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The curve above is the graph of a sinusoidal function. It goes through the point (0, 3) and (8, 3). Find a sinusoidal function that matches the given graph. If needed, you can enter =3.1416... as 'pi' in your answer, otherwise use at least 3 decimal digits.f(x)=

To find a sinusoidal function that matches the given graph, we need to determine the amplitude, period, phase shift, and vertical shift of the sinusoidal function. 

The general form of a sinusoidal function is:

\[
f(x) = A \sin(B(x - C)) + D
\]

Where:
- \(A\) is the amplitude (the peak value),
- \(B\) relates to the period \(T\) by the formula \(B = \frac{2\pi}{T}\),
- \(C\) is the phase shift (horizontal shift),
- \(D\) is the vertical shift.

### Step 1: Determine the amplitude \(A\)
The amplitude is the vertical distance from the midline to a peak (or trough). From the graph, the curve seems to oscillate around a central value (the midline), but since this value isn’t explicitly given, we'll analyze the vertical distance between a peak and the nearest trough.

### Step 2: Determine the period \(T\)
The period is the horizontal length of one complete cycle. The graph shows that the function repeats its pattern every 8 units (from \(x = 0\) to \(x = 8\)).

Therefore, \(T = 8\), and:

\[
B = \frac{2\pi}{T} = \frac{2\pi}{8} = \frac{\pi}{4}
\]

### Step 3: Determine the phase shift \(C\)
The graph appears to start from a peak at \(x = 0\). Since the standard sine function \(y = \sin(x)\) starts from 0, we need to shift the sine function horizontally so that it starts from a peak.

Given that a sine function reaches its peak at \(x = 0\) with no phase shift, \(C = 0\).

### Step 4: Determine the vertical shift \(D\)
The graph oscillates symmetrically around \(y = 3\). Therefore, the vertical shift \(D = 3\).

### Step 5: Write the function
Given these parameters:

\[
f(x) = A \sin\left(\frac{\pi}{4}(x)\right) + 3
\]

To find \(A\), note that the maximum and minimum values of the graph appear to be equally spaced above and below the midline \(y = 3\). We need the peak and trough values to estimate \(A\).

- If the peak appears to be at \(y = 5\), and the trough at \(y = 1\), then \(A = 2\) because the amplitude is the distance from the midline to either the peak or the trough.

Thus, the sinusoidal function that fits the graph is:

\[
f(x) = 2 \sin\left(\frac{\pi}{4}(x)\right) + 3
\]

Would you like further details on any of these steps, or do you have any additional questions?

### Questions:
1. How is the period of a sinusoidal function related to the parameter \(B\) in the equation?
2. What role does the amplitude \(A\) play in the graph of a sinusoidal function?
3. How can you determine the vertical shift \(D\) from the graph of a sinusoidal function?
4. Why is it important to determine the phase shift \(C\) when matching a sinusoidal graph?
5. How would the function change if the sinusoidal graph started from a trough rather than a peak?

### Tip:
Remember that the sine and cosine functions are closely related—any sine function can be written as a cosine function with an appropriate phase shift and vice versa.

Learn how to find a sinusoidal function that matches a given graph passing through points (0, 3) and (8, 3). Understand the concepts of amplitude, period, phase shift, and vertical shift in sinusoidal functions. Suitable for Grades 10-12 students studying trigonometry and sinusoidal functions.

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