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The graph of a sinusoidal function has a maximum point at
(0,5) and then has a minimum point at
(2pi,-5).
Write the formula of the function, where
x is entered in radians.

f(x)=

◻

The graph of a sinusoidal function has a maximum point at $\newline$ $(0,5)$ and then has a minimum point at $\newline$ $(2\pi,-5)$ . $\newline$ Write the formula of the function, where $\newline$ $x$ is entered in radians. $\newline$ $f(x)=$ $\newline$ $\square$

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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

\newline

(0,5)

and then has a minimum point at

\newline

(2\pi,-5)

.

\newline

Write the formula of the function, where

\newline

x

is entered in radians.

\newline

f(x)=

\newline

\square

Calculate Amplitude: The amplitude is half the distance between the maximum and minimum values. $\newline$ Amplitude = $(5 - (-5)) / 2 = 10 / 2 = 5$ .
Determine Period: The period is the distance between a maximum and the next minimum, which is $2\pi$ . $\newline$ Period = $2\pi$ .
Find B: To find B, use the formula $\text{Period} = \frac{2\pi}{B}$ . $2\pi = \frac{2\pi}{B}$ $B = 1$ .
Identify Vertical Shift: Since the maximum is at $(0,5)$ , the graph is a cosine function shifted up by $5$ . $\newline$ $D = 5$ .
Determine $A$ : The function has not been horizontally or vertically flipped because it starts at a maximum and goes to a minimum, so $A$ is positive. $A = 5$ .
Identify Horizontal Shift: There is no horizontal shift because the maximum is at $x=0$ , so $C = 0$ .
Write Equation: Write the equation using the values found for $A$ , $B$ , $C$ , and $D$ . $f(x) = 5\cos(1x + 0) + 5.$

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