To write an equation that models the points given: [tex]\((0, 6)\)[/tex], [tex]\(\left(\frac{\pi}{2}, 7\right)\)[/tex], [tex]\((\pi, 8)\)[/tex], [tex]\(\left(\frac{3\pi}{2}, 7\right)\)[/tex], and [tex]\((2\pi, 6)\)[/tex], follow these steps:
1. Identify the Pattern:
- Graph the points and observe their distribution.
- Note the pattern:
- The y-values increase to a maximum at [tex]\(y = 8\)[/tex] and decrease back to the minimum [tex]\(y = 6\)[/tex].
- This pattern is characteristic of a sinusoidal function (like sine or cosine).
2. Determine Characteristics:
- Midline: This is the central line around which the function oscillates. It can be found by calculating the average of the maximum and minimum values:
[tex]\[
\text{Midline} = \frac{8 + 6}{2} = 7
\][/tex]
- Amplitude: This represents the distance from the midline to the maximum or minimum. It's the difference between the maximum value (8) and the midline (7):
[tex]\[
\text{Amplitude} = 8 - 7 = 1
\][/tex]
- Period: The period is the length of one complete cycle of the sinusoidal function. From the points, you can see that the points repeat at [tex]\(x = 0\)[/tex] and [tex]\(x = 2\pi\)[/tex], so the period is [tex]\(2\pi\)[/tex].
3. Choose the Function Type:
- Because the pattern starts at the midline at [tex]\(x = 0\)[/tex] and rises to a peak, a cosine function is suitable (as cosine traditionally starts from the highest midpoint value in its basic form).
- The general form of the cosine function is [tex]\(y = a \cos(bx - c) + d\)[/tex], where:
- [tex]\(a\)[/tex] is the amplitude (which we determined is 1).
- [tex]\(d\)[/tex] is the vertical shift or midline (which we determined is 7).
- [tex]\(b\)[/tex] relates to the period of the function and is calculated as [tex]\(b = \frac{2\pi}{\text{Period}} = \frac{2\pi}{2\pi} = 1\)[/tex].
4. Write the Equation:
- Since the cosine function suits our pattern and the midline is 7 with an amplitude of 1, the equation of the sinusoidal function can be written as:
[tex]\[
y = \cos(x) + 7
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex] are:
- [tex]\(a = 1\)[/tex] (the amplitude)
- [tex]\(d = 7\)[/tex] (the vertical shift or midline)
The final equation modeling the points on the graph is:
[tex]\[
y = \cos(x) + 7
\][/tex]
