Problem 96 - Algebra and Trigonometry

Question

In Exercises 94–97, 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

The end behavior of the graph of the function \(f(x)=-x^{4}+8 x^{3}+4 x^{2}+2\), is that as x approaches both infinity and negative infinity, f(x) approaches negative infinity. This is seen when graphing the function using a graphing utility.

Step 1: Understand the Polynomial

The given function is a polynomial function, specifically of degree 4. Polynomial functions of odd degrees have ends that go in opposite directions: one end of the graph will approach negative infinity while the other end will approach positive infinity. However, for polynomial functions of even degrees, like in this case, both ends of the graph go in the same direction: they both approach positive infinity or both approach negative infinity. The direction is determined by the sign of the highest degree term, -x^4 in this case. Since the sign is negative, both ends of the graph should approach negative infinity.

Step 2: Use a Graphing Utility to Graph the Function

Use any graphing utility, either online, or a graphing calculator, to graph the function \(f(x)=-x^{4}+8 x^{3}+4 x^{2}+2\). Make sure to set the viewing window large enough to see the overall behavior of the graph, especially the ends of the graph that extend to infinity or negative infinity.

Step 3: Observe the End Behavior

Once the graph is plotted, observe the direction the graph takes as it extends towards positive and negative infinity along the x-axis. Directions that the graph takes will match the earlier prediction based on the polynomial type and leading coefficient. Both ends should be proceeding towards negative infinity as x approaches both infinity and negative infinity.

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions can help us visualize how they behave across different x-values. A polynomial function is an expression that involves only non-negative integer powers of the variable. In our exercise, the function is given as \(f(x) = -x^4 + 8x^3 + 4x^2 + 2\), which is a polynomial of degree 4. This degree indicates that the function could have up to four roots or x-intercepts.

To graph a polynomial function, it's crucial to understand its different features:

• The degree of the polynomial and the sign of the leading coefficient determine the overall shape or end behavior.
• Polynomials may have humps or turns depending on their degree, indicating local maxima or minima.
• The roots or x-intercepts are where the graph crosses or touches the x-axis.
• The y-intercept is simply the constant term when \(x = 0\).

While graphing manually is possible, using technological tools can give a more accurate representation, helping identify points that might not be easily calculable by hand.

End Behavior of Polynomials

The end behavior of a polynomial graph describes how the function behaves as \(x\) approaches positive or negative infinity. This behavior mainly depends on the degree of the polynomial and the sign of its leading coefficient.

For polynomial functions of:

• Even Degrees: Both ends of the graph move in the same direction. If the leading coefficient is positive, both ends extend towards positive infinity. If it's negative, as with \(-x^4 ...,\) they both extend negative infinity.
• Odd Degrees: The ends move in opposite directions. With a positive leading coefficient, the graph rises to positive infinity on one end and falls to negative infinity on the other.

In our example, since we have a degree 4 polynomial with a negative leading term, the end behavior shows that as \(x\) approaches both positive and negative infinity, \(f(x)\) approaches negative infinity. This knowledge helps in sketching the rough shape of the graph even before plotting.

Using Graphing Utilities

Graphing utilities are powerful tools to visualize polynomial functions, especially complex ones that are difficult to sketch by hand. There are several types of graphing tools available such as online graphing calculators and software that can plot functions instantly.

When using a graphing utility:

• Input the full polynomial equation, ensuring all terms are correct.
• Adjust the viewing window to allow the best perspective of the graph, including watching how it behaves at the extremes.
• Examine critical characteristics like x-intercepts, y-intercept, and turning points. Source specific values for these points to understand their exact location.

By plotting the function \(f(x) = -x^4 + 8x^3 + 4x^2 + 2\), you can easily confirm the degree and leading coefficient prediction by observing the end behavior. This technological aid allows for more flexibility in exploring mathematical functions, helping students grasp the concepts of polynomial behaviors more intuitively.