Download Trig Graphs Packet - White Plains Public Schools

Algebra 2/Trig: Trigonometric Graphs
(SHORT VERSION)
In this unit, we will…
 Learn the properties of sine and cosine curves: amplitude, frequency, period, and midline.
 Determine what the parameters a, b, and d of the function
and
do to the basic graph of sine and cosine
 Determine the relationship between the period and the frequency of a trig function
 Identify the graph of tangent
 Identify the graphs of secant, cosecant, and cotangent
 Graph (for real) sine and cosine curves and a system of those equations.
Name:______________________________
Teacher:____________________________
Pd: _______
1
Table of Contents
Day 1: Graph Sine and Cosine Curves
SWBAT: Know and Graph Sine and Cosine Curves
Pgs. 3 – 7 in Packet
HW:
Pgs. 8 – 10 in Packet
Day 2: Graphing with Properties of Trig Graphs
SWBAT: Graph Sine and Cosine Curves using Properties
Pgs. 11 – 14 in Packet
HW:
Pgs. 15 – 17 in Packet
Day 3: Graphing with Properties of Trig Graphs and System of Trig Graphs
SWBAT: Graph Sine and Cosine Curves using Properties and Systems of Trig Graphs
Pgs. 18 – 21 in Packet
HW:
Pgs. 22 – 24 in Packet
Day 4: Graphing Sine and Cosine Curves with Vertical Shifts
SWBAT: Graphing Sine and Cosine Curves with Vertical Shifts
Pgs. 25 – 28 in Packet
HW:
Pgs. 29 – 31 in Packet
Day 5: Graphing Sine and Cosine Curves with Vertical/Horizontal Shifts
SWBAT: Graphing Sine and Cosine Curves with Vertical/Horizontal Shifts
Pgs. 32 – 35 in Packet
HW:
Pgs. 36 – 38 in Packet
Day 6: Graphing Reciprocal Trig Functions
SWBAT: Graphing Reciprocal Trig Functions
Pgs. 39 – 42 in Packet
HW:
Pgs. 43 – 46 in Packet
Day 7: Graphing Inverse Trig Functions
SWBAT: Graphing Inverse Trig Functions
Pgs. 47 – 50 in Packet
HW:
Pgs. 51 – 54 in Packet
Day 8: Test
2
NAME:___________________________________________________
Algebra 2/Trig – Graphing Sine and Cosine CLASSWORK
DATE:____________
PERIOD:__________
A sine wave, or sinusoid, is the graph of the sine function in trigonometry. In addition to mathematics, this function
also occurs in other fields of study such as science and engineering. This wave pattern also occurs in nature as seen
in ocean waves, sound waves and light waves. Even average daily temperatures for each day of the year resemble
this wave.
The cosine wave is also said to be sinusoidal because of the cofunction relationship
THE GRAPHING VOCABULARY

Amplitude = the magnitude (height) of the oscillation (wave) of a sinusoidal function. Sometimes it is referred to
as the "peak from center" of the graph

Period = One complete repetition of the pattern is called a cycle. The period of a function is the horizontal
length of one complete cycle.

Frequency = the number of cycles it completes in a given interval. This interval is generally
for the sine and cosine curves.
radians (or 360o)
THE GRAPHS OF SINE AND COSINE
Graphs of trigonometric functions can be produced in degrees or in radians.
The graphs appearing here will be done in radians.


KEY POINTS FOR GRAPHING SINE AND COSINE
These key points below are important when graphing any SINE or COSINE graph!
Five key points in one period of each graph: the intercepts, maximum points, and minimum points
Degrees 0˚
Radians
90˚
0

2
180˚
270˚
360˚

3
2
2
Sin θ
Cos θ
Basic Sine and Cosine Curves

called a “wave” because of its rolling wave-like
appearance (also referred to as oscillating)

Amplitude = 1 (height)

Period =

Frequency = 1 (# of cycles in

Domain =

Range =
(length of one cycle)
radians)

called a “wave” because of its rolling wave-like
appearance (also referred to as oscillating)

Amplitude = 1 (height)

Period =

Frequency = 1 (# of cycles in

Domain =

Range =
(length of one cycle)
radians)
GRAPHING TRIG FUNCTIONS ON YOUR CALCULATOR
**Sketch each of the following using your graphing calculator to help you**



