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.