Question

write the equation of the trigonometric function shown in the graph.

Explanation:

Step1: Identify base function form

This is a cosine function, general form: $y = A\cos(Bx) + C$

Step2: Find vertical shift (midline)

Midline $C = \frac{\text{max } y + \text{min } y}{2} = \frac{5 + 1}{2} = 3$

Step3: Calculate amplitude $A$

Amplitude $A = \text{max } y - C = 5 - 3 = 2$

Step4: Determine period and $B$

Period $T = 2\pi$ (distance between peaks: $2\pi - 0 = 2\pi$).
$B = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1$

Step5: Assemble the equation

Substitute $A=2$, $B=1$, $C=3$ into the general form.

Answer:

$y = 2\cos(x) + 3$