Before starting, change the MODE of your calculator to RADIANS.
To view the correct graph window, select ZOOM7: ZTrig
.
You can also choose to graph the curves in a restricted window, by selecting the WINDOW button.
WARM-UP: PRACTICE GRAPHING IN DEGREES
MODEL #1: Graph y = sin x from

Key points:
MODEL #2: Graph y = cos x from

Key points:
REGENTS PRACTICE: GRAPHING IN RADIANS
PRACTICE #1: Graph y = cos x from

Key points:
PRACTICE #2: Sketch the graph y = sin x from

Key points:
REGENTS PRACTICE: GRAPHING WITH AMPLITUDE
PRACTICE #1: Graph y = 2cos x from

Key points:
PRACTICE #2: Sketch the graph y = -3sin x from
Key points:
NAME:___________________________________________________
Algebra 2/Trig – Graphing Sine and Cosine HOMEWORK
1. Graph y = sin x from
2. Sketch the graph y = cos x from
3. Sketch the graph y = 2 sin x from
DATE:____________
PERIOD:__________
REGENTS REVIEW
3. Solve the equation
5. Solve
algebraically for all values of x.
by completing the square, and express the answer in simplest form.
6. What is the solution set for the equation
1)
3)
2)
4)
?
7. The roots of the equation
are
1) real and irrational
2) real, rational, and equal
3) real, rational, and unequal
4) imaginary
8. The graph below represents the function
function.
9. The inverse of the function
. State the domain and range, in interval notation, of this
is
1)
2)
3)
4)
10. Write a quadratic equation that has
as one of its roots.
NAME:___________________________________________________
Algebra 2/Trig – Graphing with Properties of Trigonometry Graphs CLASSWORK
DATE:____________
PERIOD:__________
LEARNING GOAL: Graphing with Properties of Trigonometric Graphs

Amplitude =



Frequency =

Period =
Identify the amplitude, frequency, and period of the following graphs:
In general, for the function y = a sin bx and y = a cos bx
Amplitude
Frequency
y = a sin bx
or
y = a cos bx y = a sin bx
or
y = a cos bx y = a sin bx

Period
or
y = a cos bx
State the amplitude, period, and frequency of each graph below:
y  3sinx 
1 
y   sin x 
2 
1
y  cos 2 x 
2
ADVICE FOR GRAPHING SINE AND COSINE
1. Identify the key points for SINE or COSINE.
2. Identify all properties of the trig graph.
3. Graph one cycle of the wave by using the value of the period.
4. Use the given interval to determine if you will keep the wave, double the wave, or take half of the wave.
MODEL PROBLEM 1: Graph y =
sin x from 0  x  2
MODEL PROBLEM 2: Graph y = cos 2x from
MODEL PROBLEM 3: Graph y =
cos x from
MODEL PROBLEM 4: Graph y = 3 sin x from
PRACTICE: Graph y =
cos x from
NAME:___________________________________________________
Algebra 2/Trig – Graphing with Properties of Sine and Cosine Graphs HOMEWORK
1. Graph y  3 sin2x from
2. Sketch the graph
y   cos
1 from
x
2
DATE:____________
PERIOD:__________
3. Graph
from
5. What is the amplitude of the graph of the equation
?
1)
2) 2
3)
4)
6. An object that weighs 2 pounds is suspended in a liquid. When the object is depressed 3 feet from its equilibrium
point, it will oscillate according to the formula
, where t is the number of seconds after the object is
released. How many seconds are in the period of oscillation?
1)
3) 3
2)
4)
REGENTS REVIEW
7. The length S that a spring will stretch varies directly with the weight F that is attached to the spring. If a
spring stretches 20 inches with 25 pounds attached, how far will it stretch with 15 pounds attached?
8. Solve for x:
2 x  5  3 14
9. Solve the inequality  3 6  x  15 for x. Graph the solution on the line below.
10. Solve algebraically for x:
NAME:___________________________________________________
Algebra 2/Trig – Properties of Trig Graphs and System of Trig Graphs CLASSWORK
DATE:____________
PERIOD:__________
LEARNING GOAL #1: Properties of Trig Graphs (Find the Equation Give the Graph)
or
Amplitude =
Frequency=
Period =
ADVICE FOR WRITING TRIG EQUATIONS FROM GRAPHS
1. Identify if it is
or
from the y-intercept of the graph.
2. Use the properties of trig graphs (amplitude, frequency, and period) to fill in for a and b
1. Write an equation that represents the following
graph.
2. Write an equation to represent the graph below.
3. Write an equation that is represented by this graph.
4. Which equation is represented by the accompanying
graph?
LEARNING GOAL #2: Graphing Systems of Trig Equations
ADVICE FOR GRAPHING SYSTEMS OF EQUATIONS
1. Identify all properties for each trig graph (key points, amplitude, frequency, period).
2. Graph the trig function that has the larger period first!
3. Identify the location of the smaller period on the x-axis, and then graph the other trig function
4. Check your graphs with the calculator (radian mode)
5. Make sure you answer the question! (identify the number of intersections, etc.)
Graphing Systems of Trigonometric Equations with Different Periods
Model Problem A: Graph the functions y = -2 cos (x) and y = sin 2x from 0  x  2 on the same set of axis.
Then identify the number of points that satisfy the equation -2 cos (x) = sin 2x between 0  x  2 .
Model Problem B: On the same set of axes, sketch and label the graphs of y = 2 sin x and y = cos 2x in the interval
0  x  2 .
x
At

