Algebra II Trigonometry
NOTES
Chapter 13
(GraphingTrigonometric
Functions)
(2014)
Graphing Unit Algebra Trigonometry
Page 49
Graphing Unit Algebra Trigonometry
Page 50
Assignment #11:
Name: __________________________
Graph the A: function, then using a different color, graph the B: function.
Describe the transformation, as a horizontal translation right or left, a
vertical translation up or down, a horizontal compression or stretch or a
vertical compression or stretch. Then, answer the questions below each
graph.
1) A: f(x) = x2
B: f(x) = x2 + 5
Parent
Translation (right4)
(-2,4)
(0,0)
(2,4)
(-2+4,4)= (-2,4)
(0+4,0)= (4,0)
(2+4,4)= (6,4)
i) Describe the Transformation:
Translate VERTICALLY up 5 add 5 to
each y value.
ii) Don’t graph, but describe the transformation between:
f(x) = sin(x) and f(x) = sin(x) + 2
Vertically translate UP 2
iii) So, given the general form of f(x) = sinx + d, what would you say ‘d’
does to the function.
‘d’ vertically translates, up or down.
Graphing Unit Algebra Trigonometry
Page 51
2) A: f(x) = (x)2 B: f(x) = (x - 4)2
Parent
Translation (right4)
(-2,4)
(0,0)
(2,4)
(-2+4,4)= (-2,4)
(0+4,0)= (4,0)
(2+4,4)= (6,4)
i) Describe the Transformation:
TRANSLATE HORIZONTALLY to the right 4
ii) Don’t graph, but describe the transformation between:
f(x) = sin(x) and f(x) = sin(x - 90o)
TRANSLATE HORIZONTALLY to the right 90o
iii) So, given the general form of f(x) = sin(x – c), what would you say
‘c’ does to the function.
TRANSLATE HORIZONTALLY to the right or
left.
Graphing Unit Algebra Trigonometry
Page 52
3) A: f(x) = |x|
B: f(x) = 3|x|
i) Describe the Transformation:
Vertically STRETCH by 3
ii) Don’t graph, but describe the transformation between:
f(x) = sin(x) and f(x) = 3sin(x)
Vertically stretch by 3
iii) So, given the general form of f(x) = asin(x), what would you say
‘a’ does to the function.
It will vertically stretch the function
Graphing Unit Algebra Trigonometry
Page 53
1
4
4) A: f(x) = (x)2 B: f(x) = ( x)2
i) Describe the Transformation:
Horizontally Stretch by 4
ii) Don’t graph, but describe the transformation between:
f(x) = sin(x) and f(x) = sin(3x)
Horizontally compress by 3
iii) So, given the general form of f(x) = sin(bx), what would you say
‘b’ does to the function.
‘b’
either horizontally stretches or
compresses.
(If 0<b<1, then stretch…. If b> 1 compresses.)
{end of assignment}
Graphing Unit Algebra Trigonometry
Page 54
Linguini Trig Graphs
Materials:
 7 pieces of linguini
Piece of construction paper
Protractor
Setup: Groups of 3 people
y = cosx
0o
y = sinx
1. Using a protractor to determine angle measures, mark every 15 degrees around the circle
using the positive x axis as 0o. Label each mark with its positive angle measure.
2. Drop perpendicular for each of the angle measures.
3. Label each of the x-axes as angle in degrees (15, 30, 45, …..).
Activity: You are now going to create the graphs of y = sin(x) and y = cos(x), using the unit circle
and right triangles.
Procedure:
1. Use a piece of spaghetti to represent the length of the radius of the circle. This linguini’s
length represents 1 “linguini unit”. Break the linguini to be the length of the radius
2. Place the radius at the center to 0o. Transfer this length to the top graph at 0o on the y =
cosx graph. Make a point at (0,1)
Graphing Unit Algebra Trigonometry
Page 55
3. Since this is a unit circle, the length of the horizontal leg of the triangle is equal to
cos(150). Move the piece of linguini that was the horizontal leg of the triangle to the top
axes where you are graphing cosine.
4. Place the piece of linguini perpendicular to the x-axis at 150, with one end of it on the xaxis and the other above the axis. Make a dot on the paper at the top of the piece of
linguini to show the length of the horizontal leg of the 150 triangle.
4. Repeat steps 2 and 3 for each of the angle measures…. (remember what happens when
you go left horizontally )
5. Now, looking at the y = sinx graph….. What is the vertical component at 0o? Place a point
there.
6. Since this is a unit circle, the length of the vertical leg of the triangle is equal to sin(150).
Break a piece of linguini for each vertical component. Move the piece of linguini that was
the vertical leg of the triangle to the bottom axes where you are graphing sine.

