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Sketch two periods of the graph for the following function. f(x) = 6 sec(5x) Identify the stretching factor and period. Identify the asymptotes in the displayed domain of the graph you selected above. (Enter your answers as a comma-separated list of equations

Sketch two periods of the graph for the following function. f(x) = 6 sec(5x) Identify the stretching factor and period. Identify the asymptotes in the displayed domain of the graph you selected above. (Enter your answers as a comma-separated list of equations

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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Sketch two periods of the graph for the following function.

f(x) = 6 sec(5x)
Identify the stretching factor and period.
Identify the asymptotes in the displayed domain of the graph you selected above. (Enter your answers as a comma-separated list of equations.)
The image features four graphs of trigonometric functions, plotted on a grid with both x and y axes marked. Each graph is within its own quadrant, depicting variations of the tangent function with different transformations. ### Graph Details: 1. **Top Left Graph:** - **Function:** \(y = \tan(x)\) - **Characteristic:** The graph exhibits typical properties of the tangent function with vertical asymptotes at \(x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}\) and repeats every \(\pi\) units. - **Behavior:** The function increases from negative to positive infinity as it approaches each asymptote. 2. **Top Right Graph:** - **Function:** A vertically flipped version of the tangent function - **Characteristic:** Vertical asymptotes are present at similar intervals, seen in the transformation of the function. - **Behavior:** The graph decreases from positive to negative infinity approaching each asymptote, indicating a reflection over the x-axis. 3. **Bottom Left Graph:** - **Function:** Another variation of the tangent function with vertical stretching - **Characteristic:** Similar vertical asymptotes are present; the graph seems more compressed horizontally. - **Behavior:** The ascending and descending curves show a steeper slope than the standard tangent function, suggesting a transformation that affects the period or amplitude. 4. **Bottom Right Graph:** - **Function:** A horizontally compressed version of the tangent function - **Characteristic:** The asymptotic behavior occurs more frequently over the same span, indicating a horizontal compression. - **Behavior:** The curve retains the general shape of a tangent function but occurs more frequently within the same horizontal distance. ### Grid and Axes: - Each graph is plotted on a grid with both axes ranging from \(-40\) to \(40\) for the y-axis and approximately \(-\frac{2\pi}{5}\) to \(\frac{2\pi}{5}\) for the x-axis. - The grid lines offer a clear reference for understanding the periodic nature of the functions and help in identifying the points where the tangent approaches its asymptotic limits.
Transcribed Image Text:The image features four graphs of trigonometric functions, plotted on a grid with both x and y axes marked. Each graph is within its own quadrant, depicting variations of the tangent function with different transformations. ### Graph Details: 1. **Top Left Graph:** - **Function:** \(y = \tan(x)\) - **Characteristic:** The graph exhibits typical properties of the tangent function with vertical asymptotes at \(x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}\) and repeats every \(\pi\) units. - **Behavior:** The function increases from negative to positive infinity as it approaches each asymptote. 2. **Top Right Graph:** - **Function:** A vertically flipped version of the tangent function - **Characteristic:** Vertical asymptotes are present at similar intervals, seen in the transformation of the function. - **Behavior:** The graph decreases from positive to negative infinity approaching each asymptote, indicating a reflection over the x-axis. 3. **Bottom Left Graph:** - **Function:** Another variation of the tangent function with vertical stretching - **Characteristic:** Similar vertical asymptotes are present; the graph seems more compressed horizontally. - **Behavior:** The ascending and descending curves show a steeper slope than the standard tangent function, suggesting a transformation that affects the period or amplitude. 4. **Bottom Right Graph:** - **Function:** A horizontally compressed version of the tangent function - **Characteristic:** The asymptotic behavior occurs more frequently over the same span, indicating a horizontal compression. - **Behavior:** The curve retains the general shape of a tangent function but occurs more frequently within the same horizontal distance. ### Grid and Axes: - Each graph is plotted on a grid with both axes ranging from \(-40\) to \(40\) for the y-axis and approximately \(-\frac{2\pi}{5}\) to \(\frac{2\pi}{5}\) for the x-axis. - The grid lines offer a clear reference for understanding the periodic nature of the functions and help in identifying the points where the tangent approaches its asymptotic limits.
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