Problem 40 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(h(x) = 3 - |x|\)

Step-by-Step Solution

Answer

The graph of \(h(x) = 3 - |x|\) is a V-shaped graph that has been flipped upside down with a vertex at the point (0,3). After selecting a proper window, from -5 to 5 on the x-axis and from -3 to 5 on the y-axis, the function can be effectively plotted.

Step 1: Understand the function

This is an absolute value function, which is a function of the form \(f(x) = |x|\). The graph of an absolute value function typically looks like a 'V'. The input inside the absolute value can shift or flip this 'V', while a factor outside can stretch or squish it, and adding or subtracting a number outside the absolute value can shift it up or down. Therefore, the function \(h(x) = 3 - |x|\) will look like a 'V' that has been flipped and shifted upward.

Step 2: Graphing the function

In the graphing utility, input \(h(x) = 3 - |x|\). You should see a V-shaped graph.

Step 3: Choosing an appropriate viewing window

Set the domain (x-values) to span from -5 to 5. This should be enough to see the important parts of the graph. Set the range (y-values) to span from -3 to 5. This should allow you to see the whole 'V', including the vertex point and how it extends indefinitely in the negative x directions. You can adjust these values to see more or less of the graph as needed.

Key Concepts

Absolute Value Function

Understanding the absolute value function is essential for mastering various mathematical concepts and applications. An absolute value function is defined by the expression \(f(x) = |x|\), where \(|x|\) denotes the absolute value of \(x\), which is \(x\)'s distance from zero on the number line, regardless of direction. In other words, whether \(x\) is positive or negative, the absolute value function will return a non-negative result.

For the given function \(h(x) = 3 - |x|\), it represents a transformation of the basic absolute value function. This particular function implies a vertical shift upwards by 3 units, as well as a reflection across the x-axis. Subsequent stretches, compressions, or horizontal shifts can further modify the graph, but the core 'V' shape remains consistent across all absolute value functions.

Graphing Utility

A graphing utility is a powerful tool for visualizing mathematical functions, particularly useful for students and educators alike. When graphing the absolute value function \(h(x) = 3 - |x|\), using a graphing utility simplifies the process. It takes care of computations and offers an immediate visual representation of the function. Most graphing utilities allow for input of the function directly in the same format as it appears in your textbook.

You can manipulate various aspects of the graph, such as zooming in or out and adjusting the viewing window for better understanding the behavior of the function. Advanced graphing utilities also enable you to trace along the curve, calculate key points like intercepts and maxima or minima, and analyze asymptotic behavior.

Viewing Window

Selecting an appropriate viewing window is a critical step in graphing functions because it determines how much of the graph you will see. The viewing window includes the range of x-values (domain) and y-values (range) shown on the screen. To graph \(h(x) = 3 - |x|\) effectively, you need to include enough of the x-axis and y-axis to display the significant features of the graph—the vertex, intercepts, and the general 'V' shape that extends outward.

Typically, for simple functions, setting the domain from -5 to 5 and the range from -3 to 5 provides a good starting point. You may adjust this window to focus on particular areas of interest, for example, where the function intersects the axes or to illustrate the behavior of the function as \(x\) grows very large in either direction.

V-shaped Graph

The graph of an absolute value function is iconic for its 'V' shape. This shape results from the fact that the absolute value function measures distance from zero, creating a pivot at the origin, with equal and opposite slopes extending in both directions from this pivot point. For \(h(x) = 3 - |x|\), this 'V' has been flipped upside down due to the negative sign and shifted up by 3 units.

The sharp point of the 'V', known as the vertex, occurs where the value inside the absolute value is zero. In this case, the vertex would be at \(x=0\). As \(x\) becomes increasingly positive or negative, the graph extends indefinitely along a straight line, creating two rays that form the arms of the 'V'. No matter how the function is transformed, the 'V' shape is always maintained, though the direction it points and its location on the coordinate plane do change.