1. Amplitude, Period, & Phase Shift
6.2 Trig Functions
Amplitude, Period, & Phase Shift
3 ways we can change our graphs

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
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
6. The cycle repeats itself indefinitely in both directions of the x-axis.
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Properties of Sine and Cosine Functions

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. -1 1
sin x x
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 x
y = sin x
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Sine Function

4. Graph of Tangent Function: Periodicθ
tan θ
−π/2
π/2
One period: π
Vertical asymptotes
where cos θ = 0 θ
tan θ
−π/2 −∞
−π/4 −1
π/4 1
π/2 ∞
−3π/2
3π/2

5. Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cot θ θ
tan θ ∞
π/4 1
π/2
3π/4 −1 π −∞
−3π/2 -π
−π/2
π/2 π
3π/2

6. Cosecant is the reciprocal of sine
Vertical asymptotes
where sin θ = 0
csc θ θ
−3π
−2π −π π 2π 3π
sin θ
One period: 2π

7. Secant is the reciprocal of cosine
One period: 2π π 3π
−2π 2π −π
−3π θ
sec θ
cos θ
Vertical asymptotes
where cos θ = 0

8. Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [-π,4] on your x-axis
max
x-int
min 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
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Example: y = 3 cos x

9. If |a| > 1, the amplitude stretches the graph vertically.
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 x
y = 2sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
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Amplitude

10. If k > 1, the graph of the function is shrunk horizontally.
The period of a function is the x interval needed for the function to complete one cycle.
For k  0, the period of y = a sin kx is
For k  0, the period of y = a cos kx is also
For k  0, the period of y = a tan kx is
If k > 1, the graph of the function is shrunk horizontally. y x
period:
period: 2
If 0 < k < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2
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Period of a Function

11. 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 x
y = sin (–x)
Use the identity sin (–x) = – sin x
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 x
Use the identity cos (–x) = – cos x
y = cos (–x)
y = cos (–x)
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Graph y = f(-x)

12. Use the identity sin (– x) = – sin x:
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin kx with k > 0
Use the identity sin (– x) = – sin x:
y = 2 sin (–3x) = –2 sin 3x
period: 2 3 =
amplitude: |a| = |–2| = 2
Calculate the five key points. 2 –2
y = –2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
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Example: y = 2 sin(-3x)

13. The Graph of y = Asin(Kx - C)
The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to x = -C/K. The number – C/K is called the phase shift.
amplitude = | A|
period = 2 /K. y
y = A sin Kx
Amplitude: | A| x
Starting point: x = -C/K
Period: 2/B
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14. Example
Determine the amplitude, period, and phase shift of y = 2sin(3x-)
Solution:
Amplitude = |A| = 2
period = 2/K = 2/3
phase shift = -C/K = /3 to the right
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15. Example cont. y = 2sin(3x- )
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16. Amplitude Period: 2π/k Phase Shift: -c/k Vertical Shift
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17. State the periods of each function:1. 2.
4π or 720°
π/2 or 90°

18. State the phase shift of each function:1. 2.
Right phase shift 45°
Left phase shift -90°

19. State the amplitude, period, and phase shift of each function:1. 2. 3. 4. 5. 6.
A = 4, period = 360°,
Phase shift = 0°
A = 4, period = 720°,
Phase shift = 0°
A = NONE, period = 45°,
Phase shift = 0°
A = NONE, period = 90°,
Phase shift = π/2 Right
A = 2, period = 180°,
Phase shift = 0°
A = 3, period = 360°,
Phase shift = 90° Right

20. State the amplitude, period, and phase shift of each function:1. 2.
A = 10, period = 1080°,
Phase shift = 900° Right
A = 243, period = 24°,
Phase shift = 8/3°

21. Write an equation for each function described:
1.) a sine function with amplitude 7, period 225°, and
phase shift -90°
2.) a cosine function with amplitude 4, period 4π, and phase
shift π/2
3.) a tangent function with period 180° and phase shift 25°

22. Graph each function: 1.) 2.)
1.) )
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