Question

What is the cosine equation of the function shown?

Enter any phase shift as its smallest multiple from the fundamental period.

Answer 1

for these problems, you want to think about the "basic" function and how your new function is changed

they give you cos(x) as the basic function. cos(x) has period 2pi. your function also has period 2pi (notice the peaks at 9pi/4 and pi/4. subtracting this gives us 2pi). so there's no horizontal transformation in front of x. so you know nothing is being multiplied to x

Answer 2

now, normally cos is "centered" at y = 0, but in this case, it's centered at y = -4 (draw a horizontal line at y = -4 to confirm this is the midpoint between the highest and lowest peaks)

this means cos has been vertically translated down 4 units, so you have a -4 at the very end

additionally, because the amplitiude (distance from the center to either peak) is 4, you can multiply by 4 in front of the cos

Answer 3

finally, recall how cos has a peak at (0,1), but here, the peak is at (pi/4, 0). that means it's been translated to the right pi/4 units, so we subtract pi/4 from x. (remember, with horizontal translations, positive is to the left and negative is to the right)