SOLUTION: What is the period of y = 3cos ( 4x + π/2) + 5?

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Question 1142058: What is the period of y = 3cos ( 4x + π/2) + 5?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
period = 2 * pi / frequency

frequency = 2 * pi / period

your equation is y = 3 * cos (4x + pi/2) + 5

since the general form of the equation is y = a * cos(b * (x - c)) + d, you can rewrite the equation to be:

y = 3 * cos(4 * (x + pi/8) + 5

your frequency is 4.

that makes your period equal to 2 * pi / 4 = pi / 2

there is a vertical shift of 5.

this makes the center line equal to y = 5 on the graph.

the amplitube is 3.

that makes the graph go from y = 8 to y = 2.

the horizontal shift is pi/8.

i'll get to that later.

your equation, without the horizontal shift, is:

y = 3 * cos(4 * x) + 5

the graph of that equation looks like this.

with the horizontal shift, the equation is:

y = 3 * cos(4 * (x - pi/8)) + 5

the graph of that equation looks like this.

before the shift, the high point of the cosine function was at x = 0.

after the shift, the high point of the cosine function was at x = -pi/8.

this confirms that the graph of the equation was shifted to the left by pi/8.

if you try to find the value of y using your calculator, make sure that the calculator is set to radians.

the equation, and the graph, assumes x is in radians

this probably way more than you needed to know, but hopefully it is instructive.

the solution to your problem is:

frequency = 4.

period = 2 * pi / 4.

that makes period equal to pi/2, or 1/2 * pi, whichever way you want to show it.

a period of 1/2 * pi means you get 4 full cycles of the cosine function in the normal period of 2 * pi.

the graph of that looks like this:

Answer by ikleyn(52858) (Show Source):
You can put this solution on YOUR website!
.

If you are given the function y = A*cos(bx + c) + d and when you are asked to determine the period of this function, then - the value of the amplitude "A" does not matter for the answer; - the value of the horizontal shift "c" does not matter for the answer; - the value of the vertical shift "d" does not matter for the answer. The only value, which does matter to answer this question, is the coefficient "b" at variable x. The formula to find the period "T" then is b*T = , (1) which gives you the answer the period T is equal T = . It is because is the period of the function cos(x). In your case, the period T = = .