To write a sine function that has the specified characteristics, we need to understand the general formula of a sine function and how the given parameters affect it.
The general form of a sine function is:
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
Here:
- [tex]\( A \)[/tex] is the amplitude of the sine wave.
- [tex]\( \frac{2\pi}{B} \)[/tex] is the period of the sine wave.
- [tex]\( C \)[/tex] is the horizontal shift (also called phase shift).
- [tex]\( D \)[/tex] is the vertical shift (determines the midline).
Given the parameters:
- Midline [tex]\( y = 2 \)[/tex]: This means [tex]\( D = 2 \)[/tex].
- Amplitude [tex]\( 4 \)[/tex]: This means [tex]\( A = 4 \)[/tex].
- Period [tex]\( \pi \)[/tex]: This implies [tex]\( \frac{2\pi}{B} = \pi \)[/tex].
To find [tex]\( B \)[/tex], we solve:
[tex]\[ \frac{2\pi}{B} = \pi \][/tex]
[tex]\[ B = \frac{2\pi}{\pi} \][/tex]
[tex]\[ B = 2 \][/tex]
There is no mention of any horizontal shift, so [tex]\( C = 0 \)[/tex].
Putting all these together, we substitute [tex]\( A = 4 \)[/tex], [tex]\( B = 2 \)[/tex], [tex]\( C = 0 \)[/tex], and [tex]\( D = 2 \)[/tex] into our general form of the sine function:
[tex]\[ y = 4 \sin(2(x - 0)) + 2 \][/tex]
This simplifies to:
[tex]\[ y = 4 \sin(2x) + 2 \][/tex]
So, the sine function with a midline of [tex]\( y = 2 \)[/tex], an amplitude of 4, and a period of [tex]\( \pi \)[/tex] is:
[tex]\[ y = 4 \sin(2x) + 2 \][/tex]
