14.1 Graph Sine, Cosine and Tangent Functions
Let's graph y = sinx and
(0,1)
y =cosx by filling out the tables
below.
(­1,0)
(1,0)
x
sinx
(0,­1)
0
1
π/2
y = sinx
π
3π/2 2π
π/2 π
3π/2
2π
­1
x
0
π/2
π
3π/2
2π
cosx
1
y = cosx
π/2 π
3π/2
2π
­1
Apr 29­3:37 PM
Properties of y = sinx and cosx
-The domain of each function is ______________.
-The range of each function is ___________.
-The ____________ of each function is half the difference
of the maximum and minimum.
-Each function is ___________, which means its graph has a
repeating pattern. The shortest repeating portion of the graph
is called the ___________. The horizontal length of each
cycle is called the __________.
-The period of each function is ______.
Apr 29­3:49 PM
1
Examples: Determine the amplitude and period of each function
graphed below.
1.)
5
π/4
π/2 3π/4
π 5π/4 3π/2
-5
2.)
π
2π
4π
-π
Apr 29­3:59 PM
Amplitude and Period: The amplitude and period of the graphs
y = asinbx and y = acosbx are as follows:
Amplitude = a
Period = 2π
b
Examples: Graph the following.
1.) y = 4sinx
2.) y = cos4x
Apr 29­3:53 PM
2
Examples: Graph the following.
1.) y = 2sin¼x
2.) y = 2cosπx
Apr 29­4:10 PM
Let's graph y = tanx by filling out the table below.
(0,1)
(­1,0)
(1,0)
(0,­1)
x
0
π/4
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
tanx
1
π/2 π
3π/2
2π
­1
Apr 29­4:16 PM
3
Period and Vertical Asymptotes: The period and vertical
asymptotes of the graph of y = atanbx are as follows:
- The period is π.
b
- The vertical asymptotes are at odd multiples of π .
2b
Examples: Graph one period of the functions below.
2.) y = 4tan2πx
1.) y = 2tan3x
Apr 29­4:19 PM
14.2 Translate and Reflect Trig Graphs
(0,1)
Let's graph y = -sinx and
y =-cosx by filling out the tables
(­1,0)
(1,0)
below.
x
-sinx
(0,­1)
1
0
π/2
y = sinx
π
3π/2 2π
π/2 π
3π/2
2π
­1
x
0
π/2
π
3π/2
2π
-cosx
1
y = cosx
π/2 π
3π/2
2π
­1
Apr 29­3:37 PM
4
Along with reflections, graphs of trig functions can also
translate left/right and up/down.
Translations of Sine and Cosine Graphs
To graph y = asin b(x - h) + k or y = acos b(x - h) + k, follow these
steps:
1.) Identify the amplitude a , the period 2π/b, the horizontal
shift h,the vertical shift k and note any reflection.
2.) Draw the horizontal line y = k, which is called the midline.
3.) Find the five key points by translating the key points of
y = asinbx and y = acosbx in the following order:
-horizontally h units
-reflect (if necessary)
-vertically k units
4.) Draw the graph through the five translated key points.
Apr 29­5:27 PM
Examples:
1.) Graph y = sin4x + 3
2.) y = 4cos(x - π)
Apr 29­5:42 PM
5
3.) y = sin2(x + 2π) - 3
4.) y = -2sin(1/2)(x - π)
Apr 29­5:44 PM
Examples:
1.) Write a cosine equation that represents the graph.
1
π/2
-π/4
π
-1
2.) Write a sine equation that represents the graph.
2
1
-4π
4π
Apr 29­5:50 PM
6
Name: ________________
14.1-14.2 Homework
Graph the following trig functions. Label!
2.) y = -cos2x
1.) y = 2sinx
3.) Fill in the blank.
The graphs of the functions y = sinx and y = cosx both have
a ________ of 2π. They both have an ____________ of 1.
Apr 29­6:21 PM
4.) Write the equation of the graph below.
π
2π
5.) Graph y = -4sinx. Label!
Apr 29­6:23 PM
7
6.) Graph one period of
y = 4tanπx. Label!
7.) Graph one period of
y = 3tan2x. Label!
Fill in the blanks.
8.) The graph of y = cos2(x - 3) is the graph of y = cos2x
translated ____ units to the right.
The graph of y = cos2x + 1 is the graph of y = cos2x
translated ____ units up.
Apr 29­6:28 PM
9.) Graph y = 3cos(x + 3π/4) - 1. Label!
10.) Write a sine equation for the graph below.
4π
8π
Apr 29­6:33 PM
8