Try to use as few spaghetti lengths as possible-they can be reused often!
CHECKPOINT! Check in with your teacher to make sure you have created the graphs correctly.
7. Connect the dots on each of the graphs to create a smooth curve.
8. Complete the lab write-up. You will need to turn in one per pair.
Graphing Unit Algebra Trigonometry
Page 56
Names:
_______
_
____________________________
Trig graphs and Linguini Follow-up Questions
Answer the following questions to help clarify patterns seen and concepts learned during the lab.
1. Explain the sine curve that you graphed, referring to the following questions. Write your
answer in paragraph form.
A.
Where is the graph at 0 degrees? {We sometimes say “where does sine begin}
Why? Include the unit circle in your reasoning?
B.
Did the values increase or decrease right after 0o? Explain the progression of the
graph from 0 degrees to 360 degrees in general…..
C.
What was the maximum value of sine and at what degree marks did it occur?
Why? Include the unit circle in your reasoning?
D.
What was the minimum value of sine and at what degree marks did it occur? Why?
Include the unit circle in your reasoning?
E.
Where were the x-intercepts? {Give coordinates}
F.
Which values of the domain will result in a positive range? Negative? Why?
G.
What is the basic period of a sine graph? (How long did it take to start repeating?)
Graphing Unit Algebra Trigonometry
Page 57
2. Explain the cosine curve that you graphed, referring to the following questions.
A. Where is your cosine graph at 0 degrees? Why? Include the unit circle in your
reasoning?
B. Where did it go from there? Explain the progression of the graph from 0 degrees to
360 degrees.
C. What was the maximum value of cosine and at what degree marks did it occur? Why?
Include the unit circle in your reasoning?
D. What was the minimum value of cosine and at what degree marks did it occur? Why?
E. Where were the x-intercepts? {Give coordinates}
F. When is the domain positive? Negative? Why?
{Turn in one write-up per pair}
Graphing Unit Algebra Trigonometry
Page 58
Assignment #12 (after the Linguini Lab)
1) Graph each table of values on the circle and then on the coordinate plane
Identify the range, maximum value, minimum value and period ( How long it takes to begin to repeat itself.)
Be Careful to construct your graph to scale and so that the entire graph can fit on the page.
x
0
y
0
0.5

30 
45 
2
0.71=
2
3
0.87=
2
1
0.87
60 
90 
120 
135 
2
0.71=
2
0.5
0
-0.5
150 
180 
210 
225 
45
90
270
180
-1
-0.71= -
240 
270
300 
315 
330 
360 
2
2
3
-0.87= 2
-1
-0.87
-0.71
-0.5
0
1

90o
1
Maximimum: ( _________ , __________ )
-1< y <1
Range: _____________________________
Graphing Unit Algebra Trigonometry
270o
-1
Minimum: ( ____________ , ___________ )
360o
Period: _____________________
Page 59
360
2) Graph each table of values on the coordinate plane
Identify the range, maximum value, minimum value and period ( How long would it take to begin to repeat itself even if it kept going.)
Be Careful to construct your graph to scale and so that the entire graph can fit on the page. Answer questions based on the graph you
drew.
x
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360
y
0
0.26
0.38
0.5
0.71
0.87
0.92
0.97
1
0.96
0.92
0.87
0.71
0.5
0.38
0.26
0
1
90o
180o
270o
360o
-1
This is a sine graph… at least part of one.
Knowing what the general shape (from the linguini activity) is, what would the period of this
graph be?
o
The period would be 720
Graphing Unit Algebra Trigonometry
Page 60
3) Graph each table of values on the coordinate plane from 00 and try to go until 450o. {Use the linguini pattern….}
Identify the range, maximum value, minimum value and period ( How long it takes to begin to repeat itself.)
Be Careful to construct your graph to scale and so that the entire graph can fit on the page.
x
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360
390
405
420
450
y
0
1
1.42
1.74
2
1.74
1.42
1
0
-1
-1.42
-1.74
-2
-1.74
-1.42
-1
0
2
90o
180o
270o
360o
-2
o
Maximimum: ( _________
, __________ )
90
-2< y <2
2
Range: _____________________________
Graphing Unit Algebra Trigonometry
270o
-2
Minimum: ( ____________ , ___________ )
360o
Period: _____________________
Page 61
4) Fill in the chart: (using the unit circle and your calculator)
x
Exact value of cos(x) Approximate value of
cos(x)
0o
30o
1
3
2
2
2
1
2
45o
600
90o
120o
135o
150o
1800
210o
225o
240o
270o
3000
315o
330o
360o
390o
4050
0
1

