Graphing Sine and Cosine
Functions
TRIGONOMETRY, 4.0: STUDENTS GRAPH
FUNCTIONS OF THE FORM F(T)=ASIN(BT+C)
OR F(T)=ACOS(BT+C) AND INTERPRET A, B,
AND C IN TERMS OF AMPLITUDE,
FREQUENCY, PERIOD, AND PHASE SHIFT.
Graphing Sine and Cosine Functions
Objectives
Key words
Graph the equations of
sine and cosine
functions given the
amplitude, period,
phase shift, and vertical
translation
2. Write equations given a
graph.
3. Graph compound
functions
 Midline
1.
 Amplitude
 Maximum
 Minimum
 Period
 Sine curve
 Cosine curve
 Phase shift
Quick check!
 Can you find the distance between two numbers?
 Can you find the midpoint between two numbers?
1: Graphing Sine and Cosine Functions
Order does matter!
Draw the vertical shift,
k, and graph the
midline y=k. Use a
solid line.
2. Draw the amplitude,
𝐴 . Use dashed lines to
indicate the maximum
and minimum values of
the function.
1.
y=A sin[B(θ-h)]+k
y=A cos[B(θ-h)]+k
3. Draw the period of
2𝜋
,
𝐵
the function,
and
graph the appropriate
sine or cosine curve.
4. Draw the phase shift,
h, and translate the
graph accordingly.
1: Graphing Sine and Cosine Functions
State the amplitude, period, phase shift, and vertical
shift for y = 4cos(x / 2 + π) - 6. Then graph the
function.
1: Graphing Sine and Cosine Functions
State the amplitude, period, phase shift, and vertical
shift for y = 4cos(x / 2 + π) - 6. Then graph the
function.
 Amplitude is 4
 Period is 4π
 Phase shift is -2π
 Vertical shift is -6
1: Graphing Sine and Cosine Functions
State the amplitude, period, phase shift, and vertical
shift for y = 2cos(x / 4 + π) - 1. Then graph the
function.
1: Graphing Sine and Cosine Functions
State the amplitude, period, phase shift, and vertical
shift for y = 2cos(x / 4 + π) - 1. Then graph the
function.
 Amplitude is 2
 Period is 8π
 Phase shift is -4π
 Vertical shift is -1
2: Write Equations of Sine and Cosine
Order does matter!
Determine the
vertical shift, k, from
the midline y=k.
2. Determine the
amplitude, 𝐴 . From
the maximum and
minimum values of
the function.
1.
y=A sin[B(θ-h)]+k
y=A cos[B(θ-h)]+k
3. Determine the period
2𝜋
,
𝐵
of the function,
from one complete
interval.
4. Determine the phase
shift, h, from either
sine and/or cosine.
2: Write Equations Example
State the amplitude, period, phase shift, and
vertical shift for the graph of:
2: Write Equations Example
State the amplitude, period, phase shift, and
vertical shift for the graph of:
 The amplitude is 2 or 2. The period is
 The phase shift is −
𝜋
1
2
2𝜋
1
2
or 4.
or -2. The vertical shift is +3
 y = 2 cos ( /2 + ) + 3
or
 y = 2 cos (1/2( + 2)) + 3
2: Write Equations Example
YOU TRY! State the amplitude, period, phase
shift, and vertical shift for the graph of:
2: Write Equations Example
YOU TRY! State the amplitude, period, phase
shift, and vertical shift for the graph of:
 Vertical shift is 0, midline y=0
 Amplitude is 3
 Period is 2 π/3
 Phase shift is π/3
 f(x) = 3cos(3x + π)
3: Graph Compound Functions
Types of Compound
Functions
 Compound functions
may consist of sums or
products of
trigonometric
functions or other
functions.
For Example:
 𝑦 = sin 𝑥 ∙ cos 𝑥
 Product of trigonometric
functions
 𝑦 = cos 𝑥 + 𝑥
 Sum of a trigonometric
function and a linear
function.
3: Graph Compound Functions
Graph y = x + sin x.
3: Graph Compound Functions
Graph y = x + sin x.
 First create a table of each graph: y = x or y = sin x
x
sin x x + sin x
0
0
0
/2 + 1
1
/2
2.57
0

 3.14
3/2 - 1
3/2 -1
3.71
0
2
2 6.28
5/2 + 1
1
5/2
8.85
3: Graph Compound Functions
YOU TRY: Graph y = x + cos x.
 First create a table of each graph: y = x or y = cos x
3: Graph Compound Functions
YOU TRY: Graph y = x + cos x.
 First create a table of each graph: y = x or y = cos x
x
cos x x + cos x
0
1
1
0
/2
1.57
/2
-1

 -1 2.14
0
3/2 4.71
3/2
2 +1
1
2
7.28
5/2
0
5/2
7.85
Conclusion
Summary
Assignment
 Now you know how to
 6.5 Translations of Sine
graph sinusoidal
functions
 Ask questions while
you finish the
assignment
 Finish missing work
 Exam Thursday/Friday
and Cosine Functions

pg383#(14-20 ALL, 21-37
ODD, 42,45 EC)
 Problems not finished
will be left as
homework.