2 , find the difference: 2 sin x – cos 2x
PRACTICE: Graph the functions y = cos
1
x and y = -3 sin x from
2
Then identify the number of points that satisfy the equation cos
on the same set of axis.
1
x = -3 sin x between
2
.
NAME:___________________________________________________
Algebra 2/Trig – Properties of Graphs and Systems of Graphs HOMEWORK
1. Sketch the graph of
2. Sketch the graph of
DATE:____________
PERIOD:__________
on the interval 0 ≤ x ≤ . State the amplitude, frequency and the period.
on the interval 0 ≤ x ≤ 2. State the amplitude, frequency and the period.
3. Sketch the graph of y =
sin 2x on the interval - ≤ x ≤ . State the amplitude, frequency and the period.
4. Sketch, on the same set of axes, the graphs of y = 2 cos x and
as x varies from 0 to 2.
Determine the number of values of x between 0 and 2 that satisfy the equation 2 cos x =
5. Which is an equation of the graph shown below?
7. The accompanying diagram shows a section of a
sound wave as displayed on an oscilloscope.
Which equation could represent this graph?
1)
1)
2)
2)
3)
3)
4)
4)
6. Write an equation of the graph shown below.
8. What is the period of the function
?
1)
3)
2)
4)
9. What is the minimum value of
?
1)
2)
3)
4)
in the equation
NAME:______________________________________________
Algebra 2/Trig – Graphing with Vertical Shifts CLASSWORK
DATE:____________
PERIOD:__________
AIM: Graphing Trigonometric Graphs with Vertical Shifts

Vertical shifts can move your sinusoid curve either up or down!
The green graph moved up by 2
The green graph moved down by 1
or

