Problem 1 Answer the following questions and graph the function $$ y=2 \cos (x-\pi) $$ - Amplitude: $$ \qquad $$ - Period: $$ \qquad $$ and Xscale: $$ \qquad $$ - Vertical Shift: $$ \qquad $$ Horizontal Shift $$ \qquad $$ - Starting point $$ \qquad $$ and Ending Point $$ \qquad $$

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Certainly! Let's analyze and graph the function $$ y = 2 \cos (x - \pi) $$ by answering each of the provided questions.

### Function Analysis

The given function is in the form:
$$ y = A \cos(B(x - C)) + D $$
where:
- $$ A $$ = Amplitude
- $$ B $$ affects the period ($$ \text{Period} = \frac{2\pi}{B} $$)
- $$ C $$ = Horizontal Shift
- $$ D $$ = Vertical Shift

For $$ y = 2 \cos (x - \pi) $$:
- $$ A = 2 $$
- $$ B = 1 $$
- $$ C = \pi $$
- $$ D = 0 $$

### Answers

- **Amplitude:** $$ \boxed{2} $$

- **Period:** $$ \boxed{2\pi} $$  
  **X-scale:** $$ \boxed{1} $$  
  (Since $$ B = 1 $$, there's no horizontal scaling applied.)

- **Vertical Shift:** $$ \boxed{0} $$  
  **Horizontal Shift:** $$ \boxed{\pi} $$  
  (The graph is shifted $$ \pi $$ units to the right.)

- **Starting Point:** $$ \boxed{\pi} $$  
  **Ending Point:** $$ \boxed{3\pi} $$  
  (One complete cycle starts at $$ x = \pi $$ and ends at $$ x = 3\pi $$.)

### Graph of $$ y = 2 \cos (x - \pi) $$

Below is a sketch of the graph over one period:

```
y
|
2 |          *                   *
  |        *   *               *   *
  |      *       *           *       *
0 |----*-----------*-------*-----------*---- x
  |      *       *           *       *
-2|        *   *               *   *
  |          *                   *
|
```

**Key Features:**
- **Amplitude:** The graph oscillates between $$ y = 2 $$ and $$ y = -2 $$.
- **Period:** One complete cycle from $$ x = \pi $$ to $$ x = 3\pi $$.
- **Horizontal Shift:** The standard cosine graph $$ y = \cos(x) $$ is shifted $$ \pi $$ units to the right.
- **Vertical Shift:** There is no vertical shift; the center line is at $$ y = 0 $$.
- **Starting Point:** The maximum point starts at $$ x = \pi $$, $$ y = 2 $$.
- **Ending Point:** The graph returns to the maximum point at $$ x = 3\pi $$, $$ y = 2 $$.

### Summary

| Feature           | Value        |
|-------------------|--------------|
| **Amplitude**     | 2            |
| **Period**        | $$ 2\pi $$    |
| **X-scale**       | 1            |
| **Vertical Shift**| 0            |
| **Horizontal Shift**| $$ \pi $$    |
| **Starting Point**| $$ \pi $$    |
| **Ending Point**  | $$ 3\pi $$    |

Feel free to plot these points and features on a coordinate plane to visualize the graph accurately!
- **Amplitude:** 2 - **Period:** $$ 2\pi $$ - **X-scale:** 1 - **Vertical Shift:** 0 - **Horizontal Shift:** $$ \pi $$ - **Starting Point:** $$ \pi $$ - **Ending Point:** $$ 3\pi $$

Problem 1 Answer the following questions and graph the function \( y=2 \cos (x-\pi) \) - Amplitude: \( \qquad \) - Period: \( \qquad \) and Xscale: \( \qquad \) - Vertical Shift: \( \qquad \) Horizontal Shift \( \qquad \) - Starting point \( \qquad \) and Ending Point \( \qquad \)

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