https://file.aiquickdraw.com/mathbot/file/1726441534904h7j8jckb.jpg

Looking at the graph in the image, I observe the following:

1. The graph represents a sinusoidal function, likely a cosine or sine function due to its wave-like behavior.
2. The graph appears to have a **period** of \(4\pi\) because the pattern repeats from \(-2\pi\) to \(2\pi\).
3. The **amplitude** is the difference between the maximum and minimum values divided by 2. The maximum value is 6 and the minimum value is 4, so:
   \[
   \text{Amplitude} = \frac{6 - 4}{2} = 1
   \]
4. The **midline** of the function is the average of the maximum and minimum values, which is:
   \[
   \text{Midline} = \frac{6 + 4}{2} = 5
   \]
5. Since the graph looks like a cosine function starting from the midline at \(x = 0\) and the shape corresponds to a downward peak, we are likely dealing with a cosine function with a phase shift.

Thus, a possible formula for the graph could be of the form:
\[
f(x) = A \cos(Bx + C) + D
\]
Where:
- \(A = 1\) (amplitude),
- \(B = \frac{2\pi}{\text{period}} = \frac{2\pi}{4\pi} = \frac{1}{2}\),
- \(D = 5\) (midline).

To account for the phase shift and reflection, the function is likely a cosine that has been reflected across the x-axis, so the phase shift would align the graph with cosine's negative starting behavior.

Therefore, the formula could be:
\[
f(x) = -\cos\left(\frac{1}{2}x\right) + 5
\]

Would you like more details on how we arrived at this formula? Here are some related questions:

1. How do we determine the period of a sinusoidal function?
2. What role does the amplitude play in modifying the graph of a function?
3. How do you determine whether to use a sine or cosine function based on a graph?
4. What does a phase shift mean in trigonometric functions?
5. How does reflecting a cosine function affect its formula?

**Tip**: When determining the equation of a sinusoidal function, always check the period, amplitude, and midline before looking at the phase shift and reflection.

Explore how the cosine function can be adjusted with transformations to match specific graphs. This page covers amplitude, period, and midline modifications, illustrated through the example f(x) = -cos(0.5x) + 5. Perfect for high school students learning trigonometry.

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