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?

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!