Download Graphing Sine and Cosine

Unit 7: Trigonometric Functions
Graphing the Trigonometric Function
Graphing Trigonometric Functions

Amplitude: the maximum or minimum vertical
distance between the graph and the x-axis.
Amplitude is always positive
2
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
4
y = 2 sin x
y = 4 sin x
Graphing Trigonometric
Functions

Period: the number of degrees or radians we must
graph before it begins again.
4
The period of a function is the x interval needed for the
function to complete one cycle.
For b  0, the period of y = a sin bx is 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y  sin 2
period: 2
period: 
y  sin x x


2
If b > 1, the graph of the function is shrunk horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
The sine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where y = sin θ. As the particle moves
through the four quadrants, we get four pieces of the sin graph:
I. From 0° to 90° the y-coordinate increases from 0 to 1
II. From 90° to 180° the y-coordinate decreases from 1 to 0
III. From 180° to 270° the y-coordinate decreases from 0 to −1
IV. From 270° to 360° the y-coordinate increases from −1 to 0
sin θ
y
90°
135°
45°
II
I
I
0°
180°
II I
II
x
0
90°
180°
IV
225°
III
315°
270°
Interactive Sine Unwrap
360° θ
270°
IV
θ
sin θ
0
0
π/2
1
π
0
3π/2
−1
2π
0
Sine is a periodic function: p = 2π
sin θ
−3π
−2π
−π
0
π
2π
One period
2π
sin θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
sin θ is an odd function; it is symmetric wrt the origin.
  Domain, sin(−θ) = −sin(θ)
3π θ
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2
1

3
2
2
5
2
x
The cosine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where x = cos θ. As the particle moves
through the four quadrants, we get four pieces of the cos graph:
I. From 0° to 90° the x-coordinate decreases from 1 to 0
II. From 90° to 180° the x-coordinate decreases from 0 to −1
III. From 180° to 270° the x-coordinate increases from −1 to 0
IV. From 270° to 360° the x-coordinate increases from 0 to 1
y
cos θ
90°
135°
45°
II
I
I
0°
180°
II I
IV
225°
315°
270°
IV
x
0
90°
270°
180°
II
III
θ
360°
θ
cos θ
0
1
π/2
0
π
−1
3π/2
0
2π
1
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
cos x
1
2
0
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = cos x
y
3

2



1

2
2
1

3
2
2
5
2
x
Cosine is a periodic function: p = 2π
cos θ
θ
−3π
−2π
−π
0
π
One period
2π
cos θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
cos θ is an even function; it is symmetric wrt the y-axis.
  Domain, cos(−θ) = cos(θ)
2π
3π
Properties of Sine and Cosine graphs
1. The domain is the set of real numbers
2. The rage is set of “y” values such that -1≤ y ≤1
3. The maximum value is 1 and the minimum value
is -1
4. The graph is a smooth curve
5. Each function cycles through all the values of the
range over an x interval or 2π
6. The cycle repeats itself identically in both
direction of the x-axis
12
Given : A sin Bx
Sine Graph
 Amplitude = IAI
 period = 2π/B
Example:
y=5sin2X
› Amp=5
π/2
π/4
› Period=2π/2
=π
3π/4
π
Given : A sin Bx
 Cosine Graph
 Amplitude = IAI
 period = 2π/B
 Example:
y=2cos 1/2 X
› Amp= 2
2π
› Period= 2π/(1/2)
4π
π
3π
4π
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
x
y = 3 cos x
y

(0, 3)
2
1

0
3
2
0
max

-3
x-int min
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
2
( 3 , 0)
2
3
4 x
Use basic trigonometric identities to graph y = f (–x)
Example : Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
Example : Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
b
Calculate the five key points.
x
0
y = –2 sin 3x
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2

(  ,-2)
6
2
2
3

2
(  , 0) 2
3
( , 0)
3
5
6

x
More Transformations
We have seen two types of transformations on trig
graphs: vertical stretches and horizontal stretches.
There are three more: vertical translations (slides),
horizontal translations, and reflections (flips).
More Transformations
Here is the full general form for the sine function:
y  k  a sin bx  h 
Just as with parabolas and other functions, h and k
are translations:

h slides the graph horizontally (opposite of sign)

k slides the graph vertically
Also, if a is negative, the graph is flipped vertically.
More Transformations
 To graph a sine or cosine graph:
1.
Graph the original graph with the correct
amplitude and period (like section 14.1).
2. Translate h units horizontally and k units
vertically.
3. Reflect vertically at its new position if a is negative
(or reflect first, then translate).
Examples
 Describe how each graph would be transformed:
1.
y  2  sin x
2.


y  cos x  
2

3.
y  2  sin( x   )
Examples
 State the amplitude and period, then graph:
y  2  cos( x)
x
-2π
2π
Examples
 State the amplitude and period, then graph:


y   sin  x  
2

x
-2π
2π

Examples
State the amplitude and period, then graph:
1
y  2  sin x
2
x
-2π
2π
Examples
 Write an equation of the graph described:

The graph of y = cos x translated up 3 units,
right π units, and reflected vertically.