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8 D.The graph y = P(x) is shown. Which of the following could be the equation for the graph of P(x)? Explain how you know. A. P(x) = -x* + 8x2 + 16 B. P(x) = x'-8x2 + 16 C. P(x) = 2x - 8x + 16 D. P(x) = -2x- 8x' + 16

8 D.The graph y = P(x) is shown. Which of the following could be the equation for the graph of P(x)? Explain how you know. A. P(x) = -x* + 8x2 + 16 B. P(x) = x'-8x2 + 16 C. P(x) = 2x - 8x + 16 D. P(x) = -2x- 8x' + 16

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Graph Analysis and Polynomial Equation Identification **Question:** The graph \( y = P(x) \) is shown below. Which of the following could be the equation for the graph of \( P(x) \)? **Explain** how you know. **Options:** A. \( P(x) = -x^4 + 8x^2 + 16 \) B. \( P(x) = x^4 - 8x^2 + 16 \) C. \( P(x) = 2x^5 - 8x^3 + 16 \) D. \( P(x) = -2x^5 - 8x^3 + 16 \) **Graph Description:** The provided graph is a polynomial plot on a Cartesian plane. The graph starts from the top-left corner, dips towards the x-axis, and shows a double peak with the significant points turning back at the x-axis before it rises again towards the top-right corner. **Graph Characteristics:** - The polynomial graph appears to have multiple turning points, suggesting it is a higher-degree polynomial. - The overall shape is symmetric around the y-axis which hints at an even function when considering only even-powered terms. **Steps to Determine the Equation:** 1. **Identify Symmetry and Degree:** - The graph shows symmetry about the y-axis, indicating it only contains even-powered terms. 2. **Evaluate Turning Points:** - The presence and behavior of multiple local minima/maxima restrict our equation to certain forms. 3. **Analyzing the Options:** - Option A: \( P(x) = -x^4 + 8x^2 + 16 \) - This equation's symmetry and degree fit the observed graph. - Option B: \( P(x) = x^4 - 8x^2 + 16 \) - Although symmetrical, it contradicts the downwards opening nature of the graph leading with \( -8x^2 \). - Option C: \( P(x) = 2x^5 - 8x^3 + 16 \) - Contains an odd degree term \( x^5 \), which would break the symmetry. - Option D: \( P(x) = -2x^5 - 8x^3 + 16 \) - Similarly, the odd degree \( x^5 \) term negates symmetry. **
Transcribed Image Text:### Graph Analysis and Polynomial Equation Identification **Question:** The graph \( y = P(x) \) is shown below. Which of the following could be the equation for the graph of \( P(x) \)? **Explain** how you know. **Options:** A. \( P(x) = -x^4 + 8x^2 + 16 \) B. \( P(x) = x^4 - 8x^2 + 16 \) C. \( P(x) = 2x^5 - 8x^3 + 16 \) D. \( P(x) = -2x^5 - 8x^3 + 16 \) **Graph Description:** The provided graph is a polynomial plot on a Cartesian plane. The graph starts from the top-left corner, dips towards the x-axis, and shows a double peak with the significant points turning back at the x-axis before it rises again towards the top-right corner. **Graph Characteristics:** - The polynomial graph appears to have multiple turning points, suggesting it is a higher-degree polynomial. - The overall shape is symmetric around the y-axis which hints at an even function when considering only even-powered terms. **Steps to Determine the Equation:** 1. **Identify Symmetry and Degree:** - The graph shows symmetry about the y-axis, indicating it only contains even-powered terms. 2. **Evaluate Turning Points:** - The presence and behavior of multiple local minima/maxima restrict our equation to certain forms. 3. **Analyzing the Options:** - Option A: \( P(x) = -x^4 + 8x^2 + 16 \) - This equation's symmetry and degree fit the observed graph. - Option B: \( P(x) = x^4 - 8x^2 + 16 \) - Although symmetrical, it contradicts the downwards opening nature of the graph leading with \( -8x^2 \). - Option C: \( P(x) = 2x^5 - 8x^3 + 16 \) - Contains an odd degree term \( x^5 \), which would break the symmetry. - Option D: \( P(x) = -2x^5 - 8x^3 + 16 \) - Similarly, the odd degree \( x^5 \) term negates symmetry. **
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