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Use the graph of f to sketch each graph. y (0, 5) (-3,0), 3.0) -5 (-6, -4) (6, -4)

Use the graph of f to sketch each graph. y (0, 5) (-3,0), 3.0) -5 (-6, -4) (6, -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 displays four graphs representing different functions on coordinate planes, marked by various curves intersecting along the x and y axes. ### Function: The function given is \( y = \frac{1}{2}f(x) \). ### Graph Descriptions: 1. **Top Left Graph:** - The graph presents a curve that is relatively flat, centered around the x-axis, showing compression of the original function \( f(x) \). - The curve's amplitude is reduced by a factor of \(\frac{1}{2}\), indicating vertical compression. 2. **Top Right Graph:** - This graph displays a vertically stretched curve with emphasized peaks. - The graph does not seem to match the description of \( y = \frac{1}{2}f(x) \) because it suggests a vertical stretch rather than compression. 3. **Bottom Left Graph:** - Shows a symmetrical curve centered vertically with respect to the y-axis. - The graph indicates vertical compression as expected from multiplying \( f(x) \) by \(\frac{1}{2}\). 4. **Bottom Right Graph:** - Displays a similar pattern to the top left with compressed peaks, aligning with the function \( y = \frac{1}{2}f(x) \). - The overall appearance matches the vertical compression effect described. Each graph has a circle below it, presumably for selecting which graph corresponds best to the function given. ### Educational Notes: Vertical compression occurs when a function \( f(x) \) is multiplied by a factor between 0 and 1, such as \(\frac{1}{2}\), effectively reducing the amplitude of the original function without affecting the x-values.
Transcribed Image Text:The image displays four graphs representing different functions on coordinate planes, marked by various curves intersecting along the x and y axes. ### Function: The function given is \( y = \frac{1}{2}f(x) \). ### Graph Descriptions: 1. **Top Left Graph:** - The graph presents a curve that is relatively flat, centered around the x-axis, showing compression of the original function \( f(x) \). - The curve's amplitude is reduced by a factor of \(\frac{1}{2}\), indicating vertical compression. 2. **Top Right Graph:** - This graph displays a vertically stretched curve with emphasized peaks. - The graph does not seem to match the description of \( y = \frac{1}{2}f(x) \) because it suggests a vertical stretch rather than compression. 3. **Bottom Left Graph:** - Shows a symmetrical curve centered vertically with respect to the y-axis. - The graph indicates vertical compression as expected from multiplying \( f(x) \) by \(\frac{1}{2}\). 4. **Bottom Right Graph:** - Displays a similar pattern to the top left with compressed peaks, aligning with the function \( y = \frac{1}{2}f(x) \). - The overall appearance matches the vertical compression effect described. Each graph has a circle below it, presumably for selecting which graph corresponds best to the function given. ### Educational Notes: Vertical compression occurs when a function \( f(x) \) is multiplied by a factor between 0 and 1, such as \(\frac{1}{2}\), effectively reducing the amplitude of the original function without affecting the x-values.
**Graph Analysis:** The task is to use the graph of the function \( f \) to sketch each graph. **Graph Details:** - **Axes:** The graph is plotted on a standard Cartesian coordinate system with the horizontal x-axis and the vertical y-axis. - **Key Points:** - The vertex of the graph is at the point \( (0, 5) \), indicating it is a parabola opening downwards. - The graph intersects the x-axis at the points \( (-3, 0) \) and \( (3, 0) \). - The graph passes through the points \( (-6, -4) \) and \( (6, -4) \). **Graph Shape:** The shape is symmetric around the y-axis, illustrating a parabola that opens downwards. The peak is at \( (0, 5) \), which is the highest point on the graph. The parabola intersects the x-axis at \( (-3, 0) \) and \( (3, 0) \), making these the roots or the solutions of the equation \( f(x) = 0 \). **Conclusion:** This is a standard quadratic function with a downward-facing parabola, characterized by its symmetry, a vertex at \( (0, 5) \), and roots at \( (-3, 0) \) and \( (3, 0) \). The points \( (-6, -4) \) and \( (6, -4) \) highlight the function's behavior outside of this central section.
Transcribed Image Text:**Graph Analysis:** The task is to use the graph of the function \( f \) to sketch each graph. **Graph Details:** - **Axes:** The graph is plotted on a standard Cartesian coordinate system with the horizontal x-axis and the vertical y-axis. - **Key Points:** - The vertex of the graph is at the point \( (0, 5) \), indicating it is a parabola opening downwards. - The graph intersects the x-axis at the points \( (-3, 0) \) and \( (3, 0) \). - The graph passes through the points \( (-6, -4) \) and \( (6, -4) \). **Graph Shape:** The shape is symmetric around the y-axis, illustrating a parabola that opens downwards. The peak is at \( (0, 5) \), which is the highest point on the graph. The parabola intersects the x-axis at \( (-3, 0) \) and \( (3, 0) \), making these the roots or the solutions of the equation \( f(x) = 0 \). **Conclusion:** This is a standard quadratic function with a downward-facing parabola, characterized by its symmetry, a vertex at \( (0, 5) \), and roots at \( (-3, 0) \) and \( (3, 0) \). The points \( (-6, -4) \) and \( (6, -4) \) highlight the function's behavior outside of this central section.
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