write a sine function that has an amplitude of 4, a midline of y = 2 and a period of 1/2. answer attempt 1…

Explanation:

Step1: Recall the general form of sine - function

The general form of a sine function is $y = A\sin(Bx - C)+D$, where $A$ is the amplitude, $B$ is related to the period by $T=\frac{2\pi}{|B|}$, $C$ is the phase - shift, and $D$ is the vertical shift (mid - line).

Step2: Determine the value of $A$

Given that the amplitude is $4$, so $A = 4$.

Step3: Determine the value of $D$

The mid - line is $y = 2$, so $D=2$.

Step4: Determine the value of $B$

The period $T=\frac{1}{2}$. Since $T=\frac{2\pi}{|B|}$, we have $\frac{1}{2}=\frac{2\pi}{|B|}$. Solving for $B$: [ \begin{align*} |B|& = 4\pi\ \end{align*} ] We can choose $B = 4\pi$ (assuming no phase - shift, $C = 0$).

Step5: Write the sine function

Substitute $A = 4$, $B = 4\pi$, $C = 0$, and $D = 2$ into the general form $y = A\sin(Bx - C)+D$. We get $f(x)=4\sin(4\pi x)+2$.

Answer:

$f(x)=4\sin(4\pi x)+2$