2
2

2
3

2
-1
3
2
2

2
1

2

0
1
2
2
2
3
2
1
3
2
2
2
{End of Assignment}
Graphing Unit Algebra Trigonometry
Page 62
Summarizing the Linguini Lab as a class:
1.
What is the period of the sine curve? That is, after how many degrees does the
graph start to repeat?
2.
360o
In relation to ONE period, where do the ‘zeros/x intercepts’ occur?
0, 180 , 360 (Beginning, Middle and End)
3.
In relation to ONE period, where does the Maximum and Minimum occur?
Max = 90
(1/4 of a period)
4.
What is the period of the cosine function? That is, after how many degrees does the
graph start to repeat?
5.
360o
In relation to ONE period of a cosine function, where do the ‘zeros/x intercepts’
occur?
6.
Min = 270
(3/4 of a period)
¼ and ¾ of a period
In relation to ONE period of a cosine function, where does the Maximum and
Minimum occur?
Max = 0o , 360o
(Beginning and End)
Graphing Unit Algebra Trigonometry
Min = 180o
(Middle)
Page 63
Summarizing with terminology: y = asinb(x-c) + d
Amplitude is one-half the distance from the maximum value of y to the minimum value
of y. Which variable in the general equation affects the amplitude
y = sinx
1
3
y = 3sinx
2
2
a
y = ½ sinx
½
2
-½
(Vertical Stretched
or compressed !!!!)
--1
-3
2
): How long a graph takes to repeat itself?
b
How long does a standard sine and cosine graph take to start repeating itself?
Which variable in the general equation affects the period?
Period (
y = sinx
y = sin2x
b
y = sin ½ x
(horizontal Stretched
or compressed !!!!)
2
2
2
Phase shift : Shifting the graph horizontally.
Remember a standard positioned sine graph starts a cycle at the origin.
Which variable in the general equation affects the phase shift?
c
(phase shift or horizontal shift !!!!)
Vertical Shift : Shifting the graph vertically.
Remember a standard positioned sine graph, has midline at y = 0.
Which variable in the general equation affects the vertical shift?
d
(vertical shift!!!!)\
Graphing Unit Algebra Trigonometry
Page 64
Basic Shapes:
sine:
cosine:
Red graph shows one period of cosine graph. (dotted shows sine graph)
Amplitude and Period
#1-2: Find the max. value, Coordinates of a maximum point, the min value, coordinates of a minimum
point ,amplitude and period for each graph below.
1.
3
2
1
-1

2
4
6
5
-2
3
Minimum Value -1
Ampl.= 2
Maximum Value:
Coordinates of Max: (
 3
3 -1
Coordinates of Min: ( ,
Period:
Graphing Unit Algebra Trigonometry
4
,
)
,
)
Page 65
2.
6
4
2


4
2
-2

3
2
2  9
4
-4
-6
Maximum Value:
6
Coordinates of Max: (
3
,6 )
4
Minimum Value:
-6
Coordinates of Min: (
7
, 6 )
4
Ampl.=
6
Period= {look max to max:
Graphing Unit Algebra Trigonometry
11 3

}=2
4
4

Page 66
3) y
= sinx
Max =
1
Min. =
-1
Amplitude =
1
2
Period =
***************************************************************************
4) Given the general Equation: y = a sin( bx ),
‘a’ ( the number in front of the function ) affects the
‘b’ ( the number in front of the variable ) affects the
AMPLITUDE
PERIOD.
***************************************************************************
#5-7: Do NOT use your calculator. Use your brain to guess the values.
5) y = 2 sin(x)
Max =
2
Min. = -2
Amplitude =
2
Period =
360o or 2
6) y = -4sin( 2x )
Max =
7) y = 3sin (
4
1
x)
2
Min. = -4
Amplitude =
Amplitude =
3
Graphing Unit Algebra Trigonometry
4
Period = 180
Period = 720 or
or

4
Page 67
8) Graph the following: y = 3sin (
4
1
x)
2
3
, graph on the interval of 0 to 4  .
-3
Period:
Max:
Min
{To determine critical values, take your period ( fifth value ), divide it by 2 ( third value ), divide
that by 2 ( second value ), and finally, use the increments to determine the fourth value.}
x
1
3sin ( x )
2
0
0
3
0
-3
0
1
2
3
4
3
1
2
3
4
-3
Generalizations for graphing SINE curves:
Notice, what happens when the sine graph is at its beginning, middle and end of a period?
Zeros (if on x axis) = Midline points,.
What happens between two zeros?
The graph is either at a maximum or
a minimum.
If the ‘a’ is positive, first go to the
MAXIMUM,
If the ‘a’ is negative, first go to the MINIMUM.
Graphing Unit Algebra Trigonometry
Page 68
8) Graph the following: y = 3sin (
4
1
x)
2
3
, graph on the interval of 0 to 4  .
-3
Period:
Max:
Min
{To determine critical values, take your period ( fifth value ), divide it by 2 ( third value ), divide
that by 2 ( second value ), and finally, use the increments to determine the fourth value.}
x
1
3sin ( x )
2
0
0
3
0
-3
0
1
2
3
4
3
1
2
3
4
-3
Generalizations for graphing SINE curves:
Notice, what happens when the sine graph is at its beginning, middle and end of a period?
Zeros (if on x axis) = Midline points,.
What happens between two zeros?
The graph is either at a maximum or
a minimum.
If the ‘a’ is positive, first go to the
MAXIMUM,
If the ‘a’ is negative, first go to the MINIMUM.
Graphing Unit Algebra Trigonometry
Page 69
Use your generalizations from the previous page to graph the following sine graphs.
9)
y = 4sin(x)
max. =
4
min.=
-4
period =
2
4
3
2
1
1

2
2
-4
10) y = 2sin(3  )
max. = 2
min.= -2
period =
2
3
period =

2
2
 
6 3

2
2
4
3
2
3
-2
11) y = -2sin(4  )
max. = 2
min.= -2
2
-2
  3 
8 4 82 2
Graphing Unit Algebra Trigonometry

