Problem 19 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(f(x) = 0.8 - x\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 0.8 - x\) will represent a straight line that decreases from left to right, intersecting y-axis at point (0, 0.8).

Step 1: Identify The Type Of Function

Our given function \(f(x) = 0.8 - x\) is a linear function in slope-intercept form, where the slope is -1 and the y-intercept is 0.8. It will plot a straight line when graphed.

Step 2: Choose Appropriate Viewing Window

The linear function tends to decrease from left to right due to its negative slope; therefore, an appropriate viewing window might be -10 <= x <= 10, and -10 <= y <=10. However, this depends on the graphing utility's defaults and restrictions.

Step 3: Plot The Function

Input the function \(f(x) = 0.8 - x\) into the graphing utility. It will produce a straight line that slopes down from left to right, intersecting the y-axis at 0.8. Make sure you get a clear view of the line across the selected window.

Key Concepts

Graphing Utilities

Graphing utilities are essential tools for visualizing mathematical functions and equations. They are especially handy when dealing with linear functions like the one in this exercise. A graphing utility can be a software program, an online calculator, or even a physical graphing calculator. These tools allow you to:

• Input equations and functions to see their graphical representations.
• Adjust and manipulate the graph to better understand the function's behavior.
• Observe intersections, slopes, and y-intercepts without plotting points by hand.

When you enter a function like \( f(x) = 0.8 - x \), the graphing utility calculates and displays a graph, saving time and reducing potential errors compared to hand plotting. Make sure you are familiar with your graphing utility's basic features before diving into more complex graphing tasks. These tools can significantly enhance learning by providing a visual understanding of algebraic concepts.

Slope-Intercept Form

A linear function in slope-intercept form is represented as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For the function \( f(x) = 0.8 - x \), we can rewrite this in standard slope-intercept form as \( y = -x + 0.8 \). Here, the slope \( m \) is -1, indicating a line that decreases or goes "downhill" as you move from left to right across the graph. The y-intercept \( b \) is 0.8, which is where the line crosses the y-axis.
Understanding the slope and y-intercept allows you to quickly sketch the line of the function without needing every point plotted. The slope tells you how steep the line is, while the y-intercept gives you a starting point on the graph. This forms the basis of understanding and interpreting linear functions graphically.

Viewing Window

Choosing the right viewing window is crucial when graphing functions to ensure all important parts of the graph are visible. For linear functions like \( f(x) = 0.8 - x \), which extend infinitely in both directions, you need to choose a window that provides a clear picture of the line's behavior and key intercepts.

• The x-range might typically be set from -10 to 10 to capture plenty of the line's extent.
• The y-range can similarly be set from -10 to 10 to ensure the y-intercept and slope variations are visible.

Adjusting the viewing window allows you to focus on particular graph features or zoom out to get a broader view. Make sure that your selected window covers the line’s intercepts or crucial points of intersection with the axes, especially when demonstrating concepts or solving problems. Balancing detail with a comprehensive view can help in understanding the underlying behavior of the function you’re working with.