The document discusses graphs of sine and cosine functions. It explains that the sine graph oscillates between y-values of -1 and 1 and includes the point (0,0), while the cosine graph includes the point (0,1). The amplitude describes the graph's vertical stretch or shrink and the period describes how long it takes to repeat. Changing the coefficient in front of x changes the amplitude and period. Examples are provided of sketching sine and cosine graphs over one period and writing a function from a given graph.

1. 6.3 Graphs of Sine and Cosine
Chapter 6 Circular Functions and Their Graphs

2. Concepts and Objectives
 Be able to graph the sine and cosine functions on the
interval [−2, 2]
 Be able to identify how the graphs of the sine and cosine
change due to changes in
 Amplitude
 Period

3. Graphing the Sine Function
 Because the sine is a circular function, while  can be
any number (positive or negative), the value of sin 
only goes from –1 to 1. (It is also considered a periodic
function.)
 To graph a circular function, we can take the angle as
our x-coordinate and the corresponding coordinate from
the unit circle as our y-coordinate.

4. Graphing the Sine Function
 If we graph these values on a coordinate plane, we can
see how this function curves:
 The key to graphing the sine function is that it includes
the point 0, 0.
sin x = y

5. Graphing the Cosine Function
 The cosine function is very similar to the sine. Let’s look
at a graph of it:
 The key to graphing the cosine function is that it
includes the point 0, 1.
cos x = y

6. Changing the Basic Graph
 Amplitude describes how the basic graph is changed
vertically (taller or shorter or inverted). The amplitude
of a periodic function is half the difference between the
maximum and minimum values.
 Period describes the “cycle” of the graph—how long it
takes for the graph to start repeating.
 The period of the sine or cosine is 2.

7. Amplitude
 Compare the graphs of and
 Notice that both graphs cross the x-axis at the same
places and that both graphs peak at the same places.
siny x  n2siy x

8. Amplitude
 The graph of y = a sin x or y = a cos x, with a  0, will
have the same shape as the graph of y = sin x or y = cos x,
respectively, except with the range [–a, a]. The
amplitude is a.
 Example: What is the amplitude of ? What is
the range of the graph?

1
cos
3
y x

9. Amplitude
 The graph of y = a sin x or y = a cos x, with a  0, will
have the same shape as the graph of y = sin x or y = cos x,
respectively, except with the range [–a, a]. The
amplitude is a.
 Example: What is the amplitude of ? What is
the range of the graph?

1
cos
3
y x
1
Amplitude:
3
 
  
1 1
Range: ,
3 3

10. Period
 Compare the graphs of y = cos x and y = cos 2x.
 The two graphs have the same height, but the second
graph is “compressed”.

11. Period
 For b > 0, the graph of y = sin bx will resemble that of
y = sin x, but with period . Also, the graph of
y = cos bx will resemble that of y = cos x, but with period
 To calculate the period of a function, substitute the
coefficient of x in for b and reduce.
2
b
2
b

12. Sketching a Circular Graph
 Example: Sketch the graph of over one
period.
 2sin
2
x
y

13. Sketching a Circular Graph
 Example: Sketch the graph of over one
period.
The coefficient of x is , so the period is
Now, divide the interval [0, 4] into four equal parts and
evaluate the function at those values.
 2sin
2
x
y
1
2

 
2
4
1
2

14. Sketching a Circular Graph
 Example: Sketch the graph of over one
period.
 2sin
2
x
y
x 0  2 3 4
2
x
sin
2
x
2sin
2
x
0
0
0

2
1
–2

0
0
3
2
–1
2
2
0
0

15. Sketching a Circular Graph
 Example: Sketch the graph of over one
period.
Notice that when a is negative, the graph is reflected
across the x-axis.
 2sin
2
x
y

16. Writing a Function From a Graph
 To write a function from the graph:
 Count from the midline to a peak – this is the
amplitude (a).
 Identify the beginning and end of one period. Find
this length. To calculate b, re-write as ,
and solve for b.
 If the graph crosses the y-axis at the midline, use the
sine function; if it crosses the y-axis at a peak, use
cosine.
 Write your function.


2
period
b

17. Writing a Function From a Graph
 Example: Write a function to describe the graph below.

18. Writing a Function From a Graph
 Example: Write a function to describe the graph below.
a = 1.5
Period goes from 0 to , so
The graph crosses the
midline at the y-axis, so we
will use sine.

 

2
2b
 1.5sin2f x x

19. Classwork
 College Algebra
 Page 593: 16-40 (4), page 581: 52-66 (even),
page 567: 74-90 (even)