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Which of the following graphs represents the function g(x) = x(x + 1)²(x − 2)? g(x) 4 4 3 g(x) 4 3 O O 4 4

Which of the following graphs represents the function g(x) = x(x + 1)²(x − 2)? g(x) 4 4 3 g(x) 4 3 O O 4 4

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The image consists of two graphs of a function \( g(x) \). Each graph is plotted on Cartesian coordinate systems. ### Graph 1: - The graph is a function \( g(x) \) plotted with the x-axis (horizontal axis) ranging approximately from -6 to 6, and the y-axis (vertical axis) ranging from -7 to approximately 8. - The curve passes through various points, indicating multiple inflection points and a local minimum and maximum. - There are marked changes in the curvature of the graph. - The function dips to a local minimum around \( x = -2 \) and peaks to a local maximum before increasing steeply again. ### Graph 2: - Similarly, this graph represents the function \( g(x) \), plotted against the same x and y-axis scale as in Graph 1. - The function's behavior is quite similar to the first graph, but subtle differences may exist, such as shifts or scaling adjustments. - The curve again features points indicative of local minima and maxima, with complexity near the origin, and extending smoothly outside of these regions. ### Comparative Analysis: - Both graphs appear to depict a polynomial function with multiple critical points (local maxima and minima). - The functions show similar overall shapes and fluctuations with significant overlapping ranges. These graphs can be used to study the behavior of polynomial functions, especially focusing on their turning points and the nature of their curves around these critical points.
Transcribed Image Text:The image consists of two graphs of a function \( g(x) \). Each graph is plotted on Cartesian coordinate systems. ### Graph 1: - The graph is a function \( g(x) \) plotted with the x-axis (horizontal axis) ranging approximately from -6 to 6, and the y-axis (vertical axis) ranging from -7 to approximately 8. - The curve passes through various points, indicating multiple inflection points and a local minimum and maximum. - There are marked changes in the curvature of the graph. - The function dips to a local minimum around \( x = -2 \) and peaks to a local maximum before increasing steeply again. ### Graph 2: - Similarly, this graph represents the function \( g(x) \), plotted against the same x and y-axis scale as in Graph 1. - The function's behavior is quite similar to the first graph, but subtle differences may exist, such as shifts or scaling adjustments. - The curve again features points indicative of local minima and maxima, with complexity near the origin, and extending smoothly outside of these regions. ### Comparative Analysis: - Both graphs appear to depict a polynomial function with multiple critical points (local maxima and minima). - The functions show similar overall shapes and fluctuations with significant overlapping ranges. These graphs can be used to study the behavior of polynomial functions, especially focusing on their turning points and the nature of their curves around these critical points.
### Understanding the Function \( g(x) = x(x + 1)^2(x - 2)^2 \) **Question:** - Which of the following graphs represents the function \( g(x) = x(x + 1)^2(x - 2)^2 \)? **Graphs:** 1. **First Graph:** - **Axes:** - **x-axis:** The x-axis is labelled and scaled from -6 to 6. - **y-axis:** The y-axis is labelled and scaled from -8 to 3. - **Function Behavior:** - The plot starts in the positive y-direction, dips below the x-axis into negative y-values, rises back above the x-axis near \( x = -1 \), dips again, and finally rises steeply after \( x = 2 \). - **Intercepts:** - Intercepts the x-axis at \( x = -1 \), \( x = 0 \), and \( x = 2 \). - **Shape and Turning Points:** - Exhibits local maxima and minima. - Symmetric properties due to the squaring of terms \( (x + 1) \) and \( (x - 2) \). 2. **Second Graph:** - **Axes:** - **x-axis:** The x-axis is labelled and scaled from -5 to 6. - **y-axis:** The y-axis is labelled and scaled from -7 to 3. - **Function Behavior:** - The plot behaves similarly to the first graph, but has slight differences in the turning points' values and smoothness. - **Intercepts:** - Also intercepts the x-axis at \( x = -1 \), \( x = 0 \), and \( x = 2 \). - **Shape and Turning Points:** - Shares the general behavior of having local maxima and minima similar to the first graph. ### Explanation of the Function: The given function \( g(x) = x(x + 1)^2(x - 2)^2 \) expands to a degree 5 polynomial. Each factor indicates a root: - \( x = 0 \) - \( x = -1 \) with multiplicity 2 (double root indicating a local minimum or maximum) - \( x = 2 \) with multiplicity 2 (double
Transcribed Image Text:### Understanding the Function \( g(x) = x(x + 1)^2(x - 2)^2 \) **Question:** - Which of the following graphs represents the function \( g(x) = x(x + 1)^2(x - 2)^2 \)? **Graphs:** 1. **First Graph:** - **Axes:** - **x-axis:** The x-axis is labelled and scaled from -6 to 6. - **y-axis:** The y-axis is labelled and scaled from -8 to 3. - **Function Behavior:** - The plot starts in the positive y-direction, dips below the x-axis into negative y-values, rises back above the x-axis near \( x = -1 \), dips again, and finally rises steeply after \( x = 2 \). - **Intercepts:** - Intercepts the x-axis at \( x = -1 \), \( x = 0 \), and \( x = 2 \). - **Shape and Turning Points:** - Exhibits local maxima and minima. - Symmetric properties due to the squaring of terms \( (x + 1) \) and \( (x - 2) \). 2. **Second Graph:** - **Axes:** - **x-axis:** The x-axis is labelled and scaled from -5 to 6. - **y-axis:** The y-axis is labelled and scaled from -7 to 3. - **Function Behavior:** - The plot behaves similarly to the first graph, but has slight differences in the turning points' values and smoothness. - **Intercepts:** - Also intercepts the x-axis at \( x = -1 \), \( x = 0 \), and \( x = 2 \). - **Shape and Turning Points:** - Shares the general behavior of having local maxima and minima similar to the first graph. ### Explanation of the Function: The given function \( g(x) = x(x + 1)^2(x - 2)^2 \) expands to a degree 5 polynomial. Each factor indicates a root: - \( x = 0 \) - \( x = -1 \) with multiplicity 2 (double root indicating a local minimum or maximum) - \( x = 2 \) with multiplicity 2 (double
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