1. Warm-up:
Solve for x.
HW: Graphing Sine and Cosine Functions

2. SINE AND COSINE FUNCTIONS
4.5 GRAPHS OF
SINE AND COSINE FUNCTIONS
Objective:
Graph sine and cosine functions
Find the amplitude and period of the sine and cosine function
Translations of sine and cosine functions
Visual Aid Introduction:

3. We are interested in the graph of y = f(x) = sin x
Start with a "t" chart and let's choose values from our unit circle and find the sine values. x y 1
- 1
x y = sin x

4. y = f(x) = sin x choose more values x y = sin x join the points y 1 x
- 1
If we continue picking values for x we will start to repeat since this is periodic.

5. Here is the graph y = f(x) = sin x showing from -2 to 6
Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 y 2 2
y = sin x x
It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function)

6. What are the x intercepts?
Where does sin x = 0?
…-2, -, 0, , 2, 3, 4, 5, 6, . . . y
y = sin x x
Where is the function maximum?
Where does sin x = 1?

7. Where is the function minimum?
Where does sin x = -1? y
y = sin x x

8. Knowing transformations and the graph of y = sin x, what do you suppose y = sin x + 2 looks like?
The function value (or y value) is just moved up 2.
y = 2 + sin x y
y = sin x x

9. Knowing transformations and the graph of y = sin x, what do you suppose y = sin (x + /2) looks like?
This is a horizontal shift by –/2 (left /2) y
y = sin x x
y = sin (x + /2)

10. Knowing transformations and the graph of y = sin x, what do you suppose y = - sin (x ) +1 looks like?
This is a reflection about the x axis (shown in green) and then a vertical shift up one.
y = 1 - sin (x ) y
y = - sin x x
y = sin x

11. What would the graph of y = f(x) = cos x look like?
We could do a "t" chart and let's choose values from our unit circle and find the cosine values.
plot these points x y 1
- 1
x y = cos x
We could have used the same values as we did for sine but picked ones that gave us easy values to plot.

12. y = f(x) = cos x Choose more values. x y = cos x plot these points y 1
- 1
cosine will then repeat as you go another loop around the unit circle

13. Here is the graph y = f(x) = cos x showing from -2 to 6
Here is the graph y = f(x) = cos x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2
It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function)

14. Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

15. What are the x intercepts?
Where does cos x = 0?
Where is the function maximum?
Where does cos x = 1?
…-4, -2, , 0, 2, 4, . . .

16. Where is the function minimum?
Where does cos x = -1?
…-3, -, , 3, . . .

17. Let's try y = 3 - cos (x - /4) y = - cos x y = cos x
You could graph transformations of the cosine function the same way you've learned for other functions.
moves right /4
moves up 3
Let's try y = 3 - cos (x - /4)
y = - cos x
y = cos x
reflects over x axis
y = 3 - cos x
y = 3 - cos (x - /4)

18. amplitude of this graph is 2
What would happen if we multiply the function by a constant?
All function values would be twice as high
y = 2 sin x
amplitude is here
amplitude of this graph is 2
y = 2 sin x
y = sin x
The highest the graph goes (without a vertical shift) is called the amplitude.

19. For y = A cos x and y = A sin x, A  is the amplitude.
What is the amplitude for the following?
y = 4 cos x
y = -3 sin x
amplitude is 3
amplitude is 4

20. The last thing we want to see is what happens if we put a coefficient on the x.
y = sin 2x
y = sin 2x
y = sin x
It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

21. What do you think will happen to the graph if we put a fraction in front?
y = sin 1/2 x
y = sin x
The period for one complete cycle is twice as long or 4

22. The period T = So if we look at y = sin x the  affects the period.
This will be true for cosine as well.
What is the period of y = cos 4x?
y = cos x
This means the graph will "cycle" every /2 or 4 times as often
y = cos 4x

23. absolute value of this is the amplitude
Period is 2 divided by this

24. Graphing Calculator!

25. Sneedlegrit: Graph from-2π to 2π. Remember the rules of graphing.
sin(x + π/2) + 2
HW: Graphing Sine and Cosine Functions