3
2
2
Page 70
Given the general Equation:
y = a cos( bx ),
‘a’ ( the number in front of the function ) affects the
AMPLITUDE
‘b’ ( the number in front of the variable ) affects the
PERIOD.
12. y = cosx
Max = 1
Min. = -1
Period = 360 or 2 
Amplitude = 1
#13-15: Determine the max., min, amplitude and period for each function.
13. y = 5cosx
Max =
5
Min. =
-5
Amplitude =
5
2
Period =
14. y = -2cos5x
Max =
2
Min. =
-2
Amplitude =
2
Period =
5
Min. =
-5
Amplitude =
5
Period =
1
15. y = 5cos ( x )
3
Max =
Graphing Unit Algebra Trigonometry
2
5
6
Page 71
1
16) Graph the following function: 5cos ( x) over the interval of 0 to 6  .
3
To determine critical values, take your period ( fifth value ), divide it by 2 ( third value ), divide
that by 2 ( second value ), and finally, use the increments to determine the fourth value.
x
0
3
2
3
9
2
6
1
( x)
3
1
cos ( x)
3
0
1 3 

3 2
2
1
3  
3
1 9 3

3 2
2
1
6  2
3
1
5cos ( x)
3
1
5
0
0
1
5
0
0
1
5
5
5
3
3
2
9
2
6
Notice, what happens when the cosine graph is at its beginning AND END of a cycle?
Maximum (‘a’ is positive) or Minimum (‘a’ is negative)
What happens in the middle of the period?
What happens between a max and a min?
Graphing Unit Algebra Trigonometry
Opposite of beginning
Zero or Midline point
Page 72
Use your generalizations from the previous page to graph the following cosine graphs.
17) y = 4cos(x)
max. =
4
period = 2
-4
min.=
4
1
1
2
3
2
2
-4
Domain: all real numbers
18)
y = 2cos(3x)
max. =
2
Range: [-4, 4]
-4 < y< 4
-2
min.=
2
period = 3
2

6

3

2
2
4
3
3
2
-2
Domain: all real numbers
Graphing Unit Algebra Trigonometry
Range: [-2, 2]
Page 73
19) y = -2cos(4x)
max. =
2
min.=
-2
period =

2
2
  3 

3
8 4 8 2
2
2
-2
Domain: all real numbers
Range: [-2, 2]
2
2

2
Period =  1   1 
   
2  2
1

20) y = 3cos   X 
2

3
1
2
3
4
-3
Domain: all real numbers
Graphing Unit Algebra Trigonometry
Range: [-3, 3]
Page 74
13.4 Practice
amp = 4
period = 2
y  4sin  
amp = 1
period = 6
1 
y  1sin   
3 
y = 2sin2x
2

4
-2

2
3
4

2
amp = 1.5
period = 
amp = 2
period = 3
y  1.5sin  42X 
2 
y  2sin   
3 
amp = 2.5
period = 
amp = 4
period =
y  2.5sin  2 
y  4sin  2 
3
y = 3sinx

2
-3

2
-2
( , 2)
8

4
y=2sin4x

2
2

8
Graphing Unit Algebra Trigonometry

3
2
2
y = 2sin8x
2
2
3
( , 2)
8

-2

4
Page 75
(
1.5

12
-1.5
2
,1.5)

6


2
2
3

3
3
2

2
-2
( , 1.5)
4
2
1
2

2

3
2
2
-2
1
8
 3
4 8

2
1
 
4 2
3
4


6
 
3 2
2
3
3
2
2
-2
3
5
1

2
3
4 -1

4
 3
2 4

-5
-3
3
4
4

4
-3
 3 
2 4

10
-4
Graphing Unit Algebra Trigonometry

5
3
10

2
5
2 3
-4
Page 76
4
6.28  2
-0.05 or -.1
0.2
0.2
0.3
0.1
0.2
-0.3
-0.2
Graphing Unit Algebra Trigonometry
Page 77
Practice 13-5
The Cosine Function
Sketch the graph of each function in the interval from 0 to 2.
1. y = 2 cos 3
1
6
2. y = 3 cos
1
3
1
2
2
3

4
3
2
5
3
1

2

2
Write an equation of a cosine function for each graph.
3.
4.
y = -4cos θ
5.
y = 4 cos 2 θ
Graphing Unit Algebra Trigonometry
y = 3cos4 θ
Page 78
Find the period and amplitude of each cosine function. Identify where the
maximum value(s), minimum value(s), and zeros occur in the interval from
0 to 2.
6.
7.
Period: 2
Amplitude: 1
Max= 1, Min = -1
Zeros:
 3
Period: 
2 2
Amplitude: 4
Max= 4, Min = -4
Zeros:
,
8.
Max= 5, Min = -5
 3 5 7
,
4 4
,
4
,
Period: 2 Amplitude: 5
Zeros:
4
 3
,
2 2
Write a cosine function for each description. Assume that a > 0.
9.
amplitude = 3, period = 4
y = 3cos 1 
2
Graphing Unit Algebra Trigonometry
10. amplitude =
y=
1
, period = 
2
1
cos 2
2
Page 79
NOTE: No graphing Calculators will be allowed on the TEST.
Notes: Vertical Shifting
Unless it says otherwise, graph for 2 periods.
1. Graph: y = 2sinx + 3
2
Period =
Amp.=
2
V/S=
UP 3
5
3
1

2
3
2

Equation of Midline?
Range:
4
2
y=3
[1,5]
Domain:
All real numbers.
Give the coordinates of 3 points on the function over one period:
( 0, 3)
 
 ,5 
2 
Graphing Unit Algebra Trigonometry
 ,3
 3 
 ,1
 2 
 2 ,3
Page 80
2. Shift the function: y = 2cos(3x) – 1
Period =
2
3
Amp =
2
V/S =
DOWN 1
1

-1
6

2

3
4
3
2
3
-3
Equation of Midline?
y = -1
(I like the labels to be uninterrupted, so I scaled it by 0.5)
Domain: all real numbers
Range: [-3,1]
Give the coordinates of 3 points on the function over one period:
( 0 , 1)


 , 1 
6

Graphing Unit Algebra Trigonometry


 , 3 
3

Page 81
Notes: Phase Shifting
a) What makes the quantity = to zero?

