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The table you provided lists five key points of a sinusoidal function, and you are tasked with determining the equation of the function while avoiding a phase shift. The key points are as follows:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3\pi}{4} & \pi \\
\hline
y & 2 & 5 & 2 & -1 & 2 \\
\hline
\end{array}
\]

### Step-by-Step Analysis

1. **Amplitude**:
   The maximum \(y\)-value is 5 and the minimum \(y\)-value is -1. The amplitude \(A\) is half the distance between the maximum and minimum:
   \[
   A = \frac{5 - (-1)}{2} = 3
   \]

2. **Vertical Shift**:
   The sinusoidal function oscillates around a middle line, which is the average of the maximum and minimum \(y\)-values. This vertical shift \(D\) is:
   \[
   D = \frac{5 + (-1)}{2} = 2
   \]

3. **Period**:
   The period \(T\) is the length of one full cycle. From the points given, one cycle appears to start and end at \(y = 2\). From \(x = 0\) to \(x = \pi\), the function completes one cycle. Therefore, the period \(T\) is:
   \[
   T = \pi
   \]

   The standard period for a sine function is \(2\pi\), so to adjust for a period of \(\pi\), the frequency \(B\) is:
   \[
   B = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2
   \]

4. **Function Type**:
   Since the function starts at the midline \(y = 2\) when \(x = 0\), it suggests that the function could be a cosine function, as cosine functions start at their maximum or midline values. Thus, the function has no phase shift, and a cosine function would fit best.

### Equation of the Sinusoidal Function:
Now, combining the values for amplitude, vertical shift, period, and considering the cosine function, the equation is:
\[
y = 3\cos(2x) + 2
\]

This equation satisfies the given points and avoids a phase shift.

Would you like further explanation on any part of the process?

Here are five related questions:
1. What would the equation be if a sine function was used instead of cosine?
2. How would you find the phase shift if the function had one?
3. Can the same process be applied to other periodic functions like tangent?
4. What impact does changing the amplitude have on the graph of a sinusoidal function?
5. How does the vertical shift affect the maximum and minimum values of the function?

**Tip**: When determining the period of a sinusoidal function, check for how long it takes the function to repeat its cycle by focusing on the key points.

The table below gives the coordinates of the five key points for a sinusoidal function. Plot the points and determine the equation of the function without using a phase shift.

Learn how to determine the equation of a sinusoidal function from key points, avoiding phase shift. This solution covers key trigonometric concepts such as amplitude, vertical shift, and period, suitable for students in Grades 10-12.

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