Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**Polynomial Functions and Graphs** **Current Attempt in Progress** **Problem Statement:** *Give a possible formula of minimum degree for the polynomial \( h(x) \) displayed in the graph to the right. NOTE: Enter the exact answer.* **Polynomial Input:** \[ h(x) = \] **Graph Explanation:** The graph alongside the problem statement shows a polynomial function \( h(x) \) plotted on an \( xy \)-plane. - **Axes:** The horizontal axis represents the \( x \)-values ranging from approximately \(-4\) to \( 4 \), and the vertical axis represents the \( y \)-values ranging from approximately \(-4\) to \( 4 \). - **Curve Behavior:** The graph is a smooth, continuous curve crossing the \( x \)-axis at four points, indicating that these are the roots of the polynomial. These roots approximately occur at \( x = -3, -1, 1, 3 \). - **Turning Points:** The graph has three turning points - it changes direction suggesting maximums and minimums between the roots. **Interpretation Task:** - Based on the graph, identify the roots of the polynomial and construct a possible polynomial formula with these roots. - Consider the parities of the polynomial and deduce the correct factors that would fit the minimum degree. **Possible Polynomial Construction:** Given the roots observed at \( x = -3, -1, 1, 3 \), you can form the polynomial in the form: \[ h(x) = k(x + 3)(x + 1)(x - 1)(x - 3) \] where \( k \) is a leading coefficient which could be 1 or another constant depending on the specific vertical stretching or compressing of the polynomial graph. Students are expected to observe the critical points and use their knowledge to enter the exact mathematical form of the polynomial accurately.
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Transcribed Image Text:**Polynomial Functions and Graphs** **Current Attempt in Progress** **Problem Statement:** *Give a possible formula of minimum degree for the polynomial \( h(x) \) displayed in the graph to the right. NOTE: Enter the exact answer.* **Polynomial Input:** \[ h(x) = \] **Graph Explanation:** The graph alongside the problem statement shows a polynomial function \( h(x) \) plotted on an \( xy \)-plane. - **Axes:** The horizontal axis represents the \( x \)-values ranging from approximately \(-4\) to \( 4 \), and the vertical axis represents the \( y \)-values ranging from approximately \(-4\) to \( 4 \). - **Curve Behavior:** The graph is a smooth, continuous curve crossing the \( x \)-axis at four points, indicating that these are the roots of the polynomial. These roots approximately occur at \( x = -3, -1, 1, 3 \). - **Turning Points:** The graph has three turning points - it changes direction suggesting maximums and minimums between the roots. **Interpretation Task:** - Based on the graph, identify the roots of the polynomial and construct a possible polynomial formula with these roots. - Consider the parities of the polynomial and deduce the correct factors that would fit the minimum degree. **Possible Polynomial Construction:** Given the roots observed at \( x = -3, -1, 1, 3 \), you can form the polynomial in the form: \[ h(x) = k(x + 3)(x + 1)(x - 1)(x - 3) \] where \( k \) is a leading coefficient which could be 1 or another constant depending on the specific vertical stretching or compressing of the polynomial graph. Students are expected to observe the critical points and use their knowledge to enter the exact mathematical form of the polynomial accurately.
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