2
b) This becomes the new beginning of the cycle. Everything shifts in this direction
( left or right ? )
c) This shift is called a phase shift.
3
Amp. =


2
Period = 2
720 
720
720
P/S = LEFT

2
720 
720
720
Use the information learned in #1, to find answer the following questions.
Period =
2
Amp =

)
6
Period =
2
Amp.=
4. y = 3 sin ( 2x +  )
Period =

Amp.=
3
P/S =  LEFT
2
720o
Amp. =
1
P/S= 90 RIGHT
2. y = 3sin( x +

)
4
3. y = -2 cos ( x –
y = 3 sin 2( x +
5. y = sin(
1
 - 450 )
2

2
3
2
P/S =
P/S =

4

6
LEFT
RIGHT
)
Period =
y = sin1/2(  - 90)
Graphing Unit Algebra Trigonometry
Page 82
6. Graph the function. . y = 2sin( x –

)
2

P/S =
RIGHT
2
Period=
2
2

2
-2

4 9
2
2
7. Graph the function.
y = 3 cos ( 2x +  )
y = 3 cos 2(x +
Period =

Amp =
3
P/S =

2
LEFT

)
2
3


4

2

2
3
4

3
2
-3
Graphing Unit Algebra Trigonometry
Page 83
8) Graph the function for 2 periods. (Very hard… exposed to this, but won’t be this hard on tests)
8) Graph the function.
y = 4 sin 2(x +
Period = ,

)
3
Period =
 so ¼ a period =

Amp =

4
P/S =
3
left

4 , get common denominators between this
and the phase shift (= 12). Label the horizontal axis with these increments.
Then write each in terms of that denominator.

3

4
12

and
4

3
12
4
4
12


3


12
-4
2
12
5
12

6
8
12
2
3
11
12
20
12
5
3
Notice, we could just add the ¼ period value to our starting point, and keep doing that.
  4 3 
 


3 4
12 12 12
   3 2 
 



12 4 12 12 12 6
  2 3 5
 


6 4 12 12 12
5  5 3 8 2
 



12 4 12 12 12
3
Graphing Unit Algebra Trigonometry
Page 84
Practice:
9. Graph the function for 2 periods.
y = -4 cos ( 5x +  )
Period =
y = -4cos5(x +  )
5
2 Amp =
5
1 2 

4 5 10
4
P/S =
Left 
5
2
10
4


5



10
10

5
3
10
2
5
-4
Domain: All
real numbers
Range:
6
10
3
5
-4 < y < 4
Give the coordinates of 3 points on the function over one period:
 

  , 4 
 5

Minimum: _______________
Midline:
y=0
 0, 4
Maximum: _______________
Graphing Unit Algebra Trigonometry
Page 85
10) Graph the function for 2 periods.
y = 3sin 2(x +

)
4
Period =

Amp =
3
P/S =
Left 
4
3


4
-3

4

3
4
2

7
4
(, )
Domain: ___________________________
:
Range
[-3,3]
 0,3
Give the coordinates of a maximum : _________________


 , 3 
2

Give the coordinates of a minimum: __________________
  
  ,3
Give the coordinates of a midline point: ____________________
 4 
Graphing Unit Algebra Trigonometry
Page 86
11) Graph the function for 2 periods.

y = 2sin 3(x + ) + 4
6
Period =
2
3
Amp =
2
P/S =

6
left
V/S = Up 4
6
4
2

6


6
Equation of midline?
y=4
Give 2 coordinates of a maximum:
Give 2 coordinates of a minimum:

3

4
 0, 6 
 
 ,2
3 
(, )
Domain: ______________________
Graphing Unit Algebra Trigonometry
Range:
7
6
2
3
&
 2 
,6

3


 , 2 
[ 2, 6]
Page 87
12) Graph the function for 2 periods.
y = 3sin 4(x +

) - 2
4
Period =


2
Amp = 3
P/S =
4
LEFT
1


4


8

8

4
3
8

2
3
4
-2
-5
Graphing Unit Algebra Trigonometry
Page 88
13) Graph the function for 2 periods.
y = 4cos2(x – 20o)
Period: 180o
Right 20o
Amplitude= 4
(intervals: 45o)
25
20
65
Graphing Unit Algebra Trigonometry
110
155
200
245
290
335
Page 89
380
14) Graph the function for 2 periods.
y = - 4cos3(x –  )
6
-4


