Graph the following points on the graphing calculator:
[tex]\[ (0, 6), \left( \frac{\pi}{2}, 7 \right), (\pi, 8), \left( \frac{3\pi}{2}, 7 \right), (2\pi, 6) \][/tex]

Explain how to use the graph to write an equation to model the points' pattern. Be sure to identify the pattern of the points in your explanation, and identify the values of [tex]\( a \)[/tex] and [tex]\( k \)[/tex].



Answer :

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]