$h$ 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.
$$
h(x)=
$$

Question

$h$ 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.
$$
h(x)=
$$

Accepted Answer

∻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.⚹◊What are the values of the constants $a$, $b$, $c$, and $d$ in the cosine function $h(x)=a \cos(bx+c) + d$ given the maximum and minimum points on the graph provided?⍭ Generate me a similar question◊

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