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The graph of a sinusoidal function has a minimum point at $(0,-10)$ and then has a maximum point at $(2,-4)$ . Write the formula of the function, where $x$ is entered in radians. $\newline$ $f(x)=\square$

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Write equations of sine and cosine functions using properties

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

(0,-10)

and then has a maximum point at

(2,-4)

. 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 = $(-4 - (-10)) / 2$ $\newline$ Amplitude = $(6) / 2$ $\newline$ Amplitude = $3$
Find Vertical Shift: Find the vertical shift, $D$ . The vertical shift is the average of the maximum and minimum $y$ -values. $D = (\text{Maximum} + \text{Minimum}) / 2$ $D = (-4 + (-10)) / 2$ $D = (-14) / 2$ $D = -7$
Determine Period: Calculate the period of the function. $\newline$ The period is the distance between two consecutive minimums or maximums. Since we have one minimum at $x=0$ and the next maximum at $x=2$ , half the period is $2$ . $\newline$ Period = $2 \times 2$ $\newline$ Period = $4$
Calculate Value of B: Find the value of B, which is related to the period by the formula $\text{Period} = \frac{2\pi}{B}$ . $\frac{2\pi}{B} = 4$ $B = \frac{2\pi}{4}$ $B = \frac{\pi}{2}$
Determine Phase Shift: Determine the phase shift, $C$ . Since the minimum point is at $(0,-10)$ , and the sinusoidal function starts at a minimum when the phase shift is $0$ , we can conclude that $C = 0$ .
Write Function Equation: Write the equation of the function using the values of $A$ , $B$ , $C$ , and $D$ .
$f(x) = A \cdot \cos(Bx + C) + D$
$f(x) = 3 \cdot \cos\left(\frac{\pi}{2}x + 0\right) - 7$
Simplify the equation:
$f(x) = 3 \cdot \cos\left(\frac{\pi}{2}x\right) - 7$

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