Problem 95 - Algebra and Trigonometry

Question

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

The plot of the function \(f(x) = -2x^3 + 6x^2 + 3x - 1\) will show the function increases without bound as x approaches negative infinity and decreases without bound as x approaches positive infinity, matching the determined end behavior based on the degree and leading coefficient of the polynomial.

Step 1: Identify the Degree and Leading Coefficient of the Polynomial

The first step is to identify the degree and the leading coefficient of the polynomial. The degree of a polynomial function is the highest power of x in the function. In this case, the degree of the function is 3, because the highest power of x in the function is 3. The leading coefficient of the polynomial function is the coefficient of the term with the highest power of x. In this case, the leading coefficient is -2.

Step 2: Determine the End Behavior of the Polynomial

The end behavior of the polynomial function is determined by the degree and the leading coefficient of the polynomial. Since the degree is 3 (an odd degree) and the leading coefficient is negative, as x approaches positive infinity, the function f(x) will approach negative infinity, and as x approaches negative infinity, the function f(x) will approach positive infinity.

Step 3: Graph the Polynomial Function

Use a graphic calculator or a graphing tool to plot the function and check the end behavior. Make sure to use a viewing window that is large enough to show the end behavior of the function. You should verify that the function increases without bound as x approaches negative infinity and decreases without bound as x approaches positive infinity.

Key Concepts

End Behavior

The end behavior of a polynomial function describes how the function behaves as the input value, \(x\), approaches positive or negative infinity. Essentially, it tells us about what happens to the f(x) values in extremely large or small scenarios. Here, the degree and the leading coefficient play a crucial role.

• If the degree of the polynomial is odd and the leading coefficient is positive, the end behavior is such that as \(x\) goes to positive infinity, \(f(x)\) goes to positive infinity, and as \(x\) goes to negative infinity, \(f(x)\) goes to negative infinity.
• If the degree is odd and the leading coefficient is negative, like in our function \(f(x)=-2x^3+6x^2+3x-1\), then \(f(x)\) approaches negative infinity as \(x\) approaches positive infinity, and \(f(x)\) approaches positive infinity as \(x\) approaches negative infinity.

The end behavior essentially frames what the tails of the graph will look like.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable \(x\) in the polynomial expression. This power significantly affects the shape of the polynomial graph.

• An odd-degree polynomial will have ends that head off in opposite vertical directions; one will go up as \(x\) increases, and the other will come down as \(x\) decreases.
• An even-degree polynomial’s ends will both go up or both go down, depending on the leading coefficient.

For instance, in the function \(f(x)=-2x^3+6x^2+3x-1\), the degree is 3. Being an odd degree, it confirms the opposite end behaviors of the graph.

Leading Coefficient

The leading coefficient of a polynomial function is the number that multiplies the term with the highest degree. This number influences the stretch and the direction of the graph.

• If the leading coefficient is positive, the graph will generally rise as you move to the right along the x-axis.
• If negative, like with our leading coefficient -2, the graph will generally fall as you move to the right.

In our example, the leading coefficient being -2 suggests a downward opening behavior as \(x\) heads to positive infinity, a vital clue to predicting the graph's overall shape.

Graphing Utility

Utilizing a graphing utility to visualize a polynomial function can be extremely helpful, especially when examining end behavior and other intricate details. These utilities allow you to:

• Choose a viewing window that captures the end behavior effectively — ensuring the tails of the function are visible.
• Quickly identify intercepts, turning points, and approximations of roots.
• Zoom in or out to focus on areas of interest or see the overall behavior.

A graphing calculator or software helps to corroborate theoretical predictions, offering a reliable visual check against your mathematical expectations. By entering the equation \(f(x)=-2x^3+6x^2+3x-1\), a student can see the rise and fall as discussed in the theory, allowing for a deeper understanding of polynomial functions.