Sketch the graph of the function y=6cos(angle teta - pii/4)-3

To sketch the graph of the function \( y = 6\cos\left(\theta - \frac{\pi}{4}\right) - 3 \), let's break down its components step by step:

### 1. Basic Cosine Graph:
The basic cosine function \( y = \cos(\theta) \) has:
- Amplitude: 1
- Period: \( 2\pi \)
- No horizontal or vertical shifts.

### 2. Adjustments to the Basic Cosine Graph:

#### a. Amplitude:
In this case, the amplitude is multiplied by 6, so the graph will stretch vertically. The maximum value is \( 6 \) and the minimum is \( -6 \). 

#### b. Phase Shift:
The phase shift is \( \frac{\pi}{4} \) to the right due to the term \( \theta - \frac{\pi}{4} \). This means the graph is shifted \( \frac{\pi}{4} \) units to the right along the \( \theta \)-axis.

#### c. Vertical Shift:
The function has a vertical shift of \( -3 \), meaning the entire graph will be shifted down by 3 units.

### 3. Characteristics of the Graph:
- **Amplitude**: 6
- **Period**: The period remains \( 2\pi \), as the coefficient of \( \theta \) inside the cosine function is still 1.
- **Phase Shift**: \( \frac{\pi}{4} \) to the right.
- **Vertical Shift**: Downward by 3 units.

### 4. Important Points:
- **Maximum**: \( 6 - 3 = 3 \) occurs when \( \cos(\theta - \frac{\pi}{4}) = 1 \).
- **Minimum**: \( -6 - 3 = -9 \) occurs when \( \cos(\theta - \frac{\pi}{4}) = -1 \).
- **Midline**: The horizontal midline is \( y = -3 \).

### Sketch Instructions:
1. Start by drawing the midline at \( y = -3 \).
2. Mark the maximum point at \( y = 3 \) and the minimum point at \( y = -9 \).
3. Draw the cosine wave starting from \( \theta = \frac{\pi}{4} \), with its period being \( 2\pi \).
4. Make sure the graph completes one cycle from \( \theta = \frac{\pi}{4} \) to \( \theta = \frac{\pi}{4} + 2\pi \).

Would you like me to generate the graph, or do you need more details on any specific step?

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### Related Questions:
1. How do you find the period of any trigonometric function?
2. How does a vertical shift affect the graph of a trigonometric function?
3. What happens to the graph if we change the amplitude of the cosine function?
4. How can we determine the phase shift for any trigonometric function?
5. What is the relationship between the maximum and minimum values of a cosine function and its amplitude?

**Tip**: To sketch a trigonometric graph effectively, always identify the amplitude, period, phase shift, and vertical shift first.

Learn how to sketch the graph of the function y = 6cos(θ - π/4) - 3 with a step-by-step guide. This problem involves key trigonometric concepts such as amplitude, period, phase shift, and vertical shift. Suitable for students in Grades 10-12.

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