# Cosine: Graph

How to graph the given cosine function: amplitude and period of y = cos x and y = a cos bx, 1 example, and its solution.

## Graph: y = cos x

### Graph

This is the graph of y = sin x.

Just like y = sin x,

it goes up and down between 1 and -1.

The difference is the starting point.

y = sin x starts from the origin.

y = cos x starts from the highest point (1, 0).

### Amplitude

The amplitude is the distance

between the center axis and the farthest point

(highest point or lowest point).

See y = cos x.

The center axis is the x-axis: y = 0.

The farthest points are

y = 1 (highest point) or y = -1 (lowest point).

So the amplitude of y = cos x is 1.

### Period

The period is the width of a cycle

(= width of the repeating part).

A cycle of y = cos x is

from x = 0 to x = 2π.

So the period of y = cos x is 2π.

Just like a sine function,

cosine function is also a periodic function.

The period of y = cos x is 2π.

So cos x = cos (x - 2π).

### One Cycle

For y = cos x,

the amplitude is 1

and the period is 2π.

## Graph: y = a cos bx

### Formula

For y = a cos bx,

the amplitude is |a|

and the period is 2π/|b|.

## Example

### Example

### Solution

See y = 2 cos 3x.

The number in front of the cosine is 2.

So the amplitude is

|2| = 2.

See y = 2 cos 3x.

The number in the cosine is 3.

So the period is

2π/|3| = 2π/3.

The amplitude is 2.

The period is 2π/3.

y = 2 cos 3x does not show

any horizontal translation.

So, to draw the cycle of the cosine function,

first draw the boundaries:

y = 2, y = -2, and x = 2π/3.

Draw a cosine cycle.

Start from the highest point: (2, 0).

The middle point of the cycle is

one half of the period: π/3.

So this graph is the cycle of y = 2 cos 3x.