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 address your questions, let’s go through each part step by step.

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

The end behavior of a polynomial function is determined by the leading term of the function, which is the term with the highest power of x. The general form of a polynomial function is:

\[ h(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]

Where:

• \( n \) is the degree of the polynomial,
• \( a_n \) is the leading coefficient.

The end behavior is described as follows based on the degree (n) and the leading coefficient (\( a_n \)):

1. If \( n \) is even and \( a_n > 0 \): As \( x \to +\infty \) or \( x \to -\infty \), \( h(x) \to +\infty \). The graph rises on both ends.

2. If \( n \) is even and \( a_n < 0 \): As \( x \to +\infty \) or \( x \to -\infty \), \( h(x) \to -\infty \). The graph falls on both ends.

3. If \( n \) is odd and \( a_n > 0 \): As \( x \to +\infty \), \( h(x) \to +\infty \) and as \( x \to -\infty \), \( h(x) \to -\infty \). The graph rises on the right and falls on the left.

4. If \( n \) is odd and \( a_n < 0 \): As \( x \to +\infty \), \( h(x) \to -\infty \) and as \( x \to -\infty \), \( h(x) \to +\infty \). The graph falls on the right and rises on the left.

To precisely describe the end behavior of \( h(x) \), we'll need to know the degree of the polynomial and the sign of the leading coefficient.

B. Sketching the Graph of h(x)

1. Sketching the Zeros: Based on your earlier work, you should identify the x-intercepts (or zeros) of the polynomial. Each zero will correspond to a point where the graph intersects the x-axis. You would have already sketched the sections of the graph that exist between these zeros.

2. Add End Behavior: Using your findings from part A, you can now draw the ends of the graph:

   • If \( h(x) \) rises on both ends (even degree with a positive leading coefficient) or falls on both ends (even degree with a negative leading coefficient), you'll extend the lines up or down respectively from the last zeros you plotted.
   • If \( h(x) \) rises on the right and falls on the left (odd degree with a positive leading coefficient), you’ll draw a downward curve that moves upward on the right side.
   • If \( h(x) \) falls on the right and rises on the left (odd degree with a negative leading coefficient), you’ll draw an upward curve approaching downward on the right side.

Final Graph

1. Combine: Combine the zeros and the end behavior to produce a complete sketch of the polynomial function.
2. Smooth Curves: Make sure transitions between the zeros and the ends are smooth curves as polynomials are continuous functions without breaks or angles.

By following these steps, you should have a good representation of the polynomial function \( h(x) \) accounting for both its zeros and its end behavior.