12) y = 3 cosx + X
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Problem 12: Graphing a Cosine Function**
**Function:**
\[ y = 3 \cos \left( x + \frac{\pi}{4} \right) \]
**Explanation:**
- The given function is a cosine function where:
- Amplitude: The amplitude is \(3\), indicating that the graph will stretch vertically to reach a maximum of \(3\) and a minimum of \(-3\).
- Phase Shift: The function includes a phase shift of \(-\frac{\pi}{4}\) to the left, meaning the standard cosine wave will shift \( \frac{\pi}{4} \) units to the left on the x-axis.
- No vertical shift is present; therefore, the midline is on the x-axis.
**Cartesian Plane:**
- The graph is on a standard cartesian plane.
- The x-axis is labeled as \(x\), and the y-axis is labeled as \(y\).
- Both axes are marked with regular intervals, but specific values are not indicated.
- The dotted grid indicates where the function's values would fall relative to this grid.
**Graphing Tips:**
1. **Identify Key Points:**
- Start from \(-\frac{\pi}{4}\) on the x-axis due to the phase shift.
- Calculate key points where \(x = 0, \pi/2, \pi, 3\pi/2, 2\pi\) by considering the shift.
2. **Amplitude:**
- Ensure peaks reach \(3\) and troughs reach \(-3\).
3. **Sketch the Curve:**
- The curve should repeat every \(2\pi\) along the x-axis.
By plotting the crucial points and joining them in a smooth, wave-like pattern, you can visualize the full behavior of the cosine function on this graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9e96de-8215-4532-a7d0-6f1fdf87f241%2F6bc86070-6181-46d4-bfb5-8424062a328d%2Fctrepij_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 12: Graphing a Cosine Function**
**Function:**
\[ y = 3 \cos \left( x + \frac{\pi}{4} \right) \]
**Explanation:**
- The given function is a cosine function where:
- Amplitude: The amplitude is \(3\), indicating that the graph will stretch vertically to reach a maximum of \(3\) and a minimum of \(-3\).
- Phase Shift: The function includes a phase shift of \(-\frac{\pi}{4}\) to the left, meaning the standard cosine wave will shift \( \frac{\pi}{4} \) units to the left on the x-axis.
- No vertical shift is present; therefore, the midline is on the x-axis.
**Cartesian Plane:**
- The graph is on a standard cartesian plane.
- The x-axis is labeled as \(x\), and the y-axis is labeled as \(y\).
- Both axes are marked with regular intervals, but specific values are not indicated.
- The dotted grid indicates where the function's values would fall relative to this grid.
**Graphing Tips:**
1. **Identify Key Points:**
- Start from \(-\frac{\pi}{4}\) on the x-axis due to the phase shift.
- Calculate key points where \(x = 0, \pi/2, \pi, 3\pi/2, 2\pi\) by considering the shift.
2. **Amplitude:**
- Ensure peaks reach \(3\) and troughs reach \(-3\).
3. **Sketch the Curve:**
- The curve should repeat every \(2\pi\) along the x-axis.
By plotting the crucial points and joining them in a smooth, wave-like pattern, you can visualize the full behavior of the cosine function on this graph.
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