Transcript 14.1 Graphing Sine, Cosine and Tangent Functions

14.1 Graphing Sine, Cosine and
Tangent Functions
Algebra 2
The graph of y=sin x
The graph of y=cos x
Characteristics of y=sinx and
y=cosx
The domain is all real numbers
 The range is -1 ≤ y ≤ 1
 The function is periodic-the graph has a
repeating pattern. (Shortest repeating
pattern called a cycle and the horizontal
length is called the period) Both have a
period of 2π.

Characteristics of y=sinx and
y=cosx
The maximum value of y=sinx is M=1 and

x

 2 n
occurs when
2
 The maximum value of y=cosx is M=1
and occurs when x=2nπ.
 The minimum value of y=sinx is m=-1 and
3
x

 2 n
occurs at
2
 The minimum value of y=cosx is m=-1
and occurs when x=(2n+1)π

Characteristics of y=sinx and
y=cosx

The amplitude of both functions is
1
M  m   1
2

Amplitude is half the height of the graph.
Characteristics of y=a sin bx and
y=a cos bx

The amplitude and period of the graphs
of y = a sin bx and y = a cos bx where a
and b are nonzero numbers are as
follows.
amplitude= a
period=
2
b
Examples:

Graph the functions
◦
1
y  cos x
2
◦
1
y  sin x
2
◦
y  2 sin
x
4
Examples:

Give the amplitude, period. And five key
points of the graph of each function.
◦
y  sin x
◦
y  3 cos x
◦
x
y  2 sin
4
Definition

Frequency- the number of cycles per unit
of time (frequency is the reciprocal of
the period)
Examples:

A tuning fork vibrates with frequency
f=880 hertz (cycles per second.) You
strike the tuning fork with a force that
produces a maximum pressure of 4
pascals.
◦ Write a sine model that gives the pressure P
as a function of t (in seconds).
◦ Graph the model.
Examples:

You pluck the string of a violin so that it
vibrates with frequency f = 660 hertz
(cycles per second.) The force of the
pluck produces a maximum pressure of 2
pascals. Write a sine model that gives the
pressure P as a functions of time t (in
seconds). Then give the amplitude and
period of the function's graph.
Tangent Functions

The graph of y=tanx has the following
characteristics.
◦ The domain is all real numbers except odd

multiples of . At 2 , thegraph has vertical
2
asymptotes
◦ The range is all real numbers.
◦ The graph has a period of π
Characteristics of y=a tan bx

If a and b are nonzero real numbers, the
graph of y= a tan bx has these
characteristics.

◦ The period is
b
◦ There are vertical asymptotes at odd
multiples of 
2b
Examples:

Graph the functions.
◦ y  3 tan

2
x
1
1
◦ y  tan x
2
3
Question

How do you find the amplitude, period
and vertical asymptotes of a sine, cosine,
or tangent function from its equation?