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Posted 3 months ago

hh is a trigonometric function of the form h(x)=acos(bx+c)+dh(x)=a \cos (b x+c)+d.
Below is the graph of h(x)h(x). The function has a maximum point at (π2,5)\left(-\frac{\pi}{2}, 5\right) and a minimum point at (π4,4)\left(\frac{\pi}{4},-4\right).

Find a formula for h(x)h(x). Give an exact expression.
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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 (π2,5)\left(-\frac{\pi}{2}, 5\right) indicates that the value of dd is 5, as this is the vertical shift of the cosine function
step 3
The minimum point at (π4,4)\left(\frac{\pi}{4}, -4\right) indicates that the amplitude of the cosine function is 5(4)2=92\frac{5 - (-4)}{2} = \frac{9}{2}. Therefore, a=92a = \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 π2+π4=3π4\frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}. Since the period of a cosine function is 2πb\frac{2\pi}{b}, we have b=83b = \frac{8}{3}
step 5
To find the phase shift cc, we use the fact that the maximum of a cosine function occurs at x=cbx = -\frac{c}{b}. Substituting x=π2x = -\frac{\pi}{2} and b=83b = \frac{8}{3}, we solve for cc
step 6
Solving π2=c83-\frac{\pi}{2} = -\frac{c}{\frac{8}{3}} gives c=3π4c = \frac{3\pi}{4}
step 7
The function h(x)h(x) is therefore h(x)=92cos(83x+3π4)+5h(x) = \frac{9}{2} \cos\left(\frac{8}{3}x + \frac{3\pi}{4}\right) + 5
[1] Answer
h(x)=92cos(83x+3π4)+5h(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
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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