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The graph of a sinusoidal function has a maximum point at
(0,10) and then intersects its midline at
((pi)/(4),4).
Write the formula of the function, where
x is entered in radians.

The graph of a sinusoidal function has a maximum point at $(0,10)$ and then intersects its midline at $\left(\frac{\pi}{4}, 4\right)$ . $\newline$ 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 maximum point at

(0,10)

and then intersects its midline at

\left(\frac{\pi}{4}, 4\right)

.

\newline

Write the formula of the function, where

x

is entered in radians.

Determine Amplitude: Determine the amplitude $A$ of the function. $\newline$ Since the maximum value is $10$ and it intersects the midline at $4$ , the amplitude is the distance from the midline to the maximum value. $\newline$ $A = 10 - 4 = 6$
Find Vertical Shift: Find the vertical shift $D$ of the function. $\newline$ The midline value is $4$ , which is also the vertical shift. $\newline$ $D = 4$
Determine Period: Determine the period $T$ of the function. $\newline$ Since the function intersects the midline at $\frac{\pi}{4}$ , and this is a quarter of the period away from the maximum, the full period is $4$ times this value. $\newline$ $T = 4 \times \left(\frac{\pi}{4}\right) = \pi$
Calculate Value of B: Calculate the value of $B$ in the function $f(x) = A\cos(Bx + C) + D$ , using the period $T$ . The period $T$ is related to $B$ by the formula $T = \frac{2\pi}{B}$ . $B = \frac{2\pi}{T}$ $B = \frac{2\pi}{\pi}$ $B = 2$
Determine Phase Shift: Determine the phase shift $C$ of the function. $\newline$ Since the maximum point is at $(0,10)$ , there is no horizontal shift, and thus $C = 0$ .
Write Sinusoidal Function: Write the equation of the sinusoidal function using the values of $A$ , $B$ , $C$ , and $D$ .
$f(x) = A\cos(Bx + C) + D$
$f(x) = 6\cos(2x + 0) + 4$
$f(x) = 6\cos(2x) + 4$

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