To help you with this question about analyzing a polynomial function \( h(x) \) and graphing it based on its zeros, let me guide you through the process step by step. Since I don’t have access to your specific polynomial function from "Question 1 of Finding the Zeros," let's assume a hypothetical polynomial function to illustrate the concept.
A. Finding the Zeros and Analyzing the Intervals
Let’s suppose \( h(x) = x^3 - 6x^2 + 9x \). We can factor this cubic polynomial: \[ h(x) = x(x - 3)(x - 3) = x(x - 3)^2 \]
Step 1: Finding the Zeros
The zeros of the function \( h(x) \) can be found by setting \( h(x) = 0 \):
- \( x = 0 \)
- \( (x - 3)^2 = 0 \) gives \( x = 3 \) (with a multiplicity of 2).
Thus, the zeros are \( x = 0 \) and \( x = 3 \).
Step 2: Analyzing the Intervals
Next, we need to analyze the sign of the polynomial in the intervals defined by its zeros. The zeros divide the x-axis into the following intervals:
- \( (-\infty, 0) \)
- \( (0, 3) \)
- \( (3, \infty) \)
Select test points from each interval:
-
For \( x < 0 \) (let's say \( x = -1 \)): \[ h(-1) = (-1)((-1) - 3)^2 = -1(4) = -4 < 0 \quad (\text{below the x-axis}) \]
-
For \( 0 < x < 3 \) (let's say \( x = 1 \)): \[ h(1) = (1)(1 - 3)^2 = (1)(-2)^2 = 4 > 0 \quad (\text{above the x-axis}) \]
-
For \( x > 3 \) (let's say \( x = 4 \)): \[ h(4) = (4)(4 - 3)^2 = (4)(1)^2 = 4 > 0 \quad (\text{above the x-axis}) \]
Summary of Findings:
- Interval \( (-\infty, 0) \): Graph is below the x-axis
- Interval \( (0, 3) \): Graph is above the x-axis
- Interval \( (3, \infty) \): Graph is above the x-axis
B. Sketching the Graph
Using this information, the behavior of the function between its zeros can be sketched:
- Start at the left (as \( x \to -\infty \)), where the graph is below the x-axis.
- The graph crosses the x-axis at \( x = 0 \).
- The graph rises above the x-axis between \( 0 < x < 3 \).
- At \( x = 3 \), since it has a multiplicity of 2, the graph touches the x-axis but does not cross it, indicating it remains above the axis afterwards.
- Finally, the graph stays above the x-axis for \( x > 3 \).
To visualize this:
- Draw your axes (x and y).
- Mark the points (0, 0) and (3, 0).
- Draw the curve as described, being sure to keep it below the x-axis before 0, above between 0 and 3, touching 3 and staying above afterwards.
Conclusion
Your graph will show clear behavior based on the zeros and test points evaluated. By following this process for your specific polynomial \( h(x) \), you can determine the general shape and behavior of the graph accordingly. If you have a specific polynomial in mind, please share it, and I can provide a more tailored response.
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