# 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

### 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.