To find a possible formula for the trigonometric function given the values in the table, we will analyze the characteristics of the function step by step.
The table is as follows:
[tex]\[
\begin{array}{|r|r|r|r|r|r|r|r|}
\hline
x & 0 & 3 & 6 & 9 & 12 & 15 & 18 \\
\hline
y & 2 & -3 & 2 & 7 & 2 & -3 & 2 \\
\hline
\end{array}
\][/tex]
1. Determine the Period:
By observing the pattern of the values, we can notice that the function repeats its values every 12 units along the x-axis. Therefore, the period of the function is [tex]\(12\)[/tex].
2. Determine the Amplitude:
The amplitude of a trigonometric function is half the difference between the maximum and minimum values of [tex]\(y\)[/tex].
Maximum [tex]\(y\)[/tex] value: [tex]\(7\)[/tex] (at [tex]\(x = 9\)[/tex])
Minimum [tex]\(y\)[/tex] value: [tex]\(-3\)[/tex] (at [tex]\(x = 3\)[/tex] and [tex]\(x = 15\)[/tex])
Amplitude [tex]\(A\)[/tex] is calculated as follows:
[tex]\[
A = \frac{\text{max}_y - \text{min}_y}{2} = \frac{7 - (-3)}{2} = \frac{10}{2} = 5
\][/tex]
3. Determine the Vertical Shift (Midline):
The vertical shift, or the midline of the function, is the average of the maximum and minimum [tex]\(y\)[/tex] values.
Vertical shift [tex]\(D\)[/tex] is calculated as follows:
[tex]\[
D = \frac{\text{max}_y + \text{min}_y}{2} = \frac{7 + (-3)}{2} = \frac{4}{2} = 2
\][/tex]
4. Determine the Frequency:
The frequency [tex]\(B\)[/tex] relates to the period [tex]\(T\)[/tex] via the formula:
[tex]\[
B = \frac{2\pi}{T}
\][/tex]
Given that the period [tex]\(T\)[/tex] is [tex]\(12\)[/tex]:
[tex]\[
B = \frac{2\pi}{12} = \frac{\pi}{6} \approx 0.5236
\][/tex]
5. Determine the Phase Shift:
The phase shift will depend on the specific form of the trigonometric function we choose. In this case, the function appears to be a cosine function because it starts at a maximum value (2 at [tex]\(x=0\)[/tex]).
With this information, we do not need a phase shift ([tex]\(C = 0\)[/tex]) as the cosine function typically peaks at [tex]\(x = 0\)[/tex].
Thus, combining all these values, we can construct the equation of the function. The general form of a cosine function with these parameters is:
[tex]\[
y = A \cos(B (x - C)) + D
\][/tex]
Substituting the values we have found:
[tex]\[
y = 5 \cos\left(\frac{\pi}{6} x \right) + 2
\][/tex]
Therefore, a possible formula for the trigonometric function is:
[tex]\[
y = 5 \cos\left(\frac{\pi}{6} x \right) + 2
\][/tex]
