Graph the following functions .include two full periods .a)y=3cos(2x) and b)y=-4sin(4x) on same graph

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http://www.wolframalpha.com/input/?i=plot+y%3D3cos%282x%29+%2C+y%3D-4sin%284x%29

To graph the functions y = 3cos(2x) and y = -4sin(4x) on the same graph, we can follow these steps:

Step 1: Determine the period
The period of a function is the length of one complete cycle of the graph. For cosine and sine functions, the period is given by the formula T = 2π/b, where b is the coefficient of x.

For y = 3cos(2x), the coefficient of x is 2, so the period is T = 2π/2 = π.
For y = -4sin(4x), the coefficient of x is 4, so the period is T = 2π/4 = π/2.

Step 2: Determine the range
Since both functions involve cosine and sine, their ranges will be between -1 and 1. In this case, the range is [-3, 3] (for y = 3cos(2x)) and [-4, 4] (for y = -4sin(4x)).

Step 3: Create a table of values
To plot the graph, we need to create a table of x and y values for each function. Let's choose values across two full periods, which means we'll select values from 0 to 2π for y = 3cos(2x) and from 0 to π for y = -4sin(4x).

For y = 3cos(2x):
x | y
-----------
0 | 3
π/4 | 0
π/2 | -3
3π/4 | 0
π | 3
5π/4 | 0
3π/2 | -3
7π/4 | 0
2π | 3

For y = -4sin(4x):
x | y
----------
0 | 0
π/8 | -4
π/4 | 0
3π/8 | 4
π/2 | 0

Step 4: Plot the graph
Using the table of values, plot the points for each function on the same graph using the x and y coordinates. Connect the points with smooth curves.

Here is a visual representation of the graph:

```
|
4 |- x x
| x x
|
3 |_ x x
|x x
|
2 |___ x x
|
|
1 |
|
|
|
0 |_________________________
0 π/2 π 3π/2 2π
```

The blue curve represents y = 3cos(2x), and the red curve represents y = -4sin(4x). Both curves have two full periods within the given range.