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Which of the following graphs represents the function () ه کرد 4 f(x) f(x) 2 2x²+2x-12

Which of the following graphs represents the function () ه کرد 4 f(x) f(x) 2 2x²+2x-12

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 \[f(x) = \frac{x^2 - 4x}{2x^2 - 7x - 15}\]? **Graph Analysis:** **Graph (1):** - This graph plots the function \( f(x) \). - The x-axis ranges from -10 to 10. - The y-axis ranges from -10 to 10. - The graph has a vertical asymptote near \( x = 5 \) and appears to intersect the y-axis at \( y = 0 \). - The curve shows a hyperbolic shape, indicating possible rational function behavior. **Graph (2):** - This graph also plots the function \( f(x) \). - The x-axis ranges from -10 to 10. - The y-axis ranges from -10 to 10. - The graph similarly shows a vertical asymptote near \( x = 5 \) and intersects the y-axis at \( y = 0 \). - There is a clear change in the slope of the curve, suggesting another rational function behavior. **Explanation:** To determine which graph correctly represents the given function, analyze the behavior and characteristics of the function \[f(x) = \frac{x^2 - 4x}{2x^2 - 7x - 15}\]. First, factorize the function: - Numerator: \( x^2 - 4x = x(x - 4) \) - Denominator: \( 2x^2 - 7x - 15 = (2x + 3)(x - 5) \) Thus, the function is: \[ \frac{x(x - 4)}{(2x + 3)(x - 5)} \] Vertical asymptotes occur where the denominator equals zero: - \(2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\) - \(x - 5 = 0 \Rightarrow x = 5\) Points of intersection with the y-axis occur when \(x = 0\), which simplifies \( f(0) = 0 \). Based on these characteristics, determine which graph matches all these features to correctly represent the given function.
Transcribed Image Text:**Question:** Which of the following graphs represents the function \[f(x) = \frac{x^2 - 4x}{2x^2 - 7x - 15}\]? **Graph Analysis:** **Graph (1):** - This graph plots the function \( f(x) \). - The x-axis ranges from -10 to 10. - The y-axis ranges from -10 to 10. - The graph has a vertical asymptote near \( x = 5 \) and appears to intersect the y-axis at \( y = 0 \). - The curve shows a hyperbolic shape, indicating possible rational function behavior. **Graph (2):** - This graph also plots the function \( f(x) \). - The x-axis ranges from -10 to 10. - The y-axis ranges from -10 to 10. - The graph similarly shows a vertical asymptote near \( x = 5 \) and intersects the y-axis at \( y = 0 \). - There is a clear change in the slope of the curve, suggesting another rational function behavior. **Explanation:** To determine which graph correctly represents the given function, analyze the behavior and characteristics of the function \[f(x) = \frac{x^2 - 4x}{2x^2 - 7x - 15}\]. First, factorize the function: - Numerator: \( x^2 - 4x = x(x - 4) \) - Denominator: \( 2x^2 - 7x - 15 = (2x + 3)(x - 5) \) Thus, the function is: \[ \frac{x(x - 4)}{(2x + 3)(x - 5)} \] Vertical asymptotes occur where the denominator equals zero: - \(2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\) - \(x - 5 = 0 \Rightarrow x = 5\) Points of intersection with the y-axis occur when \(x = 0\), which simplifies \( f(0) = 0 \). Based on these characteristics, determine which graph matches all these features to correctly represent the given function.
### Understanding Functions with Graphs The following are illustrations of various functions demonstrated through graph plots to help visualize their behavior. #### Graph 1: Function \(f(x)\) - **Axes**: The graph is formatted in a standard Cartesian coordinate system. - The x-axis ranges from approximately -8 to 8. - The y-axis similarly ranges from approximately -8 to 8. - **Behavior**: - The function \(f(x)\) exhibits an asymptotic behavior as \(x\) approaches 0 from the negative side. It tends towards negative infinity. - As \(x\) approaches 0 from the positive side, the function jumps to positive infinity. - The function appears to flatten out as \(x\) moves towards both positive and negative directions, suggesting linear behavior away from the origin. #### Graph 2: Function \(f(x)\) - **Axes**: Again, this graph is plotted on a standard Cartesian coordinate system. - The x-axis ranges from approximately -8 to 8. - The y-axis similarly ranges from approximately -8 to 8. - **Behavior**: - The function \(f(x)\) showcases a steep drop as \(x\) transitions from negative to positive values near \(x = 0\), indicating a vertical asymptote. - As \(x\) approaches 0 from the negative side, the function moves towards positive infinity. - Conversely, as \(x\) approaches 0 from the positive side, the function sharply declines towards negative infinity. - Away from the y-axis, the function shows linear behavior, denoted by the relatively straight lines extending towards infinity. ### Summary By analyzing these graphs, one can interpret how functions behave near certain values and as they extend towards infinity. This visualization forms an essential tool in understanding mathematical functions and their limits, asymptotes, and general behavior across the domain.
Transcribed Image Text:### Understanding Functions with Graphs The following are illustrations of various functions demonstrated through graph plots to help visualize their behavior. #### Graph 1: Function \(f(x)\) - **Axes**: The graph is formatted in a standard Cartesian coordinate system. - The x-axis ranges from approximately -8 to 8. - The y-axis similarly ranges from approximately -8 to 8. - **Behavior**: - The function \(f(x)\) exhibits an asymptotic behavior as \(x\) approaches 0 from the negative side. It tends towards negative infinity. - As \(x\) approaches 0 from the positive side, the function jumps to positive infinity. - The function appears to flatten out as \(x\) moves towards both positive and negative directions, suggesting linear behavior away from the origin. #### Graph 2: Function \(f(x)\) - **Axes**: Again, this graph is plotted on a standard Cartesian coordinate system. - The x-axis ranges from approximately -8 to 8. - The y-axis similarly ranges from approximately -8 to 8. - **Behavior**: - The function \(f(x)\) showcases a steep drop as \(x\) transitions from negative to positive values near \(x = 0\), indicating a vertical asymptote. - As \(x\) approaches 0 from the negative side, the function moves towards positive infinity. - Conversely, as \(x\) approaches 0 from the positive side, the function sharply declines towards negative infinity. - Away from the y-axis, the function shows linear behavior, denoted by the relatively straight lines extending towards infinity. ### Summary By analyzing these graphs, one can interpret how functions behave near certain values and as they extend towards infinity. This visualization forms an essential tool in understanding mathematical functions and their limits, asymptotes, and general behavior across the domain.
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