# Sine: Graph

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

## Graph: y = sin x

### Graph

This is the graph of y = sin x.

It goes up and down between 1 and -1.

### Amplitude

The amplitude is the distance
between the center axis and the farthest point
(highest point or lowest point).

See y = sin 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 = sin x is 1.

### Period

The period is the width of a cycle
(= width of the repeating part).

A cycle of y = sin x is
from x = 0 to x = 2π.

So the period of y = sin x is 2π.

Just like y = sin x,
a function that shows repeated cycles
is called a periodic function.

For a periodic function,
if the period is p,
then f(x) = f(x - p).

Sine function is a periodic function.
And the period of y = sin x is 2π.

So sin x = sin (x - 2π).

### One Cycle

For y = sin x,

the amplitude is 1
and the period is 2π.

## Graph: y = a sin bx

### Formula

For y = a sin bx,

the amplitude is |a|
and the period is 2π/|b|.

## Example

### Solution

See y = 3 sin 2x.
The number in front of the sine is 3.

So the amplitude is
|3| = 3.

See y = 3 sin 2x.
The number in the sine is 2.

So the period is
2π/|2| = π.

The amplitude is 3.
The period is π.

y = 3 sin 2x does not show
any horizontal translation.

So, to draw the cycle of the sine function,
first draw the boundaries:
y = 3, y = -3, and x = π.

Draw a sine cycle.
Start from the origin.

The middle point of the cycle is
one half of the period: π/2.

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