To determine the midline of a sinusoidal function given its parameters, we need to understand what the midline represents. The midline is a horizontal line that runs through the middle of the wave, essentially the average value of the function over one period.
Given the information:
- Amplitude: 4
- Maximum point: [tex]\((\frac{\pi}{2}, 6)\)[/tex]
- Period: [tex]\(2 \pi\)[/tex]
We can find the midline by following these steps:
1. Recall that the amplitude of a sinusoidal function represents the distance from the midline to the maximum point (or the minimum point).
2. Given the maximum point [tex]\( ( \frac{\pi}{2}, 6 ) \)[/tex], the y-coordinate of 6 is the maximum value.
3. The midline is located halfway between the maximum and minimum points of the function. Since we're provided the maximum value (6) and the amplitude (4), we can find the midline by subtracting the amplitude from the maximum value.
[tex]\[
\text{Midline} = \text{Maximum Point} - \text{Amplitude}
\][/tex]
4. Substitute the given values:
[tex]\[
\text{Midline} = 6 - 4 = 2
\][/tex]
Therefore, the midline of the sinusoidal function is [tex]\(y = 2\)[/tex].
To graph the midline:
- Draw a horizontal line across the entire graph at [tex]\(y = 2\)[/tex]. This line represents the midline of the sinusoidal function.
By following this process, we can accurately determine and graph the midline of the given sinusoidal function.
