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The graph of a sinusoidal function intersects its midline at
(0,5) and then has a maximum point at
(pi,6).
Write the formula of the function, where
x is entered in radians.

f(x)=

The graph of a sinusoidal function intersects its midline at $(0,5)$ and then has a maximum point at $(\pi, 6)$ . $\newline$ Write the formula of the function, where $x$ is entered in radians. $\newline$ $f(x)=$

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Transformations of absolute value functions: translations and reflections

Full solution

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

(0,5)

and then has a maximum point at

(\pi, 6)

.

\newline

Write the formula of the function, where

x

is entered in radians.

\newline

f(x)=

Find Amplitude: Determine the amplitude of the function. $\newline$ Since the midline is at $y = 5$ and the maximum point is at $y = 6$ , the amplitude ( $A$ ) is the distance from the midline to the maximum, which is $6 - 5 = 1$ .
Determine Vertical Shift: Determine the vertical shift $D$ . The midline of the function represents the vertical shift. Since the function intersects the midline at $y = 5$ , the vertical shift is $D = 5$ .
Calculate Period: Determine the period of the function. The function reaches its maximum at $x = \pi$ , and since it's a sinusoidal function, the next maximum would be at $x = 2\pi$ (for a full cycle). Therefore, the period ( $T$ ) is $2\pi$ .
Calculate Value of B: Determine the value of $B$ in the function $f(x) = A\cos(Bx + C) + D$ . The period $T$ is related to $B$ by the formula $T = \frac{2\pi}{B}$ . Since $T = 2\pi$ , we have $2\pi = \frac{2\pi}{B}$ , which gives us $B = 1$ .
Determine Phase Shift: Determine the phase shift $C$ . $\newline$ Since the function intersects its midline at $(0,5)$ , there is no horizontal shift to the left or right. Therefore, the phase shift $C$ is $0$ .
Write Function Equation: Write the equation of the function using the determined values. $\newline$ We have $A = 1$ , $B = 1$ , $C = 0$ , and $D = 5$ . The function is $f(x) = A\cos(Bx + C) + D$ , which simplifies to $f(x) = \cos(x) + 5$ .

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