1. Unit 7: Trigonometric Functions
Graphing the Trigonometric Function

2. Graphing Trigonometric Functions
Amplitude: the maximum or minimum vertical distance between the graph and the x-axis. Amplitude is always positive 2

3. 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 = sin x
y = sin x
y = 2 sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Amplitude

4. Graphing Trigonometric Functions
Period: the number of degrees or radians we must graph before it begins again. 4

5. If b > 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 b  0, the period of y = a sin bx is
For b  0, the period of y = a cos bx is also
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period:
period: 2
If b > 1, the graph of the function is shrunk horizontally. y x
period: 4
period: 2
Period of a Function

6. 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
90°
180°
270°
360° I II
III IV
sin θ θ
45°
90°
135°
180°
270°
225° 0°
315° θ
sin θ
π/2 1 π
3π/2 −1 2π
Interactive Sine Unwrap

7. Sine is a periodic function: p = 2π
One period 2π 3π π
−2π −π
−3π
sin θ θ
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.
sin(−θ) = −sin(θ)

8. 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
Sine Function

9. 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
45°
90°
135°
180°
270°
225° 0°
315°
90°
180°
270°
360° IV
cos θ θ
III I II θ
cos θ 1
π/2 π −1
3π/2 2π

10. 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. 1 -1
cos 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 = cos x
Cosine Function

11. Cosine is a periodic function: p = 2π
One period 2π π 3π
−2π −π
−3π θ
cos θ
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.
cos(−θ) = cos(θ)

12. Properties of Sine and Cosine graphs
The domain is the set of real numbers
The rage is set of “y” values such that -1≤ y ≤1
The maximum value is 1 and the minimum value is -1
The graph is a smooth curve
Each function cycles through all the values of the range over an x interval or 2π
The cycle repeats itself identically in both direction of the x-axis 12

13. Sine Graph Example: y=5sin2X Amp=5 Period=2π/2 = π Given : A sin Bx π
Amplitude = IAI
period = 2π/B
Sine Graph
Example:
y=5sin2X
Amp=5
Period=2π/2
= π
π/2 π
π/4
3π/4

14. Cosine Graph Example: y=2cos 1/2 X 4 π Given : A sin Bx 4π
Amplitude = IAI
period = 2π/B
Cosine Graph
Example:
y=2cos 1/2 X
Amp= 2
Period= 2π/(1/2)
4 π 2π 4π π 3π

15. 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.
max
x-int
min 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Example: y = 3 cos x

16. 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 x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
Example : 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)
Graph y = f(-x)

17. 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 bx with b > 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)
Example: y = 2 sin(-3x)

18. 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).

19. More Transformations
Here is the full general form for the sine function:
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.

20. More Transformations To graph a sine or cosine graph:
Graph the original graph with the correct amplitude and period (like section 14.1).
Translate h units horizontally and k units vertically.
Reflect vertically at its new position if a is negative (or reflect first, then translate).

21. Examples
Describe how each graph would be transformed:

22. Examples
State the amplitude and period, then graph: x
-2π 2π

23. Examples
State the amplitude and period, then graph: x
-2π 2π

24. Examples
State the amplitude and period, then graph: x
-2π 2π

25. Examples Write an equation of the graph described:
The graph of y = cos x translated up 3 units, right π units, and reflected vertically.