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Sketch a graph of the function f(x) = - 2 sin(a)-1

Sketch a graph of the function f(x) = - 2 sin(a)-1

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 7** **Problem Statement:** Sketch a graph of the function \( f(x) = -2\sin\left(\frac{\pi}{2}x\right) - 1 \). **Graph Explanation:** The graph provided is a Cartesian coordinate system with both the x-axis and y-axis ranging from -9 to 9. The axes are divided into grid lines of one unit each for clear visualization: - The x-axis is labeled from -9 to 9. - The y-axis is labeled from -6 to 6. There is an inline input at the bottom left corner with options for "Clear All" and a drawing palette to sketch the graph digitally. **Instructions:** To graph \( f(x) \): 1. **Amplitude:** The amplitude of the function is 2. 2. **Period:** The period of the function \( \sin\left(\frac{\pi}{2}x\right) \) is \( \frac{4}{\pi} \). This can be calculated as \( \frac{2\pi}{\frac{\pi}{2}} = \frac{4}{1} \). 3. **Vertical Shift:** The vertical shift is downward by 1 unit. 4. **Reflection:** Since the sine function is multiplied by -2, it is reflected over the x-axis and stretched vertically by a factor of 2. To plot the function correctly, use points at intervals that align with the period of the sine function. For a complete cycle, include key points such as the maximum, minimum, and intersections with the midline (shifted down by 1 unit). Ensure to draw periodic oscillations that reflect the characteristics described above.
Transcribed Image Text:**Question 7** **Problem Statement:** Sketch a graph of the function \( f(x) = -2\sin\left(\frac{\pi}{2}x\right) - 1 \). **Graph Explanation:** The graph provided is a Cartesian coordinate system with both the x-axis and y-axis ranging from -9 to 9. The axes are divided into grid lines of one unit each for clear visualization: - The x-axis is labeled from -9 to 9. - The y-axis is labeled from -6 to 6. There is an inline input at the bottom left corner with options for "Clear All" and a drawing palette to sketch the graph digitally. **Instructions:** To graph \( f(x) \): 1. **Amplitude:** The amplitude of the function is 2. 2. **Period:** The period of the function \( \sin\left(\frac{\pi}{2}x\right) \) is \( \frac{4}{\pi} \). This can be calculated as \( \frac{2\pi}{\frac{\pi}{2}} = \frac{4}{1} \). 3. **Vertical Shift:** The vertical shift is downward by 1 unit. 4. **Reflection:** Since the sine function is multiplied by -2, it is reflected over the x-axis and stretched vertically by a factor of 2. To plot the function correctly, use points at intervals that align with the period of the sine function. For a complete cycle, include key points such as the maximum, minimum, and intersections with the midline (shifted down by 1 unit). Ensure to draw periodic oscillations that reflect the characteristics described above.
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