Problem 20 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(f(x) = 2.5x - 4.25\)

Step-by-Step Solution

Answer

The function \(f(x) = 2.5x - 4.25\) should be graphed on a coordinate plane with an appropriate viewing window. For instance, the intervals for both x and y could be from -10 to 10. The graph will be a straight line, showing the linear relationship between x and y.

Step 1: Identify the function

The given function is \(f(x) = 2.5x - 4.25\), which is a linear function, having the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Choose an appropriate viewing window

To view the function clearly, an appropriate interval for x and y should be chosen. For this function, since there is no restriction for the values of x, a common choice for the viewing window along x-axis could be from -10 to 10. On the y-axis, choose some interval that will allow the y-intercept (-4.25 in this case) to be seen. The interval from -10 to 10 on the y-axis as well will suffice.

Step 3: Graph the Function

Now, using the slope and the y-intercept, plot the function on the graph. To do this, start at the y-intercept point (0,-4.25). From there, use the slope to find another point on the graph. The slope is 2.5, which means for every unit increase in x, y increases by 2.5 units. Choose a few x values, find the corresponding y values using the function, and plot these points. Then, draw a straight line through these points, which is the graph of the function.

Key Concepts

Graphing Utility Usage

Graphing utilities, such as calculators or computer software, are indispensable tools when studying algebra and more complex mathematics. With these tools, students and mathematicians can visualize functions, like linear equations, and their behaviors over different intervals.

Using a graphing utility to plot the function f(x) = 2.5x - 4.25 allows you to see how changes in the x value affect the y value and to understand the concept of slope and intercepts visually. To depict the function adequately, after you’ve entered it into your graphing tool of choice, you need to adjust the viewing parameters. This ensures the most critical parts of the graph, such as the x and y intercepts, are visible on-screen.

Additionally, when experimenting with different functions, graphing utilities can handle calculations quickly and plot instantaneous changes, which is beneficial for exploring what happens to the graph under various transformations or when subject to different constraints.

Choosing Viewing Window

An appropriate viewing window is essential for understanding the graph of a function and its characteristics. In the context of a linear function such as f(x) = 2.5x - 4.25, a window that captures the y-intercept and a reasonable spread along the x-axis is preferred.

A suggested viewing window for this linear equation might range from -10 to 10 on both axes. This range not only shows the y-intercept but also provides a balanced view of the function's behavior in both the positive and negative directions of the x-axis. To optimize your viewing window, consider the specific features of the function you're graphing, and adjust the window to ensure those features are displayed effectively.

Always remember that the window should be large enough to include important features of the graph but not so large that these features become too small or diluted to be informative.

Slope-Intercept Form

The slope-intercept form of a linear equation is f(x) = mx + b, an elegant way of writing linear functions that highlights their slope and y-intercept, where m represents the slope of the line, and b indicates the point where the line crosses the y-axis. In the example function f(x) = 2.5x - 4.25, 2.5 is the slope and -4.25 is the y-intercept.

This form is especially useful for graphing because it tells you exactly where to start and how to continue plotting the function. You begin at the y-intercept, the point (0, b), and use the slope, m, to determine the steepness of the line and in which direction it travels. A positive slope indicates the line rises as it moves along the x-axis, while a negative slope denotes a fall. For the function at hand, starting at (0, -4.25), for every unit you move to the right (positive x-direction), you would move up 2.5 units to plot the next point. Connecting these points with a straight line creates the graph of your linear function.