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Describe the domain and range

Describe the domain and range

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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Describe the domain and range
**Problem Description:** **45. \( y = \frac{-1}{3}x^3 \)** **Graph Description:** The provided graph is a Cartesian coordinate system with both x and y-axes ranging from -4 to 4. Each axis has a scale marked in unit intervals. The graph plots the cubic function \( y = \frac{-1}{3}x^3 \). ### Explanation of the Graph: The function \( y = \frac{-1}{3}x^3 \) is a cubic equation, which is characterized by its general S-shaped curve. Here, the coefficient of \( x^3 \) is negative (\( -1/3 \)), which implies that the function is reflected over the x-axis compared to the standard \( y = x^3 \) graph. It scales the graph, making it less steep due to the factor of \( 1/3 \). **Key Observations:** - The curve passes through the origin (0,0). - As \( x \) increases in the positive direction, the value of \( y \) decreases, creating a downward slope. - As \( x \) decreases in the negative direction, the value of \( y \) increases, creating an upward slope. - There are no maxima or minima for this particular cubic function, as the direction of the curve continuously changes. The general behavior of the graph indicates an inflection point at the origin where the curvature changes. This type of cubic function graph is a reflection of the typical cubic function \( y = x^3 \) but scaled down and flipped upside down because of the negative fraction.
Transcribed Image Text:**Problem Description:** **45. \( y = \frac{-1}{3}x^3 \)** **Graph Description:** The provided graph is a Cartesian coordinate system with both x and y-axes ranging from -4 to 4. Each axis has a scale marked in unit intervals. The graph plots the cubic function \( y = \frac{-1}{3}x^3 \). ### Explanation of the Graph: The function \( y = \frac{-1}{3}x^3 \) is a cubic equation, which is characterized by its general S-shaped curve. Here, the coefficient of \( x^3 \) is negative (\( -1/3 \)), which implies that the function is reflected over the x-axis compared to the standard \( y = x^3 \) graph. It scales the graph, making it less steep due to the factor of \( 1/3 \). **Key Observations:** - The curve passes through the origin (0,0). - As \( x \) increases in the positive direction, the value of \( y \) decreases, creating a downward slope. - As \( x \) decreases in the negative direction, the value of \( y \) increases, creating an upward slope. - There are no maxima or minima for this particular cubic function, as the direction of the curve continuously changes. The general behavior of the graph indicates an inflection point at the origin where the curvature changes. This type of cubic function graph is a reflection of the typical cubic function \( y = x^3 \) but scaled down and flipped upside down because of the negative fraction.
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