The graph of a sinusoidal function intersects its midline at (0,1) and then has a maximum point at ((7pi)/(4),5). Write the formula of the function, where x is entered in radians. f(x)=◻

Determine Amplitude: Determine the amplitude A of the sinusoidal function. The amplitude is the distance from the midline to the maximum point. Since the midline is at y = 1 and the maximum point is at y = 5, the amplitude A is 5 - 1 = 4.Determine Vertical Shift: Determine the vertical shift D. The midline of the function is also the vertical shift D. Since the function intersects the midline at (0,1), D is 1.Determine Period: Determine the period of the sinusoidal function. The period T is the distance between two consecutive maximum points. Since we only have one maximum point, we can use the fact that the maximum point occurs at (7pi/4) and the function intersects the midline at (0,1). The next intersection with the midline after a maximum would be a quarter period away, so the period T is 4 times the x-coordinate of the maximum point, which is 4 * (7pi/4) = 7pi.Determine Value of B: Determine the value of B. The value of B is related to the period by the formula B = 2π / T. Substituting the period T = 7π, we get B = 2π / 7π = 2/7.Determine Phase Shift: Determine the phase shift C. Since the function intersects its midline at (0,1), there is no horizontal shift, so C = 0.Write Sinusoidal Function: Write the equation of the sinusoidal function. We have found: A = 4 B = 2/7 C = 0 D = 1 Substitute these values into the general form of the sinusoidal function f(x) = A cos(Bx + C) + D. f(x) = 4 cos((2/7)x + 0) + 1 f(x) = 4 cos((2/7)x) + 1

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The graph of a sinusoidal function intersects its midline at
(0,1) and then has a maximum point at
((7pi)/(4),5).
Write the formula of the function, where
x is entered in radians.

f(x)=◻

The graph of a sinusoidal function intersects its midline at $(0,1)$ and then has a maximum point at $\left(\frac{7 \pi}{4}, 5\right)$ . $\newline$ Write the formula of the function, where $x$ is entered in radians. $\newline$ $f(x)=\square$

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Algebra 2

Write equations of cosine functions using properties

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Q. The graph of a sinusoidal function intersects its midline at

(0,1)

and then has a maximum point at

\left(\frac{7 \pi}{4}, 5\right)

\newline

Write the formula of the function, where

x

is entered in radians.

\newline

f(x)=\square

Determine Amplitude: Determine the amplitude $A$ of the sinusoidal function. $\newline$ The amplitude is the distance from the midline to the maximum point. Since the midline is at $y = 1$ and the maximum point is at $y = 5$ , the amplitude $A$ is $5 - 1 = 4$ .
Determine Vertical Shift: Determine the vertical shift $D$ . The midline of the function is also the vertical shift $D$ . Since the function intersects the midline at $(0,1)$ , $D$ is $1$ .
Determine Period: Determine the period of the sinusoidal function. $\newline$ The period $T$ is the distance between two consecutive maximum points. Since we only have one maximum point, we can use the fact that the maximum point occurs at $(7\pi/4)$ and the function intersects the midline at $(0,1)$ . The next intersection with the midline after a maximum would be a quarter period away, so the period $T$ is $4$ times the $x$ -coordinate of the maximum point, which is $4 \times (7\pi/4) = 7\pi$ .
Determine Value of B: Determine the value of B. $\newline$ The value of B is related to the period by the formula $B = \frac{2\pi}{T}$ . Substituting the period $T = 7\pi$ , we get $B = \frac{2\pi}{7\pi} = \frac{2}{7}$ .
Determine Phase Shift: Determine the phase shift $C$ . Since the function intersects its midline at $(0,1)$ , there is no horizontal shift, so $C = 0$ .
Write Sinusoidal Function: Write the equation of the sinusoidal function. $\newline$ We have found: $\newline$ $A = 4$ $\newline$ $B = \frac{2}{7}$ $\newline$ $C = 0$ $\newline$ $D = 1$ $\newline$ Substitute these values into the general form of the sinusoidal function $f(x) = A \cos(Bx + C) + D$ . $\newline$ $f(x) = 4 \cos\left(\frac{2}{7}x + 0\right) + 1$ $\newline$ $f(x) = 4 \cos\left(\frac{2}{7}x\right) + 1$