Sure, let's analyze the given points and construct the equation of the sinusoidal function step-by-step.
### Step 1: Identify the midline
The midline of a sinusoidal function is the horizontal axis that the function oscillates around. It can be found by taking the average of the maximum and minimum y-values.
Given:
- Maximum point: [tex]\((0, 5)\)[/tex]
- Minimum point: [tex]\((2\pi, -5)\)[/tex]
The midline (vertical shift) [tex]\(d\)[/tex] can be calculated as:
[tex]\[
d = \frac{5 + (-5)}{2} = \frac{0}{2} = 0
\][/tex]
### Step 2: Determine the amplitude
The amplitude is the distance from the midline to either the maximum or minimum point. It can be found by taking the difference between the maximum (or minimum) value and the midline.
Given:
- Maximum value: [tex]\(5\)[/tex]
- Midline: [tex]\(0\)[/tex]
The amplitude [tex]\(A\)[/tex] is:
[tex]\[
A = 5 - 0 = 5
\][/tex]
### Step 3: Calculate the period
The period is the distance (or interval on the x-axis) required for the function to complete one full cycle. Given the positions of the maximum and minimum points, we can calculate the period as twice the distance from the maximum point to the next minimum point.
Given:
- Maximum point at [tex]\(x = 0\)[/tex]
- Minimum point at [tex]\(x = 2\pi\)[/tex]
The distance between a maximum and the next minimum point is [tex]\(2\pi\)[/tex], so:
[tex]\[
\text{Period} = 2 \times 2\pi = 4\pi
\][/tex]
### Step 4: Determine the function type and phase shift
Since we have a maximum point at [tex]\(x = 0\)[/tex], it suggests that the sinusoidal function could be a cosine function (since cosine starts at a maximum). There is no horizontal shift needed because the maximum is already at [tex]\(x = 0\)[/tex].
### Step 5: Formulate the function
The general form of a cosine function is:
[tex]\[
f(x) = A \cos\left(\frac{2\pi}{\text{Period}} x - \text{Phase Shift} \right) + d
\][/tex]
Substituting the values we have:
- Amplitude [tex]\(A = 5\)[/tex]
- Period [tex]\(4\pi\)[/tex], so [tex]\(\frac{2\pi}{\text{Period}} = \frac{2\pi}{4\pi} = \frac{1}{2}\)[/tex]
- No phase shift [tex]\((0)\)[/tex]
- Vertical shift [tex]\(d = 0\)[/tex]
Thus, the equation of the function is:
[tex]\[
f(x) = 5 \cos\left(\frac{1}{2}x \right)
\][/tex]
So, the formula of the function is:
[tex]\[
f(x) = 5 \cos\left(\frac{x}{2} \right)
\][/tex]
