Refer to the polynomial function h(x) that you wrote in Question 1 of Finding

Question

the Zeros of a Polynomial Function. Describe and provide an explanation for
the end behavior of h(x)
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 discuss the end behavior of the polynomial function \( h(x) \), we need to consider a few key properties of polynomials. The end behavior of a polynomial is determined by its leading term, which is the term with the highest power of \( x \). Here are some general points to consider:

1. Degree of the Polynomial: The degree of the polynomial is the highest exponent of \( x \) in the polynomial. If the degree is an even number, the ends of the graph will either both go up or both go down. If the degree is odd, one end will go up while the other goes down.

2. Leading Coefficient: The sign of the leading coefficient (the coefficient of the leading term) also influences the end behavior:

   • If the leading coefficient is positive:
     • For even degree, both ends of the graph will point upwards.
     • For odd degree, the left end will point downwards and the right end will point upwards.
   • If the leading coefficient is negative:
     • For even degree, both ends of the graph will point downwards.
     • For odd degree, the left end will point upwards and the right end will point downwards.

Example of Analyzing End Behavior

Assume \( h(x) \) is a polynomial such as \( h(x) = 2x^4 - 3x^3 + 5x - 7 \) (degree 4, positive leading coefficient).

• Degree: 4 (even)
• Leading Coefficient: 2 (positive)

End Behavior:

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

Graph Sketching

When sketching \( h(x) \):

1. Plot the zeros: Based on the earlier analysis in your portfolio, plot the x-intercepts of the polynomial.
2. Identify the y-intercept: Calculate and plot the point where \( x = 0 \) (i.e., \( h(0) \)).
3. Determine turning points: Between the zeros, plot the general shape of the curve based on its degree, taking care to respect the identified turning points.
4. Add the end behavior: Based on the end behavior analysis, ensure that both ends of the graph behave as determined above.

Example Sketching

For \( h(x) = 2x^4 - 3x^3 + 5x - 7 \):

• It has zeros at specific values (let's say \( x = a, b, c \) with b being real numbers).
• You would plot the zeros and then draw the curve that rises from the bottom left, intersects the x-axis at the zeros, and climbs to the top right as determined by the end behavior.

By following these steps, you will have a comprehensive description and graph sketch of \( h(x) \), including its end behavior.