Problem 33 - College Algebra

Question

Use a graphing utility to graph \(f(x)=-100|x|+100\) on the viewing window \([-5,5]\) ]. Identify the corresponding range. Show the graph.

Step-by-Step Solution

Answer

The range of the function is \([-400, 100]\).

Step 1: Understand the Function

The function given is \(f(x) = -100 |x| + 100\). This is an absolute value function, which is typically V-shaped. Here, the negative sign in front of \(|x|\) indicates that the graph will open downwards, and the +100 is a vertical shift upwards.

Step 2: Define the Domain

The domain is the set of \(x\)-values over which we want to graph the function. We will graph \(f(x) = -100|x| + 100\) for the interval \([-5, 5]\).

Step 3: Determine Key Points

To plot \(f(x) = -100|x| + 100\), find key points. Calculate \(f(x)\) at critical points: \(-5, 0,\) and \(5\).\\(f(-5) = -100(-5) + 100 = 600,\) \\(f(0) = 100,\) \\(f(5) = -100(5) + 100 = -400.\)

Step 4: Plot the Graph

Using the key points found in Step 3, plot these on the coordinate plane and connect them to form the V-shape graph. The vertex point is at (0, 100) and the arms extend downwards.

Step 5: Apply the Viewing Window

Set your graphing utility to the window \([x: -5, 5], [y: -400, 100]\). This ensures all critical points and the shape of the graph are visible for the desired range.

Step 6: Determine the Range

The range of the function is the set of possible \(f(x)\) values. From the graph, \(f(x)\) decreases from \(100\) at \(x = 0\) to \(-400\) as \(x\) approaches either \(-5\) or \(5\). Thus, the range is \([-400, 100]\).

Key Concepts

Absolute Value Function

An absolute value function is defined as a mathematical function where each input value returns its distance from zero on a number line. The general form for an absolute value function is given by \(f(x) = a|x-h| + k\). Here, the absolute value is indicated by the vertical bars around \(x\). This expression can translate to different forms depending on the values of \(a\), \(h\), and \(k\).

• \(a\) determines the opening direction and vertical stretching or compressing of the graph. If \(a\) is positive, the graph opens upwards. A negative \(a\) value means the graph opens downward, as is the case in our function \(f(x) = -100|x| + 100\).
• The vertex of the absolute value graph is influenced by \(h\) and \(k\), moving the vertex horizontally and vertically.

In our example, \(a = -100\), which reflects the graph to open downward, emphasizing the inverted V-shape of \(f(x) = -100|x| + 100\).

Domain and Range

The domain of a function refers to all possible input values (\(x\)) that the function can handle without causing undefined behavior. For the absolute value function \(f(x) = -100|x| + 100\), there are no bounds on the values \(x\) can take, making the domain all real numbers.When we talk about the range, we focus on the set of possible output values (\(f(x)\)) that the function can produce.

• For \(f(x) = -100|x| + 100\), the highest output value occurs at the vertex of the graph, which is \((0, 100)\).
• The smallest value occurs at the endpoints of the viewing window, which are at \(-400\) at \(x = -5\) and \(x = 5\).

Thus, for this function graphed within the interval \([-5, 5]\), the range is \([-400, 100]\).

Graph Interpretation

Graph interpretation involves analyzing the plotted points of a function to understand its behavior. When we interpret the graph of \(f(x) = -100|x| + 100\), several steps can be followed:

• Identify the vertex of the graph. For this function, the vertex \((0, 100)\) represents the highest point on the graph since it opens downward.
• Observe symmetry. Absolute value functions are symmetrical about the vertical line passing through the vertex (here, the y-axis).
• Assess the slope. Looking at points \((-5, 600)\) and \((5, -400)\), notice the steep decline indicating a sharp rate of change, attributed to the high magnitude of \(-100\).

By examining these elements, the graph clearly depicts the V-shape and allows us to visualize its range and critical points effectively.

Critical Points

Critical points are essential in defining the features of a function's graph. In the graph of \(f(x) = -100|x| + 100\), the critical points allow us to outline the graph accurately.For this particular graph:

• The vertex, which is the most critical point, is \((0, 100)\). This represents the peak of the graph, acting as a turning point since the graph changes direction here.
• Endpoint values help to fully display the extent of the function. Points \((-5, 600)\) and \((5, -400)\) highlight the extremities of the graph within the viewing window \([-5, 5]\).

These critical points not only offer a structure for plotting the graph accurately but also help to determine important aspects like range and points of inflection.