# Graphing Sine and Cosine _Student_

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```					          Lecture Notes – Math 1113: Pre-Calculus                Section 5.4 – Graphs of Sine and Cosine Functions

§5.4–Graphs of Sine and Cosine Functions

The graph of a sine function is a sine curve. It is also called a periodic function, in that it repeats itself
every so often. The period is the length over which the sine curve performs one cycle of the graph.

Graph of y = sin x
y



x

                                                                    



Graph of y = cos x
y



x

                                                                      



Page 1 of 6
Lecture Notes – Math 1113: Pre-Calculus            Section 5.4 – Graphs of Sine and Cosine Functions

Amplitude of Sine Curves

Typically, sine curves fluctuate between the heights -1 and 1, but sine curves can be amplified(or
dampened) to fluctuate between other values. To accomplish this change, just multiply the function
by a constant.

The amplitude of:

y  a sin x and y  a cos x

Represents half the distance between the
maximum and minimum values of the function
and is given by:

Amplitude = a

Graphing:       Sketch the graph of y  2sin x

Graphing:       Sketch the graph of y   cos x

Page 2 of 6
Lecture Notes – Math 1113: Pre-Calculus                Section 5.4 – Graphs of Sine and Cosine Functions

Period Shift of Sine Curves

The period of the sine curve is 2π. That is the distance it takes to complete one cycle of the sine curve.
This period can be shifted by multiplying the x inside of the sine (or cosine) function by a constant.

Let b be a positive real number.

The period of:     y  a sin bx and y  a cos bx

2
Is given by:       Period 
b

x
Practice Problem:     Calculate the period of y  3cos  
2

Graphing:     Sketch the graph of y  sin  2 x 

Page 3 of 6
Lecture Notes – Math 1113: Pre-Calculus                Section 5.4 – Graphs of Sine and Cosine Functions

Translations of Sine Curves

The sine curves can be moved to anywhere in the plane via a horizontal or vertical translation.

Horizontal Translation (Phase Shift)

Given the functions:     y  a sin( bx  c )     and      y  a cos( bx  c )

The constant c influences the phase shift or horizontal translation of the sine curve.

The graphs of the functions above are the same as y  a sin bx and y  a cos bx

Just shifted by the amount c / b

The number c / b is called the phase shift

1        
Practice Problem: Calculate the phase shift and the left and right endpoints of one cycle for y  sin  x  
2        3

Identify the following:
Graphing:     Sketch the graph of y  3cos( 2 x  4 )                                               Amplitude
     Period
     Left/Right endpoints
for one cycle

Page 4 of 6
Lecture Notes – Math 1113: Pre-Calculus                    Section 5.4 – Graphs of Sine and Cosine Functions

Vertical Translations

By adding a constant to the function, one can raise or lower where the curve is located.

Given the functions:      y  a sin( bx  c )  d and       y  a cos( bx  c )  d

The constant d is the vertical translation of the sine curve.

The graphs of the functions above are the same as y  a sin( bx  c ) and y  a cos( bx  c )

Just shifted up or down by a distance d.

Also, the sine curve will now fluctuate about the horizontal line y = d

Graphing:       Sketch the graph of y  3sin x  2

Page 5 of 6
Lecture Notes – Math 1113: Pre-Calculus               Section 5.4 – Graphs of Sine and Cosine Functions

The General Formula for Sine Curves

y  a sin( bx  c)  d                y  a cos( bx  c)  d

2
Amplitude = a          Period =             Phase Shift = c / b       Vertical Shift = d
b

To graph sine curves:

1. Identify the amplitude, period, phase shift and vertical shift
2. Find the left and right endpoints of one cycle of the sine curve by solving the equations:

bx  c  0 and bx  c  2

3. Then find ¼ mark, ½ mark and ¾ mark of the interval between the left and right endpoints.
4. Sketch the line y = d
5. Then draw the sine curve according to the amplitude about the line y = d

      
Graphing:      Sketch the graph of y  2 cos   x    2
      4

5.4 – Graphs of Sine and Cosine

Pg. 450
3-14, 39-46, 71-75

Page 6 of 6

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