1. Digital Lesson
Graphs of Trigonometric
Functions

2. Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the 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 of 2π .
6. The cycle repeats itself indefinitely in both directions.
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3. 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. π 3π
x 0 π 2π 2 2
sin x 0 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 y = sin x
3π π 1 π 3π 5π
− −
2 −π 2 2 π 2 2π 2
x
−1
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4. 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. π 3π
x 0 π 2π 2 2
cos x 1 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 y = cos x
3π π 1 π 3π 5π
− −
2 −π 2 2 π 2 2π 2
x
−1
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5. Example: Sketch the graph of y = 3 cos x on the interval [–π, 4π].
Find the key points; graph one cycle; then extend the graph in both
directions for the required interval.
π 3π
x 0 2 π 2 2π
y = 3 cos x 3 0 -3 0 3
max x-int min x-int max
y
(0, 3) (2π, 3)
2
−π 1 π 2π 3π 4π x
−1 π ( 3π , 0)
− 2 ( 2 , 0) 2
−3 ( π, –3)
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6. 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 there is a negative in front (a < 0), the graph is reflected in the x-
axis.
When I ask for y
amplitude I will not 4
ask what kind of
stretch it is. Instead, y = sin x π 3π
I will ask for the 2 π 2 2π x
value of the 1
amplitude. y = 2 sin x
y = – 4 sin x y = 2 sin x
reflection of y = 4 sin x y = 4 sin x
−4
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7. 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 2π .
b
For b > 0, the period of y = a cos bx is also 2π .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y = sin 2π period: 2π
period: π y = sin x x
−π π 2π
If b > 1, the graph of the function is shrunk horizontally.
y y = cos x
1
y = cos x period: 2π
2 −π π 2π 3π 4π x
period: 4π
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8. Use basic trigonometric identities to graph y = f (–x)
Example 1: 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 y = sin (–x)
Use the identity
x
sin (–x) = – sin x π 2π
y = sin x
Example 2: 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
cos (–x) = – cos x x
π 2π
y = cos (–x)
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9. Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x
period: 2 π 2π
amplitude: |a| = |–2| = 2 =
b 3
Calculate the five key points.
x π π π 2π
0 6 3 2 3
y = –2 sin 3x 0 –2 0 2 0
y
( π , 2)
2 2
π π π π 2π 5π
6 6 3 2 3 6 π x
(0, 0) ( π , 0) 2π
−2
3 ( , 0)
( π , -2)
3
6
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10. Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x = .
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x
π
x ≠ kπ + ( k ∈ Ζ ) π 3π
2
2. range: (–∞, +∞) 2 2
x
3. period: π − 3π −π
2 2
4. vertical asymptotes:
π 3π
x = , ( repeatseveryπ )
2 2
period: π
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11. Example: Find the period and asymptotes and sketch the graph
1 π π
of y = tan 2 x x=− y x=
3 4 4
1. Period of y = tan x is .
π
π
→ Period of y = tan 2 x is .
2 −
3π π 1
 ,− 
π
8  8 3 2
2. Find consecutive vertical x
asymptotes by solving for x: π 1
 ,   3π 1 
π π  8 3  ,− 
2x = − , 2x =  8 3
2 2 π π
Vertical asymptotes: x = − , x =
4 4
π π π 3π
3. Plot several points in (0, ) x − 0
2 8 8 8
1 1 1 1
y = tan 2 x − 0 −
4. Sketch one branch and repeat. 3 3 3 3
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