Full solution

Q. The graph of a sinusoidal function has a minimum point at

(0,2)

and then has a maximum point at

(3 \pi, 6)

\newline

Write the formula of the function, where

x

is entered in radians.

\newline

f(x)=\square

Calculate Amplitude: Determine the amplitude of the function. $\newline$ The amplitude is half the distance between the maximum and minimum values of the function. $\newline$ Amplitude = $(\text{Maximum} - \text{Minimum}) / 2$ $\newline$ Amplitude = $(6 - 2) / 2$ $\newline$ Amplitude = $4 / 2$ $\newline$ Amplitude = $2$
Find Vertical Shift: Find the vertical shift, $D$ . The vertical shift is the average of the maximum and minimum values of the function. $D = (\text{Maximum} + \text{Minimum}) / 2$ $D = (6 + 2) / 2$ $D = 8 / 2$ $D = 4$
Determine Period: Calculate the period of the function. $\newline$ The period is the distance between two consecutive minimum or maximum points. Since we have a minimum at $x=0$ and the next maximum at $x=3\pi$ , the period is twice this distance. $\newline$ Period = $2 \times (3\pi - 0)$ $\newline$ Period = $6\pi$
Calculate Value of B: Find the value of B using the period formula for a sinusoidal function. $\newline$ Period = $(2\pi) / B$ $\newline$ $6\pi = (2\pi) / B$ $\newline$ $B = (2\pi) / (6\pi)$ $\newline$ $B = 1 / 3$
Apply Phase Shift: Since the function has a minimum at $x=0$ , we will use a cosine function with a phase shift to reflect this. A cosine function normally has a maximum at $x=0$ , so we need to shift it by $\pi$ to make it a minimum. $\newline$ $C = \pi$
Write Function Equation: Write the equation of the function using the values found for $A$ , $B$ , $C$ , and $D$ .
$f(x) = A \cdot \cos(Bx + C) + D$
$f(x) = 2 \cdot \cos(\frac{1}{3}x + \pi) + 4$