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We are given a table of values for a trigonometric function and are asked to find a possible formula for the function.

The values of \(x\) are:
\[ 0, 4, 8, 12, 16, 20, 24 \]

The corresponding values of \(y\) are:
\[ -6, -4, -2, -4, -6, -4, -2 \]

### Step 1: Identify the periodic behavior
The values of \(y\) repeat in a pattern, suggesting the function is periodic. A possible candidate for such behavior is the cosine or sine function, as they naturally exhibit oscillatory behavior. The pattern seems to repeat every 16 units (from \(x = 0\) to \(x = 16\)).

Thus, the period \(T\) of the function could be 16.

### Step 2: Amplitude and vertical shift
The maximum value of \(y\) is \(-2\), and the minimum value is \(-6\). The amplitude \(A\) can be calculated as:
\[
A = \frac{\text{max} - \text{min}}{2} = \frac{-2 - (-6)}{2} = 2
\]

The midline (vertical shift) is the average of the maximum and minimum values:
\[
D = \frac{\text{max} + \text{min}}{2} = \frac{-2 + (-6)}{2} = -4
\]

So, the function has an amplitude of 2 and is vertically shifted down by 4 units.

### Step 3: Determine the trigonometric form
We suspect the function is either a cosine or sine function. The \(y\)-values start at a minimum at \(x = 0\), which is characteristic of a cosine function with a phase shift.

The general form for the cosine function is:
\[
y = A \cos(Bx + C) + D
\]
Where:
- \(A\) is the amplitude (2),
- \(B\) is related to the period \(T\) by \(B = \frac{2\pi}{T}\),
- \(C\) is the phase shift,
- \(D\) is the vertical shift (-4).

Since the period \(T = 16\), we have:
\[
B = \frac{2\pi}{16} = \frac{\pi}{8}
\]

### Step 4: Phase shift
The cosine function normally starts at its maximum when \(x = 0\), but here the function starts at its minimum. To adjust for this, we need a phase shift of \( \pi \) (which moves it to its minimum at \(x = 0\)).

Thus, the phase shift \(C = \pi\).

### Step 5: Final function
The possible formula for the trigonometric function is:
\[
y = 2 \cos\left(\frac{\pi}{8}x + \pi\right) - 4
\]

Would you like further clarification or have any questions?

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Here are five related questions:
1. How would the formula change if the phase shift was different?
2. What if the function was a sine function instead of cosine?
3. Can you derive the formula using sine instead of cosine?
4. How do you determine the period of a trigonometric function from data points?
5. How can we verify the correctness of this function?

**Tip:** When working with periodic functions, always examine the amplitude, period, phase shift, and vertical shift carefully. They define the behavior of the trigonometric function.

Find a possible formula for the trigonometric function whose values are in the following table:

x: 0, 4, 8, 12, 16, 20, 24
y: -6, -4, -2, -4, -6, -4, -2

Learn how to derive the formula of a trigonometric function from a set of points. This problem demonstrates key concepts like amplitude, phase shift, period, and vertical shift using a cosine function. Suitable for high school students in Grades 10-12.

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