Problem 75 - 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. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

Step-by-Step Solution

Answer

Using a graphing utility, plot the function \(f(x)=-2x^3 + 6x^2 + 3x - 1\), ensuring the viewing rectangle is large enough to show the end behavior. The function approaches negative infinity as \(x\) goes towards positive infinity and approaches positive infinity as \(x\) goes towards negative infinity. You can find additional significant points on the graph, such as the intercepts and turning points, by solving for \(x\) when \(f(x) = 0\) and calculating the derivative, respectively.

Step 1: Identify the polynomial function

The function to graph is \(f(x) = -2x^3 + 6x^2 + 3x - 1\).

Step 2: Draw the function with a graphing utility

Input the function into a graphing utility. Make sure the viewing rectangle is large enough to show the end behavior of the function. Notice the shape and position of the graph.

Step 3: Analyze end behavior

The end behavior of a function involves how the y-values of the function react as the x-values either increase without bound (go to positive infinity) or decrease without bound (go to negative infinity). In this case, since the term with the highest degree is \(x^3\) and has a negative coefficient, this function approaches negative infinity as \(x\) approaches positive infinity, and approaches positive infinity as \(x\) approaches negative infinity. This can be represented as follows: \(\lim_{{x \to +\infty}} f(x) = -\infty, \lim_{{x \to -\infty}} f(x) = +\infty\).

Step 4: Identify other significant points

Besides end behavior, other significant points of a polynomial function include intercepts and turning points. You can determine these points by setting \(f(x)\) equal to zero and solving for \(x\) (for x-intercepts), and by finding the derivative of the polynomial and setting it to zero (for turning points).

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of polynomial functions can provide a snapshot of a graph's long-term tendencies, serving as a crucial guide to its overall shape. For the given polynomial function, say, \(f(x) = -2x^3 + 6x^2 + 3x - 1\), the leading term is \(-2x^3\). This term has the most significant impact on the graph's behavior at extremely high or low values of \(x\).

The leading coefficient being negative and the degree being odd tells us that as \(x\) approaches positive infinity, the function goes towards negative infinity. Conversely, as \(x\) approaches negative infinity, the function will head towards positive infinity. Expressed mathematically, this is \(\lim_{{x \to +\infty}} f(x) = -\infty\) and \(\lim_{{x \to -\infty}} f(x) = +\infty\). This behavior creates a kind of 'swoosh' — a steep decline or incline on the graph's ends. These insights into end behavior are essential for sketching a rough outline of any polynomial's graph before plotting points or considering other features.

Using Graphing Utilities

In a digital age, graphing utilities are indispensable tools for mathematics students. These programs or applications allow for accurate and efficient graphing of complex polynomial functions like \(f(x) = -2x^3 + 6x^2 + 3x - 1\). When using these tools, it's important to input the function correctly and adjust the viewing rectangle. This ensures you can see not just the middle section, but also the critical end behavior of the function.

For instance, if the viewing rectangle is too small, you might miss where the graph crosses the axis or how it behaves as \(x\) extends towards infinity. This can lead to misunderstandings about the nature of the function. Many graphing utilities also offer features like zooming and tracing, which can help identify specific values and intercepts. Utilizing these utilities fully can save time and increase comprehension of the underlying mathematics.

Polynomial Function Features

Polynomial functions have distinct features that can be identified and analyzed to understand the function's graph better. With our exercise's polynomial, \(f(x) = -2x^3 + 6x^2 + 3x - 1\), significant features to look for include intercepts and turning points.

To find the x-intercepts, solve \(f(x) = 0\). These are the points where the graph crosses or touches the x-axis. As for the turning points, these are found where the graph changes direction, which can be determined by calculating the function's derivative and finding its roots. Turning points are helpful in sketching the intermediate shape of the polynomial graph between the end behavior traits. By combining an understanding of these features with end behavior and utilization of graphing utilities, the complete picture of a polynomial graph begins to emerge.