GRAPHING
TRIG FUNCTIONS
Algebra 2 & Trigonometry Lab
Miss Kersting
Room 206
Period 1
Name:____________________________
Day 1 – Graphing sine, cosine, and tangent
Unit Circle
Vocabulary:
y  a sin bx  c
y  sin x
Interval: 0    2
y  cos x
Interval: 0    2
y  tan x
Interval: 0    2
y  a cos bx  c
y  a tan bx  c
max  min
2
I.)
amplitude: a - the height of the graph. Formula:
II.)
frequency: b - the number of times the graph appears between 0  x  2  360
III.)
period:
IV.)
vertical shift: c – the shift that moves the graph up or down. Formula: max - amp

2
- the interval where you see one full cycle.
b
Maximum Value: vertical shift + amplitude
Minimum Value: vertical shift – amplitude
y  a sin b( x  d )  c
V.)
y  a cos b( x  d )  c
y  a tan b( x  d )  c
phase shift (horizontal shift): d – the shift that moves the graph right or left.
(remember: opposite)
p2
Directions: Find: a) amplitude, b) frequency, c) period, d) min & max, and e) vertical shift.
1.) y  3sin 2 x
2.) y  5sin 4 x
3.) y  3cos 6 x  10
Directions:
Find a.) amplitude b.) max/min c.) frequency d.) period e.) vertical shift f.) phase shift


4.) y  4 cos 2  x    2
2

p3
5.) Sketch y  3sin 2 x in the interval 0    2 .
1
6.) Sketch y  2 cos x in the interval   x   .
2
7.) What is the amplitude of the function
5
y  sin 4 x ?
3
8.) What is the period of the function
y  3sin 5 x ?
9.) What is the period of the function
y  3 tan x ?
10.) What is the frequency of a trig curve that
has a period of 3?
p4
1.)
2.)
3.)
Day 1 – Graphing sine, cosine, and tangent
HOMEWORK
Sketch the graph of y  3cos 2 x in the interval 0  x  2 .
3


Sketch the graph of y  sin 2 x in the interval   x  .
2
2
2
What is the minimum element in the range 4.)
of the function y  5  2sin  ?
p5
Between x  2 and x  2 , the graph of
the equation y  cos x is symmetric with
respect to which axis?
5.)
How many full cycles of the function
y  2 cos 3 x appear in 2 radians?
7.)
If the graphs of the equations y  2 cos x
and y  1 are drawn on the same set of
axes, how many points of intersection will
occur between 0 and 2 ?
6.)
8.)
p6


The graph of y  sin  x   is a shift in
3

which direction?
What is the range of the function
y  3cos 2 x  1 ?
Directions: Find: a) amplitude, b) frequency, c) period, d)minimum & maximum, e)vertical shift
(if any), and f) phase shift (if any).
9.) y  2sin 3 x  4
1
10.) y  cos  x   
2
11.)
x
y  sin    5
2
12.)
p7


y  2  3cos  x  
2

Day 2 - Write the trig function that is represented by a graph or a word problem
Do Now: Questions 1 & 2


1.) The function y  cos  x   is equivalent
2

to which of the following?
(1)
(2)
(3)
(4)
1
cos 3 x
2
(2) y  cos 3x
(3) y  3 cos 2 x
(4) y  3 cos x
y  sin x
y  sin   x 
y  cos x
y  cos   x 
1
(1) y  cos x
(3) y  cos 2 x
2
1
(2) y  sin x
(4) y  sin 2 x
2
Which equation is represented by the
graph below:
(1) y 
1
sin 2 x
2
(2) y  2 sin
Which equation has a period of  and an
amplitude of 3?
(1) y 
Directions: Multiple Choice.
3.) Which equation is represented by the
graph below:
5.)
2.)
1
x
2
(3) y 
4.)
Which equation is represented by the
graph below:
(1) y = -2 sin x
(2) y = sin (2x)
6.)
1
1
sin x
2
2
1
(4) y   cos 2 x
2
p8
(3) y = -sin (2x)
(4) y = 2 sin x
Which equation is represented by the
graph below:
(1)
y  2 sin
(2)
y
1
x
2
1
sin x
2
(3) y = 2 sin 2x
(4) y = 2 cos 2x
Steps for Writing Trigonometric Equations Represented by a Graph
max  min
2
2. Determine the frequency. (Remember: the number of full cycles between 0 and 360 )
3. Determine the vertical shift, using the formula: maximum – amplitude
4. Determine the phase shift. If you have a cosine curve, a phase shift only occurs if the graph
doesn’t start at the minimum or maximum. If you have a sine curve, a phase shift only
occurs if the graph doesn’t start at the vertical shift.
1. Determine the amplitude, using the formula:
Directions: Write the trigonometric equation represented by each graph.
7.)
8.)
9.)
10.)
p9
Directions: Solve each word problem.
11.) A building’s temperature, T, varies with time of day, t, during the course of 1 day, as follows:
T  8cos  t   78
The air conditioning operates when T  80F . Graph this function for 6  t  17 and
determine, to the nearest tenth of an hour, the amount of time in 1 day that the air
conditioning is on in the building. [Show all work.]
p10
Day 2 - Write the trig function that is represented by a graph or a word problem
HOMEWORK
Directions: Write the trigonometric equation represented by each graph.
1.)
2.)
3.)
4.)
p11
5.)
The times of average monthly sunrise, as shown in the accompanying diagram, over the
course of a 12-month interval can be modeled by the equation y  A cos  Bx   C .
Determine the values of A, B and C. Explain how you arrived at your values.
6.)
A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as
shown in the accompanying diagram, and moves the other end of the rope up and down,
producing a wave described by the equation y  a sin bx  c . The range of the rope’s height
above the ground is between 1 and 5 feet. The period of the wave is 4 . Write the equation
that represents this wave.
p12
Day 3 – Graphs of inverse functions with restricted domains
1.
2.
3.
1.)
Steps to Graph Inverse Trig Functions
Graph the original trig function.
Switch the x and the y values.
Restrict the domain.
Sketch the graph of g ( x)  Arc sin x , the inverse of f ( x)  sin x .
2.)
Sketch the graph of g ( x)  Arc cos x , the inverse of f ( x)  cos x .
3.)
Sketch the graph of g ( x)  Arc tan x , the inverse of f ( x)  tan x .
Remember:
 In order to get the inverse of a function you must switch x and y.
 In order for the inverse to be a function, the original function must pass the vertical and
horizontal line tests. (It must be a 1 - 1 function)
p13
Directions: Find each degree measure.
3
3
5.) y  Arc cos
6.) y  Arc sin 
2
2
7.) y  Arc tan 3
Directions: Find the exact value of each expression.


 12  
 3 
9.) cot  Arc sin    

8.) tan  Arc cos


 13  

 2 


 1 
10.) sec  Arc cos  
 2 


 2 

11.) sin  Arc sin 


2



p14
Practice Problems
12.) Which graph represents the equation
y  cos 1 x ?
13.) Which graph represents the equation
y  sin 1 x ?
p15
1.)
Day 3 - Graphs of inverse functions with restricted domains
HOMEWORK
-1
sin x is a function. What 2.) The inverse of cos x is a
3.) If arctan x is a function,
is the domain of sin x?
function. What would be
then what would be the
the domain for cos x?
domain of tan x?
Directions: Find each degree measure.
1
4.) y  Arc sin
2
5.)
Directions: Find the exact value of each expression.
8

6.) tan  Arc sin 
7.)
17 

p16

2
y  Arc cos  

 2 
4

sec  Arc cos 
17 
