$h$ is a trigonometric function of the form $h(x)=a \sin (b x+c)+d$ . $\newline$ Below is the graph $h(x)$ . The function intersects its midline at $(-4 \pi, 2.5)$ and has a maximum point at $\left(-\frac{5 \pi}{2}, 6\right)$ . $\newline$ Find a formula for $h(x)$ . Give an exact expression. $\newline$ $h(x)=\square \underline{+\underline{x}}$

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Algebra 2

Write equations of cosine functions using properties

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h

is a trigonometric function of the form

h(x)=a \sin (b x+c)+d

\newline

Below is the graph

h(x)

. The function intersects its midline at

(-4 \pi, 2.5)

and has a maximum point at

\left(-\frac{5 \pi}{2}, 6\right)

\newline

Find a formula for

h(x)

. Give an exact expression.

\newline

h(x)=\square \underline{+\underline{x}}

Midline determination: The midline of the function is the horizontal line that the function oscillates around. Since the function intersects its midline at $(-4\pi, 2.5)$ , the value of $d$ , which represents the vertical shift, is $2.5$ .
Amplitude calculation: The amplitude of the function is the distance from the midline to the maximum or minimum point. Since the maximum value of the function is $6$ and it occurs at the midline value of $2.5$ , the amplitude $a$ is $6 - 2.5 = 3.5$ .
Period analysis: The value of $b$ affects the period of the function. The period of a sine function is $\frac{2\pi}{b}$ . We do not have enough information to determine the period directly from the given points, but we can infer that since the function intersects the midline at $-4\pi$ and has a maximum at $-\frac{5\pi}{2}$ , one quarter of a period must be the distance between these $x$ -values. The distance between $-4\pi$ and $-\frac{5\pi}{2}$ is $\frac{\pi}{2}$ . Therefore, one quarter of the period is $\frac{\pi}{2}$ , and the full period is $4$ times that, which is $2\pi$ . So, $\frac{2\pi}{b}$ $0$ .
Phase shift determination: The phase shift $c$ is determined by the horizontal shift of the function. Since we know the function has a maximum at $-(5\pi/2)$ , and the sine function normally has a maximum at $\pi/2$ , we can find the phase shift by setting $-(5\pi/2) + c = \pi/2$ . Solving for $c$ gives us $c = \pi/2 - (-5\pi/2) = 3\pi$ .
Final function formulation: Putting all the values together, we get the function $h(x) = 3.5\sin(x + 3\pi) + 2.5$ .