6

6

3

2
2
3
4
3
-4
Graphing Unit Algebra Trigonometry
Page 90
Writing equations of graphs:
1. Write a sine equation using the given information: { y = asin b(x - k ) + d }
a. amp = 2, period = 90  no shifting.
360
360
 90
b
4
b
90
y = +2sin4(x)
b. amp. = 5 , period = 720  , shift up 1
y= +5sin ½ (x) + 1
c. max ( 90 , 3 ) , min ( 270 , - 3 )
y = 3 sin(x)
Max to min = ½ period = 180, so period = 360
y = -3sin(x) - 2
d. max ( 270 , 1 ) , min ( 90 , -5 )
e. Zeros for 1 period => ( 90, -10 ) , ( 180 , -10 ) , ( 270 , -10 ) , Amp = 3
y = +3sin2(x-90) - 10
f. Max. ( 45 , 7 ) and Max ( 225 , 7 ) , no vertical shift.
y = 7sin2(x)
Distance between 2 max’s = 1 period = 180
g) P/S = 90 Right
Period = 180
No V/S
amp = 6
Starts going down.
y = -6sin2(x-90)
h)
P/S = 45 Left
V/S = 7 down
Max = ( 0 , -3 )
y = 4sin2(x+45) - 7
Graphing Unit Algebra Trigonometry
Page 91
Reminder: No Calculators will be allowed on the TEST.
Tangents and Cotangents
Reminder about unit circle…. Find the tangent for key values…. 0,
(1, 0)  tan(0) 
(
    2
, , , ,
…
3
6 4 3 2
0
0
1
3 1
3
 
, )  tan   
2 2
6 3
 2 2
 
,

  tan    1
4
 2 2 
1 3
 
( , )  tan    3
2 2
3
  1
(0,1)  tan     undefined
2 0
 units
How long does it take for the numbers to start repeating? ______________
This is NEW! The normal period for sine and cosine was:
The normal period for tangent is
2 units .
 units
!!!!!
Let’s graph the coordinates of the tangent on the next page
 0, 0 
 3    
 

 ,
  ,1  , 3   , undefined  …..
 2

6 3  4  3
What would ‘undefined’ mean on a graph? Asymptote!
Graphing Unit Algebra Trigonometry
Page 92
Graph the following functions for two cycles.
Determine the period and equation for the asymptotes for each function.
Period tangent 
1) y = tan

b
1



4
2

2
-1
3
4


3
2
All real numbers
Notice the Period is: __________________ Range: _____________________
2)
y=
1
= cotx
tan x
1
-1

4

2
Graphing Unit Algebra Trigonometry
3
4

2
Page 93
Do not use a calculator to graph the functions. Graph for two periods.
3)

y = tan 3

Period: 3



6 12

12

6

4


3
5
12

2

all real numbers except (  k )
Domain: __________________________________
6 3
4)
all real numbers
Range: __________________________
y = -3 tan2x
3
-

8

4
 3 
4 8 2
3
Graphing Unit Algebra Trigonometry
3
4

Page 94
5)
Period:
y = 4tan 5x + 2

5
6
2


10
Equation of Asymptotes for one period:
x
Equation of Midline:
3
20

10

20
-2

10
,x 

5

10
Y=2
6)
y = 3 tan
1
(   )
2
Shift
 left
3
2



2

2

3
2
2
3
3


 3 
Coordinates of 3 points on a graph for one period:  , 3  ,  , 0  ,  ,3 
2

 2 
1
( )
Another equation? y = - 3 cot
2
Graphing Unit Algebra Trigonometry
Page 95
Do not use a calculator to graph the functions. Graph for two periods.
7)
y = cot 3
1
   
1
8)
12 6 4
3
2
3
y = -3 cot2x +1
4
1
2


8
4
Equation of Asymptotes: x = 0+
Equation of midline:
3 
8 2

2

k
y=1
Coordinates of a Maximum:
NONE
Graphing Unit Algebra Trigonometry
Page 96
9) y = 4cot 5(x -

)
10
4


10


20
 
20 10
3

20
5
2
5
4
Amplitude:
NONE
10) y = 3 cot

5
Period:
Coordinates of a Midline point:
 
(0, 0) or  , 0 
5 
1
 -4
2
1

2

3
2
2
4
4
7
Graphing Unit Algebra Trigonometry
Page 97
Graphing Secants and Cosecants
Graph two cycles of the graph ( 0 to ? ). Label the axes.
1)
y = sinx

2
On the graph above, draw in vertical dotted lines, wherever the graph crosses the midline ( These lines
are called vertical asymptotes. ) We can use this to graph
y = csc x. Let’s do it on the graph above.
2) First graph:y = -2sin(4x) , then use it to graph y = -2csc(4x)


8
4
Graphing Unit Algebra Trigonometry
Page 98
3)
Graph y
= cosx
for two periods
1


2
-1
3
2
2
4
On the graph above, draw vertical asymptotes, wherever the graph crosses the
midline. Use this graph to graph y = secx.
4)
First graph y= 2
cos3x, then use this graph to graph: y= 2 sec3x
2

6

3

2
-2
2
3
4
3
What is the maximum value of the sec function?
none
What is the amplitude of a secant function?
none
What is the range?
(, 2]  [2, )
Graphing Unit Algebra Trigonometry
Page 99
Graph each function from 
5)

2
to 2
y = 3csc(x)
3



2
2
3
2

5
2
2
3
    3

Give the coordinates of any 2 points on the graph:  ,3  ,  , 3  NOT: (0,0) or ( ,0)
2   2