Amplitude =

Frequency=

Period =

Vertical Shift =
ADVICE FOR GRAPHING SINE AND COSINE GRAPHS with VERTICAL SHIFTS
Model Problem #1: Sketch the graph of y = 2 sin x + 3 on the interval 0 ≤ x ≤ 2. State the amplitude, frequency,
period, and the vertical shift.
Model Problem #2: Sketch the graph of y = 4 – cos 2x on the interval 0 ≤ x ≤ 2. State the amplitude, frequency,
period and the vertical shift.
PRACTICE #1: Sketch the graph of
period, and vertical shift.
on the interval 0 ≤ x ≤ 2. State the amplitude, frequency,
WRITING EQUATIONS OF TRIG GRAPHS WITH VERTICAL SHIFTS
1. The accompanying graph represents a portion of a sound wave.
Write the trigonometric equation that represents this wave.
2. The periodic graph below can be represented by the trigonometric equation
real numbers.
where a, b, and c are
State the values of a, b, and c, and write an equation for the graph.
3. 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
. The range of the rope’s height above the ground is between 1 and 5 feet. The period of
the wave is . Write the equation that represents this wave.
NAME:______________________________________________
Algebra 2/Trig – Graphing with Vertical Shifts HOMEWORK
DATE:____________
PERIOD:__________
1. Sketch the graph of y = 2 cos x + 2 on the interval -2 ≤ x ≤ 2. State the amplitude, frequency, period and vertical
shift.
2. Sketch the graph of y = – sin 2x – 1 on the interval 0 ≤ x ≤ 2. State the amplitude, frequency, period and vertical
shift.
3. On the same set of axes, sketch and label the graphs of y = 2 sin x and y = cos 2x in the interval 0  x  2 .
Then identify the number of points that satisfy the equation 2 sin x = cos 2x between 0  x  2 .
4. The path traveled by a roller coaster is modeled by the equation y  27 sin 13x  30.
What is the maximum altitude of the roller coaster?
(1) 13
(3) 30
(2) 27
(4) 57
5. In physics class, Eva noticed the pattern shown in the accompanying diagram on an oscilloscope.
Which equation best represents the pattern shown on this oscilloscope?
1)
2)
3)
4)
6. What is the amplitude of the graph of the equation
?
(1)
(2) 2
(3) 3
(4)
7. What is the frequency of the function
(1) 5
(3) 3
(2)
(4)
?
8. A certain radio wave travels in a path represented by the equation
(1) 5
(3)
(2) 2
(4)
9. The graph below incorrectly represents the equation
graph is incorrect.
. What is the period of this wave?
. Write a mathematical explanation of why this
10. Which graph represents a sound wave that follows a curve whose period is π and that is in the form y  a sin bx ?
NAME:______________________________________________
Algebra 2/Trig – Graphing with Horizontal Shifts and Tangent Graphs CLASSWORK
DATE:____________
PERIOD:__________
AIM#1: Graphing Trigonometric Graphs with Horizontal Shifts
From the sinusoidal equation,
the horizontal shift is obtained by determining
the change being made to the x value.
The horizontal shift is C.
If the value of B is 1, the horizontal shift can
also be called a phase shift.
This expression is really
where the value of C is negative and the shift is left.
In this expression the value of C is positive and the shift is right.
𝒚 = 𝒂 𝐬𝐢𝐧 𝒃(𝒙 − 𝒄) + 𝒅
Amplitude =
Frequency=
or
Period =
𝒚 = 𝒂 𝐜𝐨𝐬 𝒃(𝒙 − 𝒄) + 𝒅
Vertical Shift =
Phase Shift =
1. State the amplitude, frequency, period, vertical shift and horizontal shift of the following functions.
1
π
y  2 sin  x   + 3
2
2
f(x) = - sin 4(x − 2) – 1
2. Write a trigonometric equation with the following properties:
1
amplitude of 3, frequency of 2, shifted down 4 units and amplitude of 2, period of 𝜋, shifted up 1 unit and
shifted to the left 3 units
shifted to the right 5𝜋 units
AIM #2: GRAPHING TANGENT
Tangent Function:
One cycle occurs between
and
.
There are vertical asymptotes at each end
of the cycle. The asymptote that occurs at
repeats every
units.
period:
amplitude: none, graphs go on forever in
vertical directions.
Note: a graphing utility, such as the one used to produce these graphs, may not show the
function approaching infinity (going on forever upward or downward). The graphs,
however, DO tend toward positive and negative infinity and do not STOP.

What are asymptotes? Based on the table below, where do the asymptotes occur for a tangent
graph between 0 and 2𝜋 ? Why?
0°
90°
180°
270°
360°
sin 
0
1
0
-1
0
cos 
1
0
-1
0
1
tan 
ADVICE FOR GRAPHING TANGENT GRAPHS
1. Write down the key points for a tangent graph
2. Plot the vertical asymptotes first in order to establish your restrictions of the graph
3. Plot the x-intercepts and then use the graphing calculator help finish the tangent curve
1. Sketch the graph 𝑦 = tan 𝑥 from −2𝜋 ≤ 𝑥 ≤ 2𝜋.
SYSTEMS OF TRIG GRAPHS WITH TANGENT
2. Graph the functions y = tan x and y = - 2sin x from −2𝜋 ≤ 𝑥 ≤ 2𝜋 on the same set of axis.
Then identify the number of points that satisfy the equation tan x = - 2sin x between −2𝜋 ≤ 𝑥 ≤ 2𝜋.
NAME: ____________________________________________
Algebra 2/Trig – Horizontal Shift and Tangent Graphs HOMEWORK
DATE:_________
PERIOD:_______
1. Which type of symmetry does the equation
y = cos x have?

