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.