1. 2.8 Phase Shifts and Sinusoidal Curve Fitting
y = Asin(ωx - φ) + B
Notice in this form we use – φ, instead of + h. So this means
if φ > 0, the graph is shifted to the RIGHT, and
if φ < 0, the graph is shifted to the LEFT.
The amount of the horizontal displacement, or phase shift, is φ/ω. How do we get this?
Recall that the period of the sine function is 2π.
sin(ωx - φ) = sin((ωx – φ) + 2π )
The period will begin with ωx – φ = 0  x = φ/ω = starting point of one cycle. We see from this that this shows that the starting point is shifted from 0 to φ/ω.
And end with ωx – φ = 2π  x = 2π/ω + φ/ω = ending point of one cycle.
Also remember that the new period is T = 2π/ω
Example 1
Find amplitude, period, phase shift of y = 3sin(2x –π)
A = 3, ω = 2, φ = π
Amplitude = A = 3
This that the y values range from -3 to 3.
Period T = 2π/ω = 2π/2 = π
This means that the whole sine cycle is squeezed into a period of π.
Phase shift = φ/ω = π/2.
x = φ/ω = starting point of one cycle = π/2.
x = 2π/ω + φ/ω = ending point of one cycle = 2π/2 + π /2 = 3π/2
Since φ > 0, the graph is shifted to the right, so start the cycle at x = φ/ω = π/2.
π/4
π/2
3π/4 π
5π/4
5π/4
3π/2 2π

2. Example 3 Finding a Sinusoidal Functions from Temperature Data
Month
Average Monhly Temp 1
29.7 2
33.4 3 39 4
48.2 5
57.2 6
66.9 7
73.5 8
71.4 9
62.3 10
51.4 11 12 31
We will fit this to the STANDARD sinusoidal function:
y = Asin(ωx - φ) + B
Step 1: To find the amplitude A, we compute
A =
Step 2: Determine the vertical shift, B, by finding the average of the highest and lowest value.
B =
Step 3: Find frequency, ω. Cycle will (hopefully) repeat itself every year, so period = 12 months.
T = Use the fact that T = 2π/ ω to solve for ω.
ω = 2π/T = 2π/12 = π/6
Step 4: Using A=21.9, ω = π/6, and B = 51.6,determine horizontal shift by choosing an arbitrary data point (x,y) from the given table and solving the equation for φ.

3. Let’s choose the first data point, x = 1 (Jan.), y = 29.7
So.. A sine function that fits the data is

4. Homework Sec. 2.8 p.206 #3-27 EOO