3
to
,
2
2
(1)
(2)
(3)
(4)
line symmetry with respect to the y-axis
line symmetry with respect to y = x
point symmetry with respect to the origin
point symmetry with respect to (π,0)
5. As x increases from
the value of sinx
(1) decreases, then increases
(2) increases, then decreases
(3) increases, only
(4) decreases, only
2. Write the equation of the graph of y = 3 sin x
translated 2 units up and right  units.
6. The graph below incorrectly represents the
1
equation y = 1.5sin x. Write a mathematical
2
explanation of why this graph is incorrect.
3. The path traveled by a roller coaster is modeled by
the equation
. What is the
maximum altitude of the roller coaster?
1) 13
2) 27
3) 30
4) 57
4. The Sea Dragon, a pendulum ride at an amusement
park, moves from its central position at rest according
to the trigonometric function
,
where t represents time, in seconds. What is the length
of the period of this ride?
(1) 5
(3) 3
(2) 6
(4) 10
7. A pair of figure skaters graphed part of their routine on a grid. The male skater’s path is represented by
1
the equation m(x) = 4cos x , and the female skater’s path is represented by the equation f(x) = −sinx.
2
On the accompanying grid, sketch both paths and state how many times the paths of the skaters intersect
between x  0 and x = 4π.
8. State the amplitude, period, and vertical shift of the function f(x) = −3 cos (2x) + 1. Then, graph the
function in the interval -π ≤ x ≤ π.
9. Sketch the graph 𝑦 = tan 𝑥 from −𝜋 ≤ 𝑥 ≤ 𝜋.
10. State the amplitude, frequency, period, vertical shift and the phase shift of the following function.
f(x) = 4 cos 3𝜋(x + 2) + 5
11. The accompanying graph represents a portion of a sound wave. The graph can be represented by the trigonometric
equation 𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐 where a, b, and c are real numbers.
State the values of a, b, and c and write an equation that represents this graph.
NAME:____________________________________________________
Algebra 2/Trig – Graphing Reciprocal Trig Functions CLASSWORK
DATE:_________
PERIOD:_______
GRAPHS OF COSECANT AND SECANT
Cosecant Function:
period:
amplitude: none, graphs go on forever in vertical directions.
There are vertical asymptotes. The asymptote that occurs at
The x-intercepts of y = sin x are the asymptotes for y = csc x.
repeats every
units.
repeats every
units.
Secant Function:
period:
amplitude: none, graphs go on forever in vertical directions.
There are vertical asymptotes. The asymptote that occurs at
The x-intercepts of y = cos x are the asymptotes for y = sec x.
ADVICE FOR GRAPHING SECANT AND COSECANT
1. Identify the reciprocal trig function relationship.
2. Plug the equation into the calculator and graph it in using ZOOM 7: ZoomTrig
3. Based on the given reciprocal trig function relationship,
graph
or
in the given interval.
3. Draw asymptotes at the x-intercepts.
4. Draw the secant or cosecant curves at the max and min values on the graph.
**DO NOT CROSS THE ASYMPTOTES!**
MODEL PROBLEM #1: Sketch the graph
from
.
MODEL PROBLEM #2: Sketch the graph
from
.
GRAPH OF COTANGENT
Cotangent Function:
One cycle occurs between 0 and
.
There are vertical asymptotes at each end of the cycle.
The asymptote that occurs at repeats every units.
period:
amplitude: none, graphs go on forever in vertical directions
Note: The graphs of y = tan(x) and y = cot(x) "face" in opposite directions.
MODEL PROBLEM #3: Sketch the graph

