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7 October 2019 1 Kidoguchi, Kenneth
sin or sin
sin or cos
y A x B y A x B
y A x B y A x B
1. Graph Functions in Standard Form:
2. Build Sinusoidal Models from Data.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Objectives
7 October 2019 2 Kidoguchi, Kenneth
2T
One cycle of y = A sin(x ), A > 0, > 0
§5.6 Phase Shift; Sinusoidal Curve Fitting
1. Graph Functions of the Form y = A sin(x ) + B
7 October 2019 3 Kidoguchi Kenneth
2T
The shift is to left if < 0 and to the right if > 0
Amplitude = | A |, Period = T = 2/, Phase Shift =/
For graphs of y = A sin(x ) or y = A cos(x ), > 0
One cycle of y = A sin(x ), A > 0, > 0, > 0
§5.6 Phase Shift; Sinusoidal Curve Fitting
1. Graph Functions of the Form y = A sin(x ) + B
7 October 2019 4 Kidoguchi, Kenneth
STEP 1: Determine the amplitude | A | and period T =2/.
STEP 2: Determine the starting point of one cycle of the graph, /.
Determine the ending point of one cycle, / 2/. Divide
the interval [/ , / 2/ ] into four subintervals, each of
length ¼ ∙2/ .
STEP 3: Use the endpoints of the subintervals to obtain five key points
on the graph.
STEP 4: Plot the five key points with a sinusoidal graph of one cycle.
Extend the graph in each direction to make it complete.
STEP 5: If B ≠ 0, apply a vertical shift.
sin or cos
sin or cos
y A x B y A x B
y A x B y A x B
Standard Form:
§5.6 Phase Shift; Sinusoidal Curve Fitting
Steps for Graphing Sinusoidal Functions in Standard Form
7 October 2019 5 Kidoguchi, Kenneth
STEP 1: Determine the amplitude | A | and period T =2/.
STEP 2: Determine the starting point of one cycle of the graph, /.
Determine the ending point of one cycle, / 2/. Divide
the interval [/ , / 2/ ] into four subintervals, each of
length ¼ ∙2/ .
Using the standard form: y = f(x) = A cos(x - ) + B
= A cos(x – /) + B
Find the amplitude, period, and phase shift of y = -3 cos(2x + /4) + 1
and graph the function.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form
7 October 2019 6 Kidoguchi, Kenneth
STEP 4: Plot the five key points with a sinusoidal graph of one cycle.
Extend the graph in each direction to make it complete.
STEP 5: If B ≠ 0, apply a vertical shift.
STEP 3: Use the endpoints of the subintervals to obtain five key points
on the graph.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form y = -3 cos(2x + /4) + 1
7 October 2019 7 Kidoguchi, Kenneth
Step 1: Determine the amplitude | A | and period T =2/
A = -3 and = 2 ⇒ T =
Step 2: Determine the starting point of one cycle of the graph, /.
Determine the ending point of one cycle, / 2/.
Divide the interval [/ , / 2/ ] into four subintervals,
each of length ¼ ∙2/ .
y = -3cos(2(x + /8) + 1 ⇒ / = /8 and / T = 7/8
so the interval of one cycle is: [/8, 7/8].
Using the standard form: y = f(x) = A cos(x - ) + B
= A cos(x – /) + B
Find the amplitude, period, and phase shift of y = -3 cos(2x + /4) + 1
and graph the function.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form
7 October 2019 8 Kidoguchi, Kenneth
3 5 7KeyPoints , 2 , ,0 , ,4 , ,0 , , 2
8 8 8 8 8
STEP 3: Use the endpoints of the subintervals to obtain five key points
on the graph.
1 2 3 4, , , ,
8 8 8 8 8
2 4 6 8, , , ,
8 8 8 8 8 8 8 8
04
8
3 5 7, , , ,
8 8 8 8
4 4 4
8
T T T T TS
Let S be the set of subintervals end points:
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form
7 October 2019 9 Kidoguchi, Kenneth
STEP 5: If B ≠ 0, apply a vertical shift.
STEP 4: Plot the five key points with a sinusoidal graph of one cycle.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form
7 October 2019 10 Kidoguchi, Kenneth
7 October 2019 11 Kidoguchi, Kenneth
Shift left /8
3cos 2 18
x
A = -3
22 T
4 8
3 cos 2 14
y x
cos 1y A x
Find the amplitude, period, and phase shift of y = -3cos(2x + /4) + 1
and graph the function.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph Sinusoidal Functions in Standard Form
7 October 2019 12 Kidoguchi, Kenneth
The specified point in the Figure is (t, y) = (2, 2). Present the analysis to
find a function y = f(t) as a sine function AND a cosine function whose graph
would appear as shown in the Figure.
§5.6 Phase Shift; Sinusoidal Curve Fitting
Graph of Sinusoidal Functions
7 October 2019 13 Kidoguchi, Kenneth
7 October 2019 14 Kidoguchi, Kenneth
7 October 2019 15 Kidoguchi, Kenneth
Assume the data can be fit to:
y = A cos(x – ) + B
§5.6 Phase Shift; Sinusoidal Curve Fitting
2. Build Sinusoidal Models from Data
7 October 2019 16 Kidoguchi, Kenneth
STEP 1: Determine | A |
73.5 29.721.9
2A
max min
2
y yA
§5.6 Phase Shift; Sinusoidal Curve Fitting
2. Build Sinusoidal Models from Data
7 October 2019 17 Kidoguchi, Kenneth
STEP 2: Determine B
73.5 29.751.6
2B
max min
2
y yB
§5.6 Phase Shift; Sinusoidal Curve Fitting
2. Build Sinusoidal Models from Data
7 October 2019 18 Kidoguchi, Kenneth
21.9cos 51.66
y x
2
12 6
STEP 3: Determine 2/T
§5.6 Phase Shift; Sinusoidal Curve Fitting
2. Build Sinusoidal Models from Data
7 October 2019 19 Kidoguchi, Kenneth
21.9cos 1 51.66
y x
STEP 4: Determine the horizontal
shift of the function by using
the period of the data.
Divide the period into four
subintervals of equal length.
Determine the x-coordinate
for the maximum of the
sinusoid and the x-
coordinate for the maximum
of the data. Use this
information to determine the
value of the phase shift, /.
§5.6 Phase Shift; Sinusoidal Curve Fitting
2. Build Sinusoidal Models from Data
7 October 2019 20 Kidoguchi, Kenneth
7 October 2019 21 Kidoguchi, Kenneth
§5.6 Phase Shift; Sinusoidal Curve Fitting
Fit a Sinusoidal Model with a Graphing Utility
7 October 2019 22 Kidoguchi, Kenneth
§5.6 Phase Shift; Sinusoidal Curve Fitting
Fit a Sinusoidal Model with a Graphing Utility
7 October 2019 23 Kidoguchi, Kenneth
§5.6 Phase Shift; Sinusoidal Curve Fitting
Fit a Sinusoidal Model with a Graphing Utility
7 October 2019 24 Kidoguchi, Kenneth
§5.6 Phase Shift; Sinusoidal Curve Fitting
Fit a Sinusoidal Model with a Graphing Utility
7 October 2019 25 Kidoguchi, Kenneth
21.145556sin(0.5495633 2.350732) 51.196763Casioy x
21.9cos 1 51.66
eyebally x
§5.6 Phase Shift; Sinusoidal Curve Fitting
Fit a Sinusoidal Model with a Graphing Utility
