write the equation of the trigonometric function shown in the graph.\n$y = 2\\cos(\\square x)+1$

Answer:

$y = 2\cos\left(\frac{2}{3}x\right)+1$

Explanation:

Step1: Recall the general form of cosine function

The general form is $y = A\cos(Bx - C)+D$. Here $A = 2$, $D = 1$, and $C = 0$.

Step2: Find the period

The period of the function $y=\cos(x)$ is $2\pi$. For the given graph, the period $T$ can be found by looking at the distance between two consecutive similar points (e.g., from $x = 0$ to $x = 3\pi$). So $T=3\pi$.

Step3: Use the period formula

The period formula is $T=\frac{2\pi}{|B|}$. Substituting $T = 3\pi$ into $T=\frac{2\pi}{|B|}$, we get $3\pi=\frac{2\pi}{|B|}$.

Step4: Solve for $B$

Cross - multiply: $3\pi|B|=2\pi$. Divide both sides by $\pi$: $3|B| = 2$. Then $|B|=\frac{2}{3}$. Since the graph is a basic cosine - type (no reflection indicated), we take $B=\frac{2}{3}$.