write the equation of the trigonometric function shown in the graph. try fractional values or π for the box…

Explanation:

Step1: Determine the vertical shift (D).

The vertical shift $D$ is the midline of the function. The maximum value of the solid curve is $1$ and the minimum value is $-1$. The midline is $D = \frac{\text{Maximum} + \text{Minimum}}{2}$. $$D = \frac{1 + (-1)}{2} = \frac{0}{2} = 0$$

Step2: Determine the amplitude (A).

The amplitude $A$ is the distance from the midline to the maximum value (or half the distance between the maximum and minimum values). $$A = \text{Maximum} - D = 1 - 0 = 1$$ Alternatively, $A = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{1 - (-1)}{2} = \frac{2}{2} = 1$.

Step3: Determine the period and the coefficient B.

The graph shows a cosine function starting at its maximum value at $x=0$. One full cycle is completed when it reaches its next maximum. The first maximum is at $(0, 1)$. The next maximum is at $(\frac{5\pi}{2}, 1)$. The period $P$ is the difference in the x-values: $$P = \frac{5\pi}{2} - 0 = \frac{5\pi}{2}$$ The relationship between the period $P$ and the coefficient $B$ for $y = \cos(Bx)$ is $P = \frac{2\pi}{|B|}$. Assuming $B > 0$: $$\frac{5\pi}{2} = \frac{2\pi}{B}$$ $$B = \frac{2\pi}{\frac{5\pi}{2}} = 2\pi \cdot \frac{2}{5\pi} = \frac{4\pi}{5\pi} = \frac{4}{5}$$

Step4: Write the equation.

The equation is in the form $y = A \cos(Bx) + D$. Substituting the values $A=1$, $B=\frac{4}{5}$, and $D=0$: $$y = 1 \cos\left(\frac{4}{5}x\right) + 0$$ $$y = \cos\left(\frac{4}{5}x\right)$$

Answer:

The equation is $y = \boxed{1} \cos\left(\boxed{\frac{4}{5}}x\right) + \boxed{0}$.