to visualize the behavior of the graph on both the positive and negative sides of x-axis as well as below the x-axis. A suggested window might be -10 to 10 for x-values and -10 to 10 for the y-values.

Step 3: Graphing the function

Enter the function into the graphing utility to graph the function \(g(x) = |x| - 5\). The graph will visually confirm the analysis from step 1, showing a 'V' shaped graph pivoting at the point (0,-5).

Key Concepts

Understanding Absolute Value Graph Transformations

Absolute value functions create distinctive V-shaped graphs, in which each point on the graph represents the absolute value of the x-coordinate at that point. Transforming these graphs can be accomplished through various alterations to the function's equation. In the exercise given, the function is expressed as
\( g(x) = |x| - 5 \).
This function represents a basic absolute value graph that has been transformed by a vertical translation. Here, the absolute value of x, \( |x| \), normally creates a V-shape with its vertex at the origin (0,0). However, the '-5' at the end of the function denotes a downward shift of the entire graph by 5 units. As a result, the vertex of the graph is now at the point (0,-5). To grasp these transformations:

• Vertical Shifts: Adding or subtracting a number from the function translates the graph up or down without altering its shape.
• Horizontal Shifts: Adding or subtracting a number inside the absolute value symbol translates the graph left or right.
• Reflections: Multiplying the entire function or the value inside the absolute value by -1 reflects the graph over the x-axis or y-axis, respectively.
• Stretching and Shrinking: Multiplying the value inside the absolute value symbol by a factor greater or less than 1 stretches or shrinks the graph vertically.

Understanding how these transformations affect the graph is essential for correctly interpreting and graphing absolute value functions.

Setting the Graphing Utility Window

Selecting an appropriate viewing window on a graphing utility is a critical step for visualizing the behavior of functions. When working with the function
\( g(x) = |x| - 5 \),
you encounter a common scenario where the function's graph has important features both above and below the x-axis, as well as on both sides of the y-axis. The ideal window should accommodate the full extent of these significant features. In this case, since the vertex of the graph is located at (0,-5), a viewing window from -10 to 10 for both x-values and y-values will provide a symmetric view that captures the vertex and the lines as they extend in both the positive and negative directions. When setting the graphing window:

• Ensure that the window includes all points of interest, such as intercepts, vertices, and asymptotes.
• Adjust the scale to avoid a distorted view that could misrepresent the function's behavior.
• Use the zoom function to refine the view and focus on particular details of the graph as needed.

A well-chosen window provides a clear and accurate representation of the function, aiding in analysis and understanding.

Graphing Functions with Technology

Incorporating technology, such as graphing calculators or computer software, is invaluable for correctly portraying and analyzing mathematical functions. For the absolute value function
\( g(x) = |x| - 5 \),
using technology helps to construct the exact V-shape graph effortlessly. When graphing with technology:

• Input the function correctly by taking note of syntax and symbols specific to the technology used.
• Utilize features such as 'Trace' to examine values at specific points on the graph, providing a deeper insight into the function's behavior.
• Make use of graphing utilities' capabilities to plot multiple functions simultaneously and observe interactions or compare behaviors.

After entering the function into the graphing utility and using the previously determined window setting, the graph displayed should confirm the 'V' shape pivoting at the point (0,-5). This visual aid reinforces the conceptual understanding of absolute value functions and their transformations, making technology an indispensable tool in the study of mathematics.