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Find the equation of the line tangent to the graph of f(x) = -3 tan (x) - 5+3 at x = -4. Submit your answer by dragging the red point such that the drawn tangent line (shown in red) matches the equation for it Provide your answer below: (-π/4, 2.712) -2T 10- 50 0 --5- 10- (117/12/3) 2T

Find the equation of the line tangent to the graph of f(x) = -3 tan (x) - 5+3 at x = -4. Submit your answer by dragging the red point such that the drawn tangent line (shown in red) matches the equation for it Provide your answer below: (-π/4, 2.712) -2T 10- 50 0 --5- 10- (117/12/3) 2T

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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### Educational Resource: Tangent Line to a Function #### Problem Statement Find the equation of the line tangent to the graph of \( f(x) = -3 \tan(x) - 5 + \frac{3\pi}{2} \) at \( x = -\frac{\pi}{4} \). Submit your answer by dragging the red point such that the drawn tangent line (shown in red) matches the equation for it. Provide your answer below: #### Detailed Explanation of the Graph The graph includes several components: 1. **Graph of the Function \( f(x) \)**: - The graph of \( f(x) = -3\tan(x) - 5 + \frac{3\pi}{2} \) is shown in blue. - The function has vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer. These vertical lines are characteristic of the tangent function. 2. **Coordinate Grid**: - The x-axis ranges from \(-2\pi\) to \(2\pi\), allowing for clear visualization of periodic behavior. - The y-axis ranges from \(-10\) to \(10\), providing ample room to see function values and tangent line behavior at specified points. 3. **Tangent Line**: - A red line represents the tangent to the function at \( x = -\frac{\pi}{4} \). - There are two points marked explicitly: - The point of tangency: \( (-\frac{\pi}{4}, 2.712) \) - Another point on the tangent line: \( \left( \frac{11\pi}{12}, -3 \right) \) ### Instructional Guidance As a student, you need to: 1. **Understand the Function**: - Recognize that the function \( f(x) = -3\tan(x) - 5 + \frac{3\pi}{2} \) is a transformation of the basic tangent function \( \tan(x) \). - The transformations include a vertical stretch by a factor of 3, a vertical translation downward by 5 units, and an additional vertical shift by \( \frac{3\pi}{2} \). 2. **Finding the Tangent Line**: - Determine the derivative of the function to find the slope
Transcribed Image Text:### Educational Resource: Tangent Line to a Function #### Problem Statement Find the equation of the line tangent to the graph of \( f(x) = -3 \tan(x) - 5 + \frac{3\pi}{2} \) at \( x = -\frac{\pi}{4} \). Submit your answer by dragging the red point such that the drawn tangent line (shown in red) matches the equation for it. Provide your answer below: #### Detailed Explanation of the Graph The graph includes several components: 1. **Graph of the Function \( f(x) \)**: - The graph of \( f(x) = -3\tan(x) - 5 + \frac{3\pi}{2} \) is shown in blue. - The function has vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer. These vertical lines are characteristic of the tangent function. 2. **Coordinate Grid**: - The x-axis ranges from \(-2\pi\) to \(2\pi\), allowing for clear visualization of periodic behavior. - The y-axis ranges from \(-10\) to \(10\), providing ample room to see function values and tangent line behavior at specified points. 3. **Tangent Line**: - A red line represents the tangent to the function at \( x = -\frac{\pi}{4} \). - There are two points marked explicitly: - The point of tangency: \( (-\frac{\pi}{4}, 2.712) \) - Another point on the tangent line: \( \left( \frac{11\pi}{12}, -3 \right) \) ### Instructional Guidance As a student, you need to: 1. **Understand the Function**: - Recognize that the function \( f(x) = -3\tan(x) - 5 + \frac{3\pi}{2} \) is a transformation of the basic tangent function \( \tan(x) \). - The transformations include a vertical stretch by a factor of 3, a vertical translation downward by 5 units, and an additional vertical shift by \( \frac{3\pi}{2} \). 2. **Finding the Tangent Line**: - Determine the derivative of the function to find the slope
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