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A portion of the parabola y=6x−x2 is shown below.  Write down the definite integral which will give the exact area under the graph and above the x-axis between  x = 0  and  x = 6

A portion of the parabola y=6x−x2 is shown below.  Write down the definite integral which will give the exact area under the graph and above the x-axis between  x = 0  and  x = 6

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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A portion of the parabola y=6x−x2 is shown below.  Write down the definite integral which will give the exact area under the graph and above the x-axis between  x = 0  and  x = 6

The graph presented is a parabola, which represents a quadratic function. The graph is plotted on a standard Cartesian coordinate system with both x and y axes labeled. ### Key Features of the Graph: - **Shape**: The graph is an upward-facing parabola. This indicates a standard quadratic relationship. - **Vertex**: The highest point of the parabola is the vertex, located at (3, 9). This suggests a maximum point for this particular quadratic function. - **Axis of Symmetry**: The graph is symmetric around the vertical line \(x = 3\), which passes through the vertex. - **Intercepts**: - **Y-Intercept**: The graph intersects the y-axis at (0, 0). - **X-Intercepts**: The parabola also crosses the x-axis at approximately (0, 0) and (6, 0). - **Grid**: The background grid helps in identifying the coordinates more easily by giving a clearer view of the relative spacing. ### Observations: This parabola represents a quadratic equation with a negative leading coefficient, which can be deduced from the downward opening of the graph. The vertex form of the equation might look something like \(f(x) = a(x - 3)^2 + 9\) where \(a\) is a negative value determining the parabola's opening direction and width. This type of graph is often used to illustrate concepts in algebra, such as the properties of quadratic functions, symmetry in functions, and their applications in real-world scenarios like projectile motion.
Transcribed Image Text:The graph presented is a parabola, which represents a quadratic function. The graph is plotted on a standard Cartesian coordinate system with both x and y axes labeled. ### Key Features of the Graph: - **Shape**: The graph is an upward-facing parabola. This indicates a standard quadratic relationship. - **Vertex**: The highest point of the parabola is the vertex, located at (3, 9). This suggests a maximum point for this particular quadratic function. - **Axis of Symmetry**: The graph is symmetric around the vertical line \(x = 3\), which passes through the vertex. - **Intercepts**: - **Y-Intercept**: The graph intersects the y-axis at (0, 0). - **X-Intercepts**: The parabola also crosses the x-axis at approximately (0, 0) and (6, 0). - **Grid**: The background grid helps in identifying the coordinates more easily by giving a clearer view of the relative spacing. ### Observations: This parabola represents a quadratic equation with a negative leading coefficient, which can be deduced from the downward opening of the graph. The vertex form of the equation might look something like \(f(x) = a(x - 3)^2 + 9\) where \(a\) is a negative value determining the parabola's opening direction and width. This type of graph is often used to illustrate concepts in algebra, such as the properties of quadratic functions, symmetry in functions, and their applications in real-world scenarios like projectile motion.
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