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

the Zeros of a Polynomial Function and find the zeros you located. Choose xvalues and determine what the graph of h(x) will look like between its zeros. In
particular, make sure to determine whether the graph will be above the x-axis
or below the x-axis in each region. Show your work.
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). Using
your work from part A, continue the graph of h(x) by scaling the y-axis and
sketching what the graph of h(x) looks like between its zeros.
Note: You will complete Graph 1 in the final section of the portfolio.

1 answer

To provide a thorough response, I’ll need to outline the steps typically involved in finding the zeros of a polynomial function, determining the intervals where the graph is above or below the x-axis, and sketching the graph based on that information.

Step 1: Finding the Zeros of h(x)

Assuming you have a polynomial \( h(x) = a(x - r_1)(x - r_2)(x - r_3) \ldots (x - r_n) \) where \( r_1, r_2, r_3, \ldots \) are the zeros of the polynomial. The zeros of the polynomial are the values of \( x \) such that \( h(x) = 0 \). Let’s assume we found zeros at \( x = r_1, r_2, \) and \( r_3 \) for illustration purposes.

Step 2: Choosing x-values

Choose points in each interval defined by the zeros and in between:

  1. Interval 1: \( (-\infty, r_1) \)
  2. Interval 2: \( (r_1, r_2) \)
  3. Interval 3: \( (r_2, r_3) \)
  4. Interval 4: \( (r_3, \infty) \)

Select test points within these intervals to evaluate \( h(x) \):

  • For Interval 1: Choose \( x = r_1 - 1 \)
  • For Interval 2: Choose \( x = (r_1 + r_2)/2 \)
  • For Interval 3: Choose \( x = (r_2 + r_3)/2 \)
  • For Interval 4: Choose \( x = r_3 + 1 \)

Step 3: Evaluating h(x) at Test Points

Calculate \( h(x) \) at each selected point to determine where the function is positive or negative:

  1. For \( x = r_1 - 1 \): \( h(r_1 - 1) \) (assume it's positive)
  2. For \( x = (r_1 + r_2)/2 \): \( h((r_1 + r_2)/2) \) (assume it's negative)
  3. For \( x = (r_2 + r_3)/2 \): \( h((r_2 + r_3)/2) \) (assume it's positive)
  4. For \( x = r_3 + 1 \): \( h(r_3 + 1) \) (assume it's positive)

Step 4: Conclusion about the Graph

Based on the signs of \( h(x) \):

  • In Interval 1: \( h(x) > 0 \) (above x-axis)
  • In Interval 2: \( h(x) < 0 \) (below x-axis)
  • In Interval 3: \( h(x) > 0 \) (above x-axis)
  • In Interval 4: \( h(x) > 0 \) (above x-axis)

Step 5: Sketching the Graph

Now, using this information, sketch the graph:

  1. Begin at the point where \( x \) crosses the x-axis at \( r_1 \) and the graph should go down into the negative region till it reaches \( r_2 \).
  2. At \( r_2 \), the graph crosses the x-axis again, and you go up into the positive region until you reach \( r_3 \).
  3. Finally, at \( r_3 \), again the graph crosses the x-axis and you can extend the graph slightly above the x-axis as \( x \) increases.

Final Adjustment

If you have more details based on which you could derive the polynomial, you can adjust the intervals and test points accordingly, making sure to capture the behavior of the function correctly in your sketch.

Now you can finalize Graph 1 in the portfolio using these details!