Graphs of Sine and Cosine Functions

Question

Write the equation for a cosine function with amplitude

3 pi, period 2, and phase shift -7

Answer 1

y = Acos(Bx+C) + D

Amplitude = |A|

Period = 2π/B

Phase Shift = |C/B| units to the right is C/B is negative

= C/B units left if C/B is positive

In the given problem:

A = 3π

A period of 2 is required, so 2 = 2π/B, which means B = π.

Interpreting a phase shift of -7 to mean 7 units to the left means we want C/B to be positive.

C/B = 7, but B = π, so C = 7π.

So the equation is:

y = 3πcos(πx + 7π) which can also be written as: y = 3πcos[π(x+7)]

Note that D = 0, so there is no vertical shift, i.e. the midline is still y = 0.

Here is a Desmos graph for confirmation:

desmos.com/calculator/arfpqjdxli