Question

Which of the following would best represent a cosine function with an amplitude of 2, a period of pi over 3, and a midline at y = 1?

f(x) = cos(x − pi over 3) + 2
f(x) = 2 cos(x − pi over 3) + 1
f(x) = −2 cos 6x + 1
f(x) = −cos 6x + 2

Answer 1

C. f(x) = – 2 cos 6x + 1

Start by determining the amplitude. Since we've deduced the amplitude is 2, the equation can include either a positive or negative 2 (since amplitude measures absolute value).

Next is the period. The equation for period P is P = (2pi)/b. If P is pi/3, then
pi/3 = (2pi)/b. Thus your b value should be 6.

Finally, the midline would be given by + 1 since adding a unit shifts the function upwards. This means that instead of the highest y value being 2 and the lowest -2, instead you'd have values of 3 and -1.
(3 – 1)/2 = 1 (midpoint theory).

Answer 2

f(x) = −2 cos 6x + 1