Problem 76 - College Algebra

Question

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{4}+8 x^{3}+4 x^{2}+2$$

Step-by-Step Solution

Answer

After inputing the function and choosing a suitable viewport, the graph shows that as \(x\) approaches both positive and negative infinity, \(f(x)\) goes to negative infinity due to the even highest power and negative leading coefficient.

Step 1: Input the Polynomial Function

First, input the given polynomial function, \(f(x) = -x^4 + 8x^3 + 4x^2 + 2\), into a graphing utility or graphing calculator. It is important to input the function accurately to ensure the correct graph is formed.

Step 2: Choose an Appropriate Viewing Rectangle

Choose a viewing rectangle that is large enough to display the end behavior of the function. Start with a standard viewing rectangle like \([-10,10]\) for \(x\) and \([-10,10]\) for \(y\). If the end behavior is not visible, adjust the viewing rectangle by either increasing the range or shifting the window.

Step 3: Graph the Function

Graph the function within the selected viewing rectangle. Pay attention to the end behavior of the graph, where the function heads to as \(x\) approaches positive and negative infinity.

Step 4: Interpret the Graph

Interpret the graph. What does the shape of the graph tell about the function? Note that the highest power in the function is 4 which is even and the leading coefficient is negative. Therefore, as \(x\) approaches both positive and negative infinity, \(f(x)\) will also go to negative infinity.

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of polynomials is crucial when graphing these functions. The end behavior describes how the function behaves as the input values, or x-values, move towards positive or negative infinity. For the example polynomial function f(x) = -x^4 + 8x^3 + 4x^2 + 2, its end behavior can be determined by looking at the leading term.

Once you know that the leading term is -x^4, you can establish that since the power is even, the function will have the same behavior on both ends, and because the leading coefficient is negative, the function will tend towards negative infinity in both directions. This is a general characteristic of polynomials with an even-degree leading term and a negative coefficient:

• Even-degree and positive coefficient: Rises to infinity on both ends.
• Even-degree and negative coefficient: Falls to negative infinity on both ends.
• Odd-degree and positive coefficient: Falls to negative infinity on one end and rises to infinity on the other.
• Odd-degree and negative coefficient: Rises to infinity on one end and falls to negative infinity on the other.

By applying this knowledge, you'll be able to predict the graph's behavior far beyond the visible plotting area on your graphing tool, aiding in comprehension of the function's characteristics.

Graphing Utility Usage

Graphing utilities, such as a graphing calculator or computer software, are incredibly helpful for visualizing polynomial functions. To get the most out of these tools when working with a function like f(x) = -x^4 + 8x^3 + 4x^2 + 2, it's essential to accurately input the function and select a suitable viewing rectangle.

This rectangle should be large enough to show the important features of the graph, particularly the end behavior. Starting with a standard viewing window and then adjusting as necessary can be a good strategy. When inputting the function, pay careful attention to signs and coefficients, as a single mistake can significantly alter the graph. After graphing, use the tool's zoom and trace features to explore the function's behavior further. These functions often also provide valuable information about key features such as intercepts, turning points and asymptotes, if applicable.

Polynomial Function Characteristics

There are multiple characteristics of polynomial functions that can be understood from their graphs. For example, consider the polynomial f(x) = -x^4 + 8x^3 + 4x^2 + 2. By graphing it, you can identify its degree, which in this case is four, and observe the number of turning points. A polynomial function of degree n can have up to n - 1 turning points.

Moreover, the graph also reveals the intercepts; where the graph crosses the x-axis are the function's roots or solutions. The Y-intercept, the point where the graph crosses the y-axis, is just the constant term of the polynomial. Interpreting the graph allows us to see the overall shape and realize, for instance, that the leading term largely influences the function's long-term behavior. These characteristics are fundamental to understanding polynomials and can be easily analyzed with a correctly defined viewing window on a graphing utility.