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Which of the following graphs represents the function g(x) = x²(x + 1)(x − 2)? 2 4 2 * 0 4 0 O 4 2 P g(x) a g(x)

Which of the following graphs represents the function g(x) = x²(x + 1)(x − 2)? 2 4 2 * 0 4 0 O 4 2 P g(x) a g(x)

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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**Question:** Which of the following graphs represents the function \( y = (x + 1)(x - 1)(x - 2)? \) **Graphs:** 1. **First Graph:** - The graph \( g(x) \) is depicted with a visible range on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - There is a clear polynomial curve with roots (where the curve intersects the x-axis) at approximately \( x = -2 \), \( x = 1 \), and \( x = 2 \). 2. **Second Graph:** - The graph \( g(x) \) is shown with a similar range on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - The curve has roots at \( x = -1 \), \( x = 1 \), and \( x = 2 \). 3. **Third Graph:** - The graph \( g(x) \) maintains the same range as the previous graphs on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - This polynomial curve intersects the x-axis exactly at the points \( x = -1 \), \( x = 1 \), and \( x = 2 \). In summary, all three graphs display polynomial functions, but only one correctly represents the specific polynomial \( y = (x + 1)(x - 1)(x - 2) \). **Explanation of the Correct Graph:** The function \( y = (x + 1)(x - 1)(x - 2) \) has roots at \( x = -1 \), \( x = 1 \), and \( x = 2 \). Therefore, the correct graph should show the curve intersecting the x-axis exactly at these points. The graph that meets this criteria is the **second graph**.
Transcribed Image Text:**Question:** Which of the following graphs represents the function \( y = (x + 1)(x - 1)(x - 2)? \) **Graphs:** 1. **First Graph:** - The graph \( g(x) \) is depicted with a visible range on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - There is a clear polynomial curve with roots (where the curve intersects the x-axis) at approximately \( x = -2 \), \( x = 1 \), and \( x = 2 \). 2. **Second Graph:** - The graph \( g(x) \) is shown with a similar range on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - The curve has roots at \( x = -1 \), \( x = 1 \), and \( x = 2 \). 3. **Third Graph:** - The graph \( g(x) \) maintains the same range as the previous graphs on the x-axis from -5 to 5 and on the y-axis from -10 to 10. - This polynomial curve intersects the x-axis exactly at the points \( x = -1 \), \( x = 1 \), and \( x = 2 \). In summary, all three graphs display polynomial functions, but only one correctly represents the specific polynomial \( y = (x + 1)(x - 1)(x - 2) \). **Explanation of the Correct Graph:** The function \( y = (x + 1)(x - 1)(x - 2) \) has roots at \( x = -1 \), \( x = 1 \), and \( x = 2 \). Therefore, the correct graph should show the curve intersecting the x-axis exactly at these points. The graph that meets this criteria is the **second graph**.
The image depicts the graph of a function \( g(x) \) on a Cartesian coordinate system. Here is a detailed explanation suitable for an educational website: **Graph Analysis: Function \( g(x) \)** **Domain and Range:** - The domain (x-values) of the function \( g(x) \) extends from \(-4\) to 5. - The range (y-values) of the function \( g(x) \) extends from \(-4\) to 5. **Behavior of the Function:** - The graph starts at the point \((-4, 3)\), which means that when \(x = -4\), \(g(x) = 3\). - From \(-4\) to \(-3\), the graph decreases, reaching a local minimum at the point approximately \((-3, -4)\). - The function then increases and crosses the x-axis at \((-1.5, 0)\), indicating a root of the function. - As \(x\) continues to increase, the function initially rises, forming a local maximum at approximately \((-1, 0.5)\). - Following this peak, the function decreases, intersecting the y-axis at \(g(0) = -6\). - The graph then continues to increase again, with an inflection point (where the graph changes concavity). It crosses the x-axis at another root, approximately at \((2.5, 0)\). - Finally, the function rises steeply at the right side of the graph ending around \((5, 4)\). **Key Features:** - The graph shows important points of intersection with the x-axis (roots of \(g(x)\)). - It includes local maxima and minima, indicating points where the function changes direction. - The function appears to have smooth, continuous curves and shows both concave upwards and concave downwards behavior. Understanding the shape and key points of the graph help in analyzing the properties of the function \( g(x) \).
Transcribed Image Text:The image depicts the graph of a function \( g(x) \) on a Cartesian coordinate system. Here is a detailed explanation suitable for an educational website: **Graph Analysis: Function \( g(x) \)** **Domain and Range:** - The domain (x-values) of the function \( g(x) \) extends from \(-4\) to 5. - The range (y-values) of the function \( g(x) \) extends from \(-4\) to 5. **Behavior of the Function:** - The graph starts at the point \((-4, 3)\), which means that when \(x = -4\), \(g(x) = 3\). - From \(-4\) to \(-3\), the graph decreases, reaching a local minimum at the point approximately \((-3, -4)\). - The function then increases and crosses the x-axis at \((-1.5, 0)\), indicating a root of the function. - As \(x\) continues to increase, the function initially rises, forming a local maximum at approximately \((-1, 0.5)\). - Following this peak, the function decreases, intersecting the y-axis at \(g(0) = -6\). - The graph then continues to increase again, with an inflection point (where the graph changes concavity). It crosses the x-axis at another root, approximately at \((2.5, 0)\). - Finally, the function rises steeply at the right side of the graph ending around \((5, 4)\). **Key Features:** - The graph shows important points of intersection with the x-axis (roots of \(g(x)\)). - It includes local maxima and minima, indicating points where the function changes direction. - The function appears to have smooth, continuous curves and shows both concave upwards and concave downwards behavior. Understanding the shape and key points of the graph help in analyzing the properties of the function \( g(x) \).
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