Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Writing an Equation for a Transformed Logarithm **Problem Statement:** Write an equation for the transformed logarithm shown below that passes through the points \((0,0)\) and \((2,-3)\). **Graph Description:** The graph is a plot on a coordinate grid. The x-axis ranges from -5 to 5, and the y-axis ranges from -5 to 5. The graph features a blue curve representing a logarithmic function, which passes through the origin \((0,0)\) and the point \((2,-3)\). - The curve starts at the point \((0,0)\) and descends towards the point \((2,-3)\), showing a transformed logarithmic behavior. - The red dashed line at \(x = -1\) likely indicates a vertical asymptote, typical of logarithmic functions, if shifted. **Interactive Component:** There is an input box labeled \(f(x) =\) where users are prompted to "Enter an algebraic expression." ### Additional Notes: - Ensure the entered equation reflects the transformations (translations and/or dilations) that align the curve with the given points. - The equation should be in logarithmic form, likely involving parameters that cause the curve to pass through the specified points. - Viewers can get assistance via a linked video resource. **Common Transformations for Logarithms:** - Horizontal shifts: \(f(x) = \log_b(x - h)\) - Vertical shifts: \(f(x) = \log_b(x) + k\) - Vertical stretching/compression: \(f(x) = a \cdot \log_b(x)\) Based on the passage through \((0,0)\) and \((2,-3)\), determine the specific transformations applied.
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Transcribed Image Text:### Writing an Equation for a Transformed Logarithm **Problem Statement:** Write an equation for the transformed logarithm shown below that passes through the points \((0,0)\) and \((2,-3)\). **Graph Description:** The graph is a plot on a coordinate grid. The x-axis ranges from -5 to 5, and the y-axis ranges from -5 to 5. The graph features a blue curve representing a logarithmic function, which passes through the origin \((0,0)\) and the point \((2,-3)\). - The curve starts at the point \((0,0)\) and descends towards the point \((2,-3)\), showing a transformed logarithmic behavior. - The red dashed line at \(x = -1\) likely indicates a vertical asymptote, typical of logarithmic functions, if shifted. **Interactive Component:** There is an input box labeled \(f(x) =\) where users are prompted to "Enter an algebraic expression." ### Additional Notes: - Ensure the entered equation reflects the transformations (translations and/or dilations) that align the curve with the given points. - The equation should be in logarithmic form, likely involving parameters that cause the curve to pass through the specified points. - Viewers can get assistance via a linked video resource. **Common Transformations for Logarithms:** - Horizontal shifts: \(f(x) = \log_b(x - h)\) - Vertical shifts: \(f(x) = \log_b(x) + k\) - Vertical stretching/compression: \(f(x) = a \cdot \log_b(x)\) Based on the passage through \((0,0)\) and \((2,-3)\), determine the specific transformations applied.
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