6)
y = -2 sec ( x )


2

2
Find the equation of the asymptotes: X =
Graphing Unit Algebra Trigonometry

3
2
2

n
2
Page 100
Graph each equation for at least two periods.
7)
y = 4 sec ( 2x )
4

4

2
3
4

2
-4
+
, 4 sec ( 2x ) 
4
{‘+’ means from the right}
Checkpoint: as x 
, 4 sec ( 2x ) 
4
{‘-’ means from the left}
Checkpoint: as x 
, 4 sec ( 2x ) 
2
Checkpoint: as x 
Checkpoint: as x 
+
, 4 sec ( 2x ) 
2
Graphing Unit Algebra Trigonometry

.
-4
. {What is the y value doing?}
{What is the y value doing?}
. {What is the y value doing?}
-4
. {What is the y value doing?}
Page 101
8)
y = -5csc(
1

3
3
2
3
)
5
9
2
6
12
-5
Graphing Unit Algebra Trigonometry
Page 102
sec/csc with shifting.
Determine the amplitude and period for each graph below.
Graph two periods of the graph ( 0 to ? ). Label the axes.
9) y = - 2 csc 4x + 3
5
3
1

8

4
3 
8 2

Give the coordinates of 2 points (for one period) on the graph.
Graphing Unit Algebra Trigonometry

3
( ,1), ( ,5)
8
8
Page 103
10) y = 2 sec3(x -

2
)
2 4


 One  fourth period=
3
6
6
 3
shift =
Right:
2 6
Common Denominator: 6
Period =
2

6
-2

3
 2 5
2 3 6

Graphing Unit Algebra Trigonometry
5 11
3 6
Page 104
11) Graph:
y = 2 sec(x


2
)+ 1

4
2
Period  2 , so one-fourth of period = 
2
4
shift

4

5
4
15
4
Right
3
1


4
1

2
Equation of midline?
3
2
13
4
5
2
y=1
Equation of asymptotes for one period?
x=
9
4
3
7
, x=
4
4
General equation of asymptotes :
where
17
4
x
3
k
4
n  Integers ( )
Graphing Unit Algebra Trigonometry
Page 105
12) y = -4csc2(x +

)
3
Period=  , one-fourth period =
Shift

3

3
4
Left, so common Denominator=12.
Critical values every
Shift




4
left.
12

3


12
4

3
12

6
Graphing Unit Algebra Trigonometry
5
12
2
3
11
12
7
6
17
12
5
3
Page 106
Negative Functions
1) Graph y= sinx and y= –sin(x) (dotted) on the same set of axes from -2  to 2  .
Try to decide which graph will match up with sin(-x)
sin(

) = _________
4
- sin(

) = _________
4
sin(-

) = _________
4
Generalization:
Review from Chapter 3:
f(-x) = -f(x) gives what type of function? ______________
Examples: Write each expression with a positive angle.
1b) sin(-40)
Graphing Unit Algebra Trigonometry
1c) -sin(-100)
Page 107
2) Graph cosx and –cos(x) on the same set of axes from -2  to 2  .
Try to decide which graph will match up with cos(-x)
cos(  ) = _________
-cos(  ) = __________
cos(-  ) = _________
Generalization:
Reminder from Chapter 3:
f(-x) = f(x) gives what type of function:
Examples {re-write with positive angles}:
2b)
sin(-

)=
6
Graphing Unit Algebra Trigonometry
2c) cos( 
5
)=
6
Page 108
3) Let’s use what we know about sine and cosine to make a generalization for sec(-x) ,
sec(x) and/or –sec(x).
Generalization:
4) Let’s use what we know about sine and cosine to make a generalization for cscx ,
csc(-x) and/or –csc(x).
Generalization:
Here are some examples: Write an equation that is true using only positive angles.
5) csc( 
7
)=
4
Graphing Unit Algebra Trigonometry
6) sec( 
4
)=
3
Page 109
7) To find our generalization for tangent, we can either graph it: using a calculator graph tanx , -tan(x)
and tan(-x) on the same set of axes.
{Set lines to be different. Let y1 be normal, y2 = thick, y3 = dot.}
Or we can verify this algebraically, using what we know of sin(-x) and cos(-x).
Generalization:
8) Use your knowledge of tan(x) to make a generalization about cot(x).
9) Graph: y = cos(-2x) + 3 for 2 periods.
What is the domain? _______________________ What is the range? _______________________
Coordinates of maximum: _______________
2 coordinates on the graph for one period: _____________ ____________
Equation of Asymptotes: _____________________
Graphing Unit Algebra Trigonometry
Page 110
10) Graph: y = tan(- x -

) for 2 periods.
4
11) Graph: y = 3sin(  – 2x) for 2 periods.
Graphing Unit Algebra Trigonometry
Page 111


14) Graph: y = -2 sec   x  + 4
2

What is the domain? _______________________ What is the range? _______________________
Coordinates of maximum: _______________
2 coordinates on the graph for one period: _____________ ____________
Equation of Asymptotes: _____________________
Graphing Unit Algebra Trigonometry
Page 112
Here are some problems tying graphing into the picture…
15) as t   +, cost  ________
(This means as t approaches  from the right, what is cos(t) doing?)
16)

