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?

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.