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1) y = 4cos 30 2) y = 3sin 0 2

1) y = 4cos 30 2) y = 3sin 0 2

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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**Graphing Trigonometric Functions** **Instructions:** Graph each function using degrees. 1. **Function 1: \( y = 4\cos 3\theta \)** - **Graph Details:** - The graph is plotted on a coordinate plane with the x-axis labeled in degrees, ranging from 0° to 360°. - The y-axis ranges from -5 to 5. - The function \( y = 4\cos 3\theta \) will have a period of 120° due to the factor of 3 in the argument of the cosine, which compresses the graph horizontally. - The amplitude is 4, reflected in the range of the function values from -4 to 4. 2. **Function 2: \( y = 3\sin \frac{\theta}{2} \)** - **Graph Details:** - The graph is on a coordinate plane with the x-axis labeled from 0° to 1080°, indicating a longer period due to the factor of \(\frac{1}{2}\). - The y-axis also ranges from -5 to 5. - The period, in this case, is extended to 720° because the argument of the sine function, \(\frac{\theta}{2}\), stretches the graph horizontally. - The amplitude is 3, so the graph will oscillate between -3 and 3. 3. **Function 3: \( y = 4\sin \frac{\theta}{4} \)** - **Graph Details:** - This graph is displayed with the x-axis ranging from 0° to 2160°. - The y-axis ranges similarly from -5 to 5. - The function \( y = 4\sin \frac{\theta}{4} \) has a period extended to 1440° due to the \(\frac{1}{4}\) factor. - The amplitude is 4, with values oscillating between -4 and 4. **Analysis:** For each function, observe how the coefficients and arguments affect the graph's amplitude and period. The cosine function starts at its maximum value, while the sine functions start at zero. Adjusting the angle modifies the speed of oscillations, affecting the wave's period. The amplitude scaling factor dictates the peak values of the wave. Understanding these transformations is key
Transcribed Image Text:**Graphing Trigonometric Functions** **Instructions:** Graph each function using degrees. 1. **Function 1: \( y = 4\cos 3\theta \)** - **Graph Details:** - The graph is plotted on a coordinate plane with the x-axis labeled in degrees, ranging from 0° to 360°. - The y-axis ranges from -5 to 5. - The function \( y = 4\cos 3\theta \) will have a period of 120° due to the factor of 3 in the argument of the cosine, which compresses the graph horizontally. - The amplitude is 4, reflected in the range of the function values from -4 to 4. 2. **Function 2: \( y = 3\sin \frac{\theta}{2} \)** - **Graph Details:** - The graph is on a coordinate plane with the x-axis labeled from 0° to 1080°, indicating a longer period due to the factor of \(\frac{1}{2}\). - The y-axis also ranges from -5 to 5. - The period, in this case, is extended to 720° because the argument of the sine function, \(\frac{\theta}{2}\), stretches the graph horizontally. - The amplitude is 3, so the graph will oscillate between -3 and 3. 3. **Function 3: \( y = 4\sin \frac{\theta}{4} \)** - **Graph Details:** - This graph is displayed with the x-axis ranging from 0° to 2160°. - The y-axis ranges similarly from -5 to 5. - The function \( y = 4\sin \frac{\theta}{4} \) has a period extended to 1440° due to the \(\frac{1}{4}\) factor. - The amplitude is 4, with values oscillating between -4 and 4. **Analysis:** For each function, observe how the coefficients and arguments affect the graph's amplitude and period. The cosine function starts at its maximum value, while the sine functions start at zero. Adjusting the angle modifies the speed of oscillations, affecting the wave's period. The amplitude scaling factor dictates the peak values of the wave. Understanding these transformations is key
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