Trig Packet Notes #2 ­ Graphing.notebook
Graphing Trig Functions
February 27, 2017
Name: ______________________
Objectives: Students will be able to graph sine, cosine and tangent functions and translations of
these functions.
y = sinx
x
(0,1)
sinx
1
0
(1,0)
(­1,0)
(0,­1)
π/2
π/2 π
π
3π/2
2π
­1
3π/2
2π
x
cosx
y = cosx
1
0
π/2
π/2 π
π
3π/2
3π/2
2π
­1
2π
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
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
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
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
Examples: Graph the following.
1.) y = 2sin¼x
2.) y = 2cosπx
Apr 29­4:10 PM
Translations/Reflections of Trig Functions
(0,1)
x
-sinx
y = -sinx
(1,0)
(­1,0)
0
(0,­1)
π/2
1
π
π/2 π
3π/2
2π
3π/2
­1
2π
x
0
π/2
π
3π/2
2π
-cosx
y = -cosx
1
π/2 π
3π/2
2π
­1
Apr 29­3:37 PM
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
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)
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
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
3.) y = sin2(x + π/2) - 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
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
Graphing Reciprocal Trig Functions
y = cscx
y = secx
Mar 1­12:16 PM
Examples
Graph.
1.) y = 2csc(x - π)
2.) y = -sec[2(x - π/2)] + 1
Mar 1­12:20 PM
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
Let's graph y = tanx by filling out the table below.
(0,1)
(1,0)
(­1,0)
(0,­1)
x
0
π/4
tanx
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
1
π/2 π
3π/2
2π
­1
Apr 29­4:16 PM
Period and Vertical Asymptotes: The period and vertical
asymptotes of the graph of y = atanbx are as follows:
- The period is π.
π
b
- The vertical asymptotes are at odd multiples of 2b
Examples Graph one period of the functions below.
1.) y = 2tan3x
2.) y = 4tan2πx
Apr 29­4:19 PM
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
y = cotx
Examples Graph.
1.) y = 2cotx + 1
Mar 1­12:16 PM
2.) y = cot(x - π/4) + 1
3.) y = -tan[2(x + π/8)] - 1
Mar 1­12:48 PM
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Trig Packet Notes #2 ­ Graphing.notebook
Graph Trig Functions Homework
February 27, 2017
Name: ________________
Graph the following trig functions. Label!
1.) y = 2sinx
2.) y = -cos2x
3.) Fill in the blank.
The graphs of the functions y = sinx and y = cosx both have
a ________ of 2 π. They both have an ____________ of 1.
Apr 29­6:21 PM
4.) Write both a sine and cosine equation of the graph below.
π
2π
5.) Graph y = -4sinx. Label!
Apr 29­6:23 PM
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Trig Packet Notes #2 ­ Graphing.notebook
6.) Graph one period of
y = 4tanπx. Label!
February 27, 2017
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/2) - 1. Label!
10.) Write a sine equation for the graph below.
4π
8π
Apr 29­6:33 PM
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
Graph.
11.) y = -4cos(x + π) - 1
12.) y = 2sin[2(x - π/2)] + 1
Mar 1­12:20 PM
Graph.
13.) y = 3sec(x + π)
14.) y = csc[4(x - π/2)] + 1
Mar 1­12:20 PM
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Trig Packet Notes #2 ­ Graphing.notebook
February 27, 2017
Graph.
15.) y = cotx - 1
16.) y = 2tan[π(x + 1/2)]
Mar 1­12:20 PM
17.) Write a sine function with a period of π, an amplitude of 3 and
a vertical shift up 2.
18.) Write a cosine function with a period of π/2, a reflection over
the x-axis, an amplitude of 4 and a vertical shift down 2.
19.) Each branch of y = secx and y = cscx is a curve. Explain why
these curves cannot be parabolas. Hint: Do parabolas have
asymptotes?
Mar 1­1:02 PM
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