To find the formula for the function $f(x)$, we will assume that the function is of a sinusoidal form, such as a sine or cosine function, because it intersects a midline and has a maximum point. The general form of a sinusoidal function is:

$f(x) = A \sin(B(x - C)) + D \quad \text{or} \quad f(x) = A \cos(B(x - C)) + D$

where:

• $A$ is the amplitude,
• $B$ is related to the period,
• $C$ is the phase shift, and
• $D$ is the vertical shift (midline).

Step 1: Determine the midline

The function intersects its midline at the point $(3, -6.5)$. This means the midline is at $y = -6.5$, so $D = -6.5$. Hence, the equation becomes:

$f(x) = A \sin(B(x - C)) - 6.5 \quad \text{or} \quad f(x) = A \cos(B(x - C)) - 6.5$

Step 2: Determine the amplitude $A$

The maximum point of the function is given as $(4, -2)$. This is the highest point of the sinusoidal curve. The distance between the maximum and the midline gives the amplitude.

• The midline is at $y = -6.5$,
• The maximum value is at $y = -2$.

The amplitude is the vertical distance between these two values:

$\text{Amplitude} = -2 - (-6.5) = 4.5$

Thus, $A = 4.5$.

Step 3: Determine the phase shift $C$

Since the function intersects the midline at $x = 3$, and we are assuming the function is either sine or cosine, we need to analyze how the graph behaves. For simplicity, let's assume a cosine function because the cosine function starts at its maximum when $x = 0$ (and a phase shift allows us to adjust that).

For a cosine function with a phase shift, the formula would be:

$f(x) = A \cos(B(x - C)) - 6.5$

From the given maximum point $(4, -2)$, and recognizing that this should correspond to a peak (where cosine is at its maximum value), we use the fact that cosine reaches its maximum at $x = C$.

Thus, $C = 4$.

Step 4: Determine the value of $B$

We can use the period of the function to determine $B$. However, we are not given the full period, so we assume the function has a standard period of $2\pi$. In this case, $B$ will be equal to 1 since it has no stretching or compression in the horizontal direction.

Step 5: Final formula for $f(x)$

Now, we can write the final expression for $f(x)$:

$f(x) = 4.5 \cos(x - 4) - 6.5$

Final Answer:

$f(x) = 4.5 \cos(x - 4) - 6.5$

Would you like to discuss the steps in more detail or need any clarification?

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Here are some related questions to expand on this topic:

1. How can you find the period of a sinusoidal function from its graph?
2. What are the differences between sine and cosine functions in terms of their graphs?
3. How do phase shifts affect the graph of a sinusoidal function?
4. How do amplitude and midline shifts relate to vertical transformations of the graph?
5. What are some methods to determine the amplitude and phase shift from a given graph?

Tip: If you want to confirm your function's correctness, plot it and check if it aligns with the provided graph's key features like the midline, maximum, and intersection points.