Question

Math

Posted 5 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 5 months ago

Solution by Steps

step 1

Identify the amplitude of the function from the graph. The amplitude is the distance from the midline to the maximum or minimum value of the function

step 2

The amplitude is given by the difference between the maximum and minimum y-values divided by 2: $a = \frac{5 - (-4)}{2} = \frac{9}{2}$

step 3

Determine the period of the function. The period is the distance on the x-axis to complete one full cycle

step 4

The period is given by the distance between two consecutive maximum points or two consecutive minimum points. From the graph, the period is $3\pi/2$

step 5

Relate the period of the function to the coefficient $b$ in the function $h(x) = a \cos(bx + c) + d$. The period $T$ is given by $T = \frac{2\pi}{b}$

step 6

Solve for $b$ using the period: $b = \frac{2\pi}{T} = \frac{2\pi}{3\pi/2} = \frac{4}{3}$

step 7

Identify the vertical shift $d$ by finding the midline of the function, which is the average of the maximum and minimum y-values

step 8

The vertical shift is $d = \frac{5 + (-4)}{2} = \frac{1}{2}$

step 9

Determine the phase shift by finding the value of $c$. The graph shows a maximum at $x = -\pi/2$, which corresponds to the cosine function's maximum at $0$

step 10

Solve for $c$ using the fact that the maximum of the cosine function is at $bx + c = 0$. With $b = \frac{4}{3}$ and $x = -\pi/2$, we have $\frac{4}{3}(-\pi/2) + c = 0$

step 11

Solve for $c$: $c = \frac{4}{3}(\pi/2) = \frac{2\pi}{3}$

step 12

Write the final function using the values of $a$, $b$, $c$, and $d$: $h(x) = \frac{9}{2} \cos\left(\frac{4}{3}x + \frac{2\pi}{3}\right) + \frac{1}{2}$

[1] Answer

$h(x) = \frac{9}{2} \cos\left(\frac{4}{3}x + \frac{2\pi}{3}\right) + \frac{1}{2}$

Key Concept

Determining the equation of a trigonometric function from its graph

Explanation

The equation of a trigonometric function can be determined by identifying the amplitude, period, phase shift, and vertical shift from its graph. These values correspond to the constants in the function's equation.

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