from
Where are the asymptotes located for the graph of
.
?
PRACTICE
1. Write a trigonometric function that matches each of the following graphs. Check your answers with a partner.
2. Sketch the graph
from
.
NAME:____________________________________________________
Algebra 2/Trig – Graphing Reciprocal Trig Functions HOMEWORK
1. Sketch the graph of
2. Sketch the graph
in the interval
from
.
.
DATE:_________
PERIOD:_______
3. Sketch the graph of
in the interval
4. Which graph represents one complete cycle of the equation
1)
3)
2)
4)
.
?
5. Which equation is represented by the graph below?
1)
2)
3)
4)
6. What is the period of the graph of
?
(A)
(B)
(C)
(D)
(E)
7. What is the range of the function , where
(A)
(B)
(C)
(D)
(E)
8. As
(A)
(B)
(C)
(D)
(E)
increases from to
, the value of
?
9. Sketch the graph
from
10. On the same set of axes, sketch the graphs of
0 ≤ x ≤ 2.
. State the amplitude, frequency, and period.
and
for the values of x in the interval
State the number of values of x in the interval 0 ≤ x ≤ 2 that satisfy the equation
.
NAME:_____________________________________________
Algebra 2/Trig – Graphs of Inverse Trig Functions CLASSWORK
DATE:_____________
PERIOD:___________
LEARNING GOAL #1: Inverse Trig Functions Using Arc Notation
Inverse Notation:

What is the value of

( )
What is the value of
( )
**Use you calculator to your advantage (degrees mode)!**
1. If
1) 1
2)
, what is the value of
?
2. What is the value of
1)
2)
3)
4)
3)
4)
?
3. If
, the value of y is
4. If
1)
1)
2)
2)
3)
3)
4)
4)
5. What is the principal value of
1)
3)
2)
4)
(
√
)?
6. If
, the value of y is
, then x is equal to
1)
3)
2)
4)
LEARNING GOAL #2: Graphs of Inverse Trig Functions

When we studied inverse functions in general, we learned that the inverse of a function can be
formed by switching the x and y values. To prove if the inverse of a given graph will be a function,
you can perform the Horizontal Line Test on the original graph.
Summary of the Graphs of Inverse Trig Functions
Inverse sine: f (x) = sin-1(x)
Domain:
Range:
Restricted Domain for y = sin x:
Inverse cosine: f (x) = cos-1(x)
Domain:
Range:
Restricted Domain for y = cos x:
Inverse tangent: f (x) = tan-1(x)
Domain:
Range:
Restricted Domain for y = tan x:
Determining Domains of Inverse Trig Functions
1. The inverse of Sin x is a function. The domain of Sin x could be
(1) {
(3) {
(2) {
(4) {
2.
(1) {
(2) {
is a function. The domain of Cos x could be
(3) {
(4) {
3. The inverse of Tan x is a function. The domain of Tan x could be
(1) {
(3) {
(2) {
(4) {
NAME:____________________________________________________
Algebra 2/Trig – Unit 15: Trigonometry V REVIEW SHEET
1. The graph of which equation has amplitude 2 and period
1)
3)
2)
4)
2. A monitor displays the graph
DATE:_________
PERIOD:_______
?
. What will be the amplitude after a dilation of 2?
1) 5
2) 6
3) 7
4) 10
3. The brightness of the star MIRA over time is given by the equation
, where x represents time
and y represents brightness. What is the period of this function, in radian measure?
4. If
, then the maximum value of
1)
2)
3)
4)
5. As x increases from
(A)
(B)
(C)
(D)
(E)
to
, the value of
is:
6.
Which equation represents the graph below?
1)
2)
3)
4)
7. Sketch and label the function
in the interval
.
8. Identify the amplitude, frequency, period, vertical shift, and horizontal shift of the trigonometric equation
(
)
9. What is the amplitude of the function shown in the accompanying graph?
1) 1.5
2) 2
3) 6
4) 12
10. On the same set of axes, sketch the graphs of
-2 ≤ x ≤ 2.
and
for the values of x in the interval
State the number of values of x in the interval -2 ≤ x ≤ 2 that satisfy the equation
.
11. Which statement regarding the inverse function is true?
1) A domain of
is
.
2) The range of
is
.
3) A domain of
is
4) The range of
is
.
.
12. The periodic graph below can be represented by the trigonometric equation
are real numbers.
State the values of a, b, and c, and write an equation for the graph.
where a, b, and c
13. If
, then k is
1) 1
2) 2
3)
4)
14. Which is an equation of the graph shown below?
1)
2)
3)
4)
15. Which of the following shows the graph of
?