https://file.aiquickdraw.com/mathbot/file/1728526725897kasplob9.jpg

The graph provided appears to resemble a **cosine function** (or sine function) that has been transformed. Let's analyze the graph to answer the given questions:

### 1. **Period of the Function**:
The period of a function is the length it takes for the function to complete one full cycle. In this case, the graph repeats its pattern between \(x = -4\) and \(x = 4\), meaning the period is \(8\) units.

### 2. **Positive Vertical Stretch Factor**:
The vertical stretch factor affects the amplitude (the maximum distance from the centerline to the peak). The graph oscillates between 5 and -5, giving an amplitude of 5. Since this is a vertical stretch of the standard cosine function, the stretch factor is \(5\).

### 3. **Vertical Shift**:
The graph appears to be centered on the y-axis (at \(y = 0\)). This means there is no vertical shift, so the vertical shift is \(0\).

### 4. **Equation of the Function**:
Given that the graph looks like a cosine function that repeats every 8 units and has an amplitude of 5 with no vertical shift, the equation can be written as:

\[
f(x) = 5\cos\left(\frac{\pi}{4}x\right)
\]

Where:
- \(5\) is the amplitude.
- \(\frac{\pi}{4}\) adjusts the period to 8, since the standard period for cosine is \(2\pi\), and dividing \(2\pi\) by the factor \(\frac{\pi}{4}\) gives a period of 8.

### Summary of Answers:
1. **Period**: 8
2. **Positive Vertical Stretch Factor**: 5
3. **Vertical Shift**: 0
4. **Equation**: \(f(x) = 5\cos\left(\frac{\pi}{4}x\right)\)

Would you like any more details or further explanation on any part of this?

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Here are 5 related questions for deeper understanding:
1. How does changing the amplitude affect the graph of a trigonometric function?
2. What happens to the period of the graph if the coefficient of \(x\) inside the cosine function is increased?
3. Can you explain the difference between sine and cosine graphs in terms of their starting points?
4. How would a horizontal shift affect the equation of the cosine function?
5. What would happen to the graph if a negative vertical shift was applied?

**Tip**: Always check for transformations like shifts, stretches, or compressions when analyzing trigonometric graphs!

Give the period, vertical stretch factor, vertical shift, and the equation for the function whose graph is shown.

Learn how to find the period, amplitude, and vertical shift of a cosine function from its graph. This guide walks through analyzing the graph and writing the equation based on key transformations, suitable for Grades 10-12.

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