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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. $\newline$ $f(x)=$ $\square$

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Write equations of sine and cosine functions using properties

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)=

\square

Amplitude calculation: The amplitude is the distance from the midline to the maximum, so it's $6 - 5 = 1$ .
Phase shift determination: Since the graph intersects the midline at $(0,5)$ , the phase shift is $0$ .
Period calculation: The function has a maximum at $\pi$ , which means the period is $2\pi$ , so $B = 1$ .
Midline identification: The midline is the $D$ value in the equation, which is $5$ .
Function equation formation: Putting it all together, we get $f(x) = A\cos(Bx + C) + D$ , where $A = 1$ , $B = 1$ , $C = 0$ , and $D = 5$ .
Final function equation: So the equation of the function is $f(x) = \cos(x) + 5$ .

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