find an equation for the sine graph, $f(x)$:\nwrite your answer in the form $f(x) = a \\sin(bx + c) + d$…

Explanation:

Step1: Identify the midline and amplitude

The maximum value is $4.5$ and the minimum value is $3.5$. The midline $D$ is the average, and amplitude $A$ is half the range. $$D = \frac{4.5 + 3.5}{2} = 4, \quad A = \frac{4.5 - 3.5}{2} = 0.5$$

Step2: Determine the period and frequency

The graph completes one cycle from $x = -\pi$ to $x = 3\pi$. The period $P$ is $4\pi$. $$B = \frac{2\pi}{P} = \frac{2\pi}{4\pi} = 0.5$$

Step3: Determine the phase shift

A standard sine wave starts at the midline and goes up. This occurs at $x = -\pi$. $$\text{Phase Shift} = -\frac{C}{B} = -\pi \implies C = \pi \cdot B = 0.5\pi$$

Step4: Construct the final equation

Substitute $A$, $B$, $C$, and $D$ into the general form $f(x) = A \sin(Bx + C) + D$. $$f(x) = 0.5 \sin(0.5x + 0.5\pi) + 4$$

Answer:

f(x) = 0.5 \sin(0.5x + 0.5\pi) + 4