Vertical Translations! Need Help Quickly. I have a test tomorrow and I still can't figure out how to do these.

Question

Find a cosine equation having the following characteristics:

amplitude= 5

period= 2pi/3

horizontal shift= -pi/6

vertical shift= 2

Please explain how you figured out how to solve the problem! Thank you in advance, you're really helping me out.

Answer 1

A cosine function in this format:

has an amplitude of A units

has a period of (2π)/B

has a horizontal shift of C units

has a vertical shift of D units

So, an amplitude of 5, period of 2π/3, horizontal shift of -π/6, and a vertical shift of 2 is represented by the following values of A, B, C, and D:

A = 5

B = (2π)/(2π/3) = 3

C = -π/6

D = 2

So the equation becomes:

f(x) or y = (5)cos[3(x - -π/6)] + 2 or, simplifying:

y = 5cos[3(x + π/6)] + 2

You could (if you wanted) also distribute the 3 that is inside the cosine function and get:

y = 5cos(3x + π/2) + 2 but I like the previous version myself.