SOLUTION: A cosine curve has a maximum point at (2,16) and the nearest minimum point to the right of this point is at (7,4). Which of the following answer choices gives an equation of this c

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Question 955032: A cosine curve has a maximum point at (2,16) and the nearest minimum point to the right of this point is at (7,4). Which of the following answer choices gives an equation of this curve?
a. +y=+6cos+%28pi%2F5%29+%28x-2%29%2B10+
b. +y=+6cos+%28pi%2F5%29+%28x%2B2%29%2B10+
c. +y=+6cos+%282pi%2F5%29+%28x-2%29%2B10+
d. +y=+6cos+%282pi%2F5%29+%28x%2B2%29%2B10+
Thank you
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you've got a maximum point at (2,16)
you've got the next minimum point at (7,4)

the formula for cosine is y = a * cosine (b * (x-c)) + d
a is the amplitude.
b is the frequency
c is the horizontal displacement
d is the vertical displacement.

the amplitude is the maximum distance from the center line of the graph.
the center line of the graph is half the distance between the maximum point and the minimum point.
the vertical displacement is the distance between the center line of the graph and the x-axis.
the frequency is equal to 2 * pi divided by the period.
the period is equal to 2 * pi divided by the frequency.

first we find your amplitude.
maximum height is 16
minimum height is 4
difference between is 12
difference divided by 2 = 6
your amplitude is 6.

next we find the center line.
the center line is halfway between the maximum point and the minimum point.
16 - 4 = 12 divided by 2 equal 6.
the center line is 6 units above the minimum point and 6 units below the maximum point.
the center line is at x = 10
10 + 6 = 16
10 - 6 = 4
center line at x = 10 is confirmed.
your center line is at x = 10.

next we find the vertical displacement.
vertical displacement is number of units that the center line is from the x-axis.
vertical displacement is therefore equal to 10.

next we find the frequency from the period or the period from the frequency.

we have a maximum point at x = 2.
we have a minimum point at x = 7
distance between min and max point is 5 units on the x-axis.
distance between min and max point is half the cycle of the cosine wave.
distance between 1 full cycle is therefore equal to 2 * 5 = 10.
that's the period.
frequency = 2 * pi / period which is equal to 2 * pi / 10 which is equal to pi/5.
your frequency is pi/5.

you now have:

a = amplitude = 6
b = frequency = pi/5
d = vertical displacement = 10.

we still need horizontal displacement.

the cosine wave is normally at its maximum point when x = 0
since its max point is at x = 2, the max point of the cosine function has been displaced 2 units to the right.
this make the max point at (x-2) because when x = 2, x-2 = 0 and the max point is there.
your horizontal displacement is equal to 2
this means that c is equal to 2 because c is the horizontal displacement.

your formula of y = a * (b * (x-c)) + d becomes:

y = 6 * cosine (pi / 5 * (x - 2)) + 10

that would correspond to selection a.

a graph of that equation is shown below:

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