Problem 99 - Algebra and Trigonometry

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}\)

Step-by-Step Solution

Answer

By plotting the functions and zooming out, we can observe that both functions have the same end behavior. As \(x\) goes to \(\pm\infty\), both \(f(x)\) and \(g(x)\) approach \(-\infty\).

Step 1: Plot the Functions

Use the graphing utility to plot both functions, \(f(x) = -x^{4}+ 2x^{3}-6x\) and \(g(x) = -x^{4}\), in the same viewing rectangle. Initial viewing might be set for \(x\) and \(y\) in the range of \([-10, 10]\)

Step 2: Zoom Out and Observe

Using the ZOOM OUT feature on the graphing utility, enlarge the view. Notice the behavior of the two graphs as they approach \(\pm\infty\).

Step 3: Analyze the End Behavior

Both functions share the same end behavior. As \(x\) goes to \(\pm\infty\), \(f(x)\) and \(g(x)\) approach \(-\infty\). Similarly, as \(x\) goes to \(\pm\infty\), \(f(x)\) and \(g(x)\) approach \(-\infty\). This shows that the two functions have identical end behavior.

Key Concepts

End Behavior of Functions

Understanding the end behavior of functions is essential for analyzing how a function behaves as the input values grow larger or smaller, particularly towards positive or negative infinity. In the given exercise, we examine polynomial functions, namely, f(x) = -x^{4}+2x^{3}-6x and g(x) = -x^{4}. The end behavior of a polynomial function is largely determined by its leading term, the term with the highest power.

When analyzing the end behavior, we often focus on the sign of the coefficient of the highest degree term. For even powers, if the coefficient is positive, both ends of the graph will rise to infinity (+), while if it's negative, both ends will fall to negative infinity (-). In our specific case, both functions have the leading term -x^4, which indicates that as x goes to positive or negative infinity, the output of both functions will trend towards negative infinity.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or computer software, are powerful tools for visualizing and understanding the behavior of functions. These utilities allow us to plot functions rapidly and accurately over different intervals. When faced with the task of comparing two functions, as in the exercise, graphing them simultaneously can be incredibly insightful.

To get the most out of graphing utilities, it's important to become familiar with their functions, such as 'ZOOM OUT' which allows us to observe long-range behavior or 'TRACE' to evaluate the function at specific x values. In our example, using 'ZOOM OUT' reveals that despite the additional terms in f(x), the end behavior of f(x) and g(x) is identical due to their shared leading term.

Polynomial Functions Comparison

When comparing polynomial functions, it is crucial to pay attention to factors such as the degree of the polynomial and the coefficients of the terms. The degree of the polynomial, which is the highest exponent of x, generally dictates the function's overall shape and its end behavior. In the case of the functions f(x) and g(x) from our exercise, both are fourth-degree polynomials with a leading coefficient of -1.

The additional terms in f(x) do affect the graph of the function but do not change the overall end behavior, which is consistent with the degree and leading coefficient. Therefore, despite f(x) having a 'bumpier' look due to the other terms, at very high or low x values both functions will look remarkably similar.

Analyzing Function Graphs

Graph analysis is a fundamental skill in mathematics that helps students understand the features of a function, beyond just its end behavior. When analyzing graphs, we look for key characteristics such as intercepts, turning points, and asymptotes. Other important aspects include intervals of increase or decrease, concavity, and relative maxima or minima.

With our given functions f(x) and g(x), though they share the same end behavior, the presence of the 2x^3 and -6x terms in f(x) introduce turning points not present in g(x). These features can be precisely identified using graphing utilities, and they are essential for understanding the complete picture of how a function behaves across its domain.