# 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

### 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.