, cos(-t)  ________
3

(Key- what is the value of cos( )?)
3
as t 
17) as t 

, sec(t)  ________
2

18) as t 
, sec(-t)  ________
2
19)

as x 
, tan(-x)  ________
2
20)
as x 

, cot(-x)  ________
2
REMINDER: No Graphing Calculators will
be allowed on the TEST.
Graphing Unit Algebra Trigonometry
Page 113
Trig Review Worksheet
Graph each function in the interval from 0 to 2.
1.
y = cot
1

2
4. y = csc   1
2. y = sec  + 2
5.


y = sin  x  
2

3.
y = sec
1

2


6. y = 3 cos  x   + 2
4

Match each function with its graph. Write the appropriate number next to the graph.
Graphing Unit Algebra Trigonometry
Page 114
7. y  sin( x)
8. y  cos( x)
9. y   sin( x)
10. y   cos( x)
11. y  sin(2 x)
12. y  cos(2 x)
13. y  2sin( x)
14. y  2 cos( x)
.
15. y  sin( x)  2
16. y  cos( x)  1
18. y  cos( x  30)
19. y  3sin(2 x)
21. y   sin 2( x  90)
22. y  3cos
17. y  sin( x  45)
1 
20. y   cos  x   1
2 
1
 x  90 
2
Use the graph of y = tan  to find each value. If the tangent is undefined at
that point, write undefined.
Graphing Unit Algebra Trigonometry
Page 115
23. tan
 3 
24. tan   
 4 

2
 
25. tan   
 4
26. tan
3
2
Identify the period and tell where the asymptotes occur, in the interval from
0 to 2, for each function.
27. y = 2 tan

2
28 y = 3 tan 2
29. y = 4 tan 
Find the amplitude and period of each function. Describe any phase shift
and vertical shift in the graph.
30. y = 3 cos x + 2


31. y = 2 sin  x  
2



33. y = sin  x  
3

34. y =
1
cos x  3
2
32. y = cos 2x + 1
35. y = cos
1
x–2
2
Identify the period of each tangent function.
36.
37.
Graphing Unit Algebra Trigonometry
38.
Page 116
Trig Graph Matching
PAGE 1
Please match the verbal descriptions from page 2 with the equations written below, and with the
appropriate graph. Use the answer sheet provided to record your answers.
______ 1. y = 3cos  + 2
_______ 13. y = 2sin 
______ 2. y = tan 
_______ 14. y = sin(  -
______ 3. y = 2sin  + 3
_______ 15. y = 3sin  + 2
______ 4. y = sin(4  )
_______ 16. y = sec 
______ 5. y = 2sin(2  )
_______ 17. y = sin(2(  -
______ 6. y = sin(  +

)
4

)
4

))
4
_______ 18. y = 2csc 
______ 7. y = 2cot 
_______ 19. y = 2sin(2(  +
______ 8. y = -3sin  + 2
_______ 20. y = -2sin  - 3
______ 9. y = -3cos2 
_______ 21. y = -2sin 
______ 10. y = -2sin 
_______ 22. y = csc 
1
3
______ 11. y = sin 
_______ 23. y = 2sin(  +
______ 12. y = 3sin 
_______ 24. y = cot 
1
2
Graphing Unit Algebra Trigonometry

))
2

)
2
Page 117
Trig Graph Matching
PAGE 2
Please match the equation from page 1 with the verbal descriptions written below, and with the
appropriate graph. Use the answer sheet provided to record your answers.
1.
2.
3.
4.
5.
6.
7.
8.
9.
The absolute value of a sine curve with an amplitude of 3.
A sine curve w/horizontal compression x factor of 4.
A sine curve with a = 3, and d = 2.
A cotangent curve where a = 2.
A cosecant curve w/amplitude of 2.
A cosine curve w/amplitude of 3, and a vertical shift up 2.
A sine curve with a = -3, and d = 2
A sine curve with an amplitude of 2.
A sine curve w/vertical stretch x factor of 2, and a vertical shift up 3

10. A sine curve with a phase shift right units.
4
11. A sine curve with an amplitude of 2, a horizontal stretch by a factor of 3, and
reflected about the x-axis.
12. A secant curve.
13. A cotangent curve.
1
14. A sine curve w/b = .
2

15. A sine curve w/b = 2 and c =
4
16. A sine curve w/a = 2, b = 2, and c = 

2
17. A cosecant curve.
1
3
18. A sine curve w/a = -2, and b = .
19. A sine curve with a phase shift left
20. A sine curve with a = 2, and c = 

2

units.
4
21. A tangent curve.
22. A sine curve with an amplitude of 2, and reflected about the x-axis.
23. A cosine w/amplitude of 3, horizontal compression x 2, and reflected about the
x-axis.
24. A sine curve w/a = 2 and b = 2
Graphing Unit Algebra Trigonometry
Page 118
Trig Graph Match Answer Sheet
Trig
Equation #
Verbal
Description #
Graphs of Trig Eq.
Letter
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Graphing Unit Algebra Trigonometry
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Graphing Unit Algebra Trigonometry
Page 120
Graphing Unit Algebra Trigonometry
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Graphing Unit Algebra Trigonometry
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Graphing Unit Algebra Trigonometry
Page 123