A Refer to the polynomial function h(x) that you wrote in Question 1 of Finding the Zeros of a Polynomial Function. Describe and provide an explanation for the end behavior of h(x)

Question

B. Find Graph 1 that you started in the first section of this portfolio (on using the factored form of a polynomial function to find the zeros of its graph) and continued in the second section of this portfolio (on sketching the portions of a polynomial function’s graph that exist between its zeros). Using your work from part A, complete the sketch of the graph of h(x) by including its end behavior.

Answer 1

To provide a detailed response, let's break this down into two parts based on your inquiry about the polynomial function \( h(x) \).

A. End Behavior of the Polynomial Function \( h(x) \)

To understand the end behavior of a polynomial function, we first need to consider its leading term, as this term dominates the function's behavior for large values of \( |x| \) (both positive and negative).

1. Degree of the Polynomial: Identify the degree of the polynomial function \( h(x) \). The degree is the highest power of \( x \) in the polynomial.

2. Leading Coefficient: Identify the leading coefficient, which is the coefficient of the leading term (the term with the highest degree).

The end behavior can be summarized as follows:

• If the degree is even and the leading coefficient is positive:

  • As \( x \to \infty \), \( h(x) \to \infty \)
  • As \( x \to -\infty \), \( h(x) \to \infty \)

• If the degree is even and the leading coefficient is negative:

  • As \( x \to \infty \), \( h(x) \to -\infty \)
  • As \( x \to -\infty \), \( h(x) \to -\infty \)

• If the degree is odd and the leading coefficient is positive:

  • As \( x \to \infty \), \( h(x) \to \infty \)
  • As \( x \to -\infty \), \( h(x) \to -\infty \)

• If the degree is odd and the leading coefficient is negative:

  • As \( x \to \infty \), \( h(x) \to -\infty \)
  • As \( x \to -\infty \), \( h(x) \to \infty \)

By determining the degree and leading coefficient of your polynomial function \( h(x) \), you can accurately describe its end behavior.

B. Complete Graph Sketch of \( h(x) \)

To complete the sketch of the graph of \( h(x) \):

1. Plotting the Zeros: From the analysis in the previous sections, plot the zeros of \( h(x) \). These are the x-values where \( h(x) = 0 \) on your graph.

2. Sketching the Portions Between Zeros: Using the behavior of the function between the zeros (as mentioned in your previous work), sketch the graph's path that connects these x-intercepts.

3. Incorporating End Behavior: Finally, use the end behavior identified in part A to draw how the graph behaves as \( x \) approaches \( \infty \) and \( -\infty \). Ensure that your sketch reflects the trends identified (rising or falling towards positive or negative infinity).

4. Final Graphing: Ensure that the graph is smooth and continuous. The overall shape should reflect the polynomial's degree and whether it is even or odd, as per the end behavior identified.

If you have specific details about the degree and leading coefficient of \( h(x) \) or the zeros, feel free to share, and I can provide a more tailored explanation!