Question

Math

Posted 3 months ago

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$h$ is a trigonometric function of the form $h(x)=a \cos (b x+c)+d$.
Below is the graph of $h(x)$. The function has a maximum point at $\left(-\frac{\pi}{2}, 5\right)$ and a minimum point at $\left(\frac{\pi}{4},-4\right)$.
Find a formula for $h(x)$. Give an exact expression.
$h(x)=$
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Identify the maximum and minimum points of the trigonometric function

step 2

The maximum point at $\left(-\frac{\pi}{2}, 5\right)$ indicates that the value of $d$ is 5, as this is the vertical shift of the cosine function

step 3

The minimum point at $\left(\frac{\pi}{4}, -4\right)$ indicates that the amplitude of the cosine function is $\frac{5 - (-4)}{2} = \frac{9}{2}$. Therefore, $a = \frac{9}{2}$

step 4

The period of the function can be determined by the distance between the maximum and minimum points on the x-axis, which is $\frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}$. Since the period of a cosine function is $\frac{2\pi}{b}$, we have $b = \frac{8}{3}$

step 5

To find the phase shift $c$, we use the fact that the maximum of a cosine function occurs at $x = -\frac{c}{b}$. Substituting $x = -\frac{\pi}{2}$ and $b = \frac{8}{3}$, we solve for $c$

step 6

Solving $-\frac{\pi}{2} = -\frac{c}{\frac{8}{3}}$ gives $c = \frac{3\pi}{4}$

step 7

The function $h(x)$ is therefore $h(x) = \frac{9}{2} \cos\left(\frac{8}{3}x + \frac{3\pi}{4}\right) + 5$

[1] Answer

$h(x) = \frac{9}{2} \cos\left(\frac{8}{3}x + \frac{3\pi}{4}\right) + 5$

Key Concept

Determining the parameters of a trigonometric function from its graph

Explanation

The amplitude, period, and phase shift of a trigonometric function can be found by analyzing the maximum and minimum points and the shape of its graph.

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