## Investigation of the Graphs y = asin(bx+c) and y = acos(bx+c) For Different Values of a, b, and c

### Jennifer Roth

First we want to look at a graph of the simplest equations; y = sin(x),(red), and y=cos(x),(green).

Where do these graphs cross the x and y axis?

Now we want to look at different values of a, b, and c for the equations y = asin(bx+c) and
y = acos(bx+c).

First we will investigate different graphs for y=asin(bx+c), then we will investigate different graphs for y=acos(bx+c), and then we will combine the two.
For each graph, think about where the different graphs crossing the x and y axis and where are the maximum and mimimum values for x and y. In other words, what changes are taking place which are caused by varying the values for a, b, and c?

#### Example 1

Graph of y=asin(bx+c) changing the value of a, and letting b = 1 and c = 0.

The graphs correspond as follows:

Red: y = sin(x)
Green: y = 1/2sin(x)
Blue: y = 2sin(x)
Brown: y = 5sin(x)

#### Example 2

Graph of y=asin(bx+c) changing the value of b, and letting a = 1 and c = 0.

The graphs correspond as follows:

Red: y = sin(x)
Green: y = sin(1/2x)
Blue: y = sin(2x)
Brown: y = sin(5x)

#### Example 3

Graph of y=asin(bx+c) changing the value of c, and letting a = 1 and b = 1.

The graphs correspond as follows:
Red: y = sin(x)
Green: y = sin(x + 1/2)
Blue: y = sin(x + 2)
Brown: y = sin(x + 5)

Now we will look at the same graphs for acos(bx+c)

#### Example 4

Graph of y=acos(bx+c) changing the value of a, and letting b = 1 and c = 0.

The graphs correspond as follows:

Red: y = cos(x)
Green: y = 1/2cos(x)
Blue: y = 2cos(x)
Brown: y = 5cos(x)

#### Example 5

Graph of y=acos(bx+c) changing the value of b, and letting a = 1 and c = 0.

The graphs correspond as follows:

Red: y = cos(x)
Green: y = cos(1/2x)
Blue: y = cos(2x)
Brown: y = cos(5x)

#### Example 6

Graph of y=acos(bx+c) changing the value of c, and letting a = 1 and b = 1.

The graphs correspond as follows:
Red: y = cos(x)
Green: y = cos(x + 1/2)
Blue: y = cos(x + 2)
Brown: y = cos(x + 5)

Now we will combine the graphs of asin(bx+c) and acos(bx+c). First we will combine the graphs for both , varying a, then b, and then c.

### Example 7

Graphs of y = asin(x) and y = acos(x)

The graphs correspond as follows:

Red: y = sin(x)
Green: y = 2sin(x)
Blue: y = cos(x)
Brown: y = 2cos(x)

### Example 8

Graphs of y = sin(bx) and y = cos(bx)

The graphs correspond as follows:

Red: y = sin(x)
Green: y = sin(2x)
Blue: y = cos(x)
Brown: y = cos(2x)

### Example 9

Graphs of y = sin(x + c) and y = cos(x + c)

The graphs correspond as follows:

Red: y = sin(x)
Green: y = sin(x+2)
Blue: y = cos(x)
Brown: y = cos(x+2)

### Example 10

It might be interesting to look at all the graphs combined.

What conclusions can be made about the affect of different values of a, b, and c on the graphs of y=asin(bx+c) and y=acos(bx+c)?