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.

Determine Amplitude: Determine the amplitude (A) of the function. 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. A = 10 - 4 = 6Find Vertical Shift: Find the vertical shift (D) of the function. The midline value is 4, which is also the vertical shift. D = 4Determine Period: Determine the period (T) of the function. Since the function intersects the midline at (pi/4), and this is a quarter of the period away from the maximum, the full period is 4 times this value. T = 4 * (pi/4) = piCalculate Value of B: Calculate the value of B in the function f(x) = Acos(Bx + C) + D, using the period T. The period T is related to B by the formula T = 2pi/B. B = 2pi/T B = 2pi/pi B = 2Determine Phase Shift: Determine the phase shift (C) of the function. 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) = Acos(Bx + C) + D f(x) = 6cos(2x + 0) + 4 f(x) = 6cos(2x) + 4

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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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Algebra 2

Write equations of sine and cosine functions using properties

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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 $D$ .
$f(x) = A\cos(Bx + C) + D$
$f(x) = 6\cos(2x + 0) + 4$
$f(x) = 6\cos(2x) + 4$