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The graph of a sinusoidal function has a minimum point at
(0,3) and then intersects its midline at
(5pi,5).
Write the formula of the function, where
x is entered in radians.

The graph of a sinusoidal function has a minimum point at $(0,3)$ and then intersects its midline at $(5 \pi, 5)$ . $\newline$ Write the formula of the function, where $x$ is entered in radians.

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Write equations of sine and cosine functions using properties

Full solution

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

(0,3)

and then intersects its midline at

(5 \pi, 5)

.

\newline

Write the formula of the function, where

x

is entered in radians.

Determine Amplitude: Determine the amplitude of the function. $\newline$ Since the graph has a minimum point at $(0,3)$ and intersects its midline at a higher value, the midline value is the average of the maximum and minimum. The given midline value is $5$ , and the minimum value is $3$ , so the amplitude ( $A$ ) is the difference between the midline and the minimum value. $\newline$ $A = \text{Midline} - \text{Minimum}$ $\newline$ $A = 5 - 3$ $\newline$ $A = 2$
Identify Vertical Shift: Identify the vertical shift $D$ . The midline value also represents the vertical shift of the function. Since the midline is at $y = 5$ , the vertical shift $D$ is $5$ . $D = 5$
Calculate Period: Calculate the period of the function. $\newline$ The function intersects its midline at $(5\pi,5)$ , which is a quarter of the period away from the minimum point at $(0,3)$ . Therefore, the period $T$ is four times the x-value of the midline intersection. $\newline$ $T = 4 \times (5\pi)$ $\newline$ $T = 20\pi$
Find Value of B: Find the value of B using the period. $\newline$ The period $T$ is related to $B$ by the formula $T = \frac{2\pi}{B}$ . We can solve for $B$ using the period we found. $\newline$ $T = \frac{2\pi}{B}$ $\newline$ $20\pi = \frac{2\pi}{B}$ $\newline$ $B = \frac{2\pi}{20\pi}$ $\newline$ $B = \frac{1}{10}$
Determine Phase Shift: Determine the phase shift $C$ . Since the minimum point is at $(0,3)$ , and we know that the cosine function starts at a maximum, we will use a sine function which starts at a minimum when there is no phase shift. Therefore, $C = 0$ . $C = 0$
Write Equation: Write the equation of the function. $\newline$ We have determined the amplitude $A$ , the vertical shift $D$ , the value of $B$ , and the phase shift $C$ . The general form of a sinusoidal function is $f(x) = A \cdot \cos(Bx + C) + D$ or $f(x) = A \cdot \sin(Bx + C) + D$ . Since we are using a sine function that starts at a minimum, we will use the sine form. $\newline$ $f(x) = A \cdot \sin(Bx + C) + D$ $\newline$ $f(x) = 2 \cdot \sin(\frac{1}{10}x + 0) + 5$

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