1. Graphing SinE and Cosine FUnctions

2. Periodic Function
A periodic function is one which repeats itself over and over (have we studied any periodic functions so far?)
Think ekg’s or wallpaper

3. The period of a periodic function is the “length of the repeat”
The amplitude of a periodic function is
Max - Min 2

4. What is the Period? The amplitude?

5. Complete the Table (Think first though)
Graph y = sin x and y = cos x by completing a table of values. Let x go from 0 to 4π in increments of π/2. Check this on your calculator. Check your window. You can use increments of π. What is the period of each function? The amplitude? The domain? The range? Make sure you have a clear diagram of each in your notes!

6. We will eventually want to graph
y = a sin (bx – c) + d and
y = a cos (bx – c) + d
So let’s try to find out how each of these variables will affect the graph.

7. What does A Do?
What do you think y = 3 sin x will look like? Check on your calculator. What do you think y = -1/2 cos x will look like? So what does |a| represent?

8. And B?
What do you think y = sin 2x will look like? The pattern should begin when 2x = 0, so x = 0. The pattern should end when 2x = 2π, so x = π So the period should be π. Graph this. So when b = 1, period = 2π. when b = 2, period = π. So although not intuitive the period = 2π b

9. Graph y = sin (x/3)
When graphing sin and cos functions, you generally label the intercepts and maximum and minimum values.

10. What about C? What should y = sin (x – π) look like?
It should be the sin graph shifted right π units.
So, c shifts the graph left or right, but we will elaborate in a minute.
This is called the phase shift.

11. Graph y = 4sin(2x + π/2)
First factor so that y = 4sin {2(x + π/4)}
The amplitude is 4, the period is π, and the phase shift is –π/4.
Graph this.

12. Instead of factoring each time, you can find the phase shift by dividing c by b.
Phase shift = c/b
Note that in the formula we subtract c.

13. Finally, what does D do?
It of course shifts the entire graph up or down. We could also say that y = 0 is the mid-line of y = sin x or y = cos x, so when shifted the mid-line becomes y = d.

14. Summary For y = a sin(bx – c) + d and y = a cos(bx – c) + d
amplitude |a|
period π/b
phase shift c/b
mid-line y = d
You might want to draw a schematic of this in your notes.

15. Give the amplitude, period, and phase shift for each graph.
f(x) = 3 sin (4x – 5)
f(x) = -2 cos (πx + 2π/3)

16. Graph f(x) = -2 cos {π x + 2π/3)}
Hint: first graph y = 2 cos π x. Then reflect and shift.

17. Find an equation for a sine function which has a minimum value at (9, -4).