Name: Teacher: Pd:€¦ · Day 3: Graphing with Properties of Trig Graphs and System of Trig Graphs...

1

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: _______

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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


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


Day 4: Graphing Sine and Cosine Curves with Vertical Shifts SWBAT: Graphing Sine and Cosine Curves with Vertical Shifts



Day 5: Graphing Sine and Cosine Curves with Vertical/Horizontal Shifts SWBAT: Graphing Sine and Cosine Curves with Vertical/Horizontal Shifts



Day 6: Graphing Reciprocal Trig Functions

SWBAT: Graphing Reciprocal Trig Functions



Day 7: Graphing Inverse Trig Functions

SWBAT: Graphing Inverse Trig Functions



Day 8: Test

NAME:___________________________________________________ DATE:____________

Algebra 2/Trig – Graphing Sine and Cosine CLASSWORK 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 radians (or 360o) for the sine and cosine curves.

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˚ 90˚ 180˚ 270˚ 360˚

Radians 0 2

2

3 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 = (length of one cycle)

Frequency = 1 (# of cycles in radians)

Domain =

Range =

called a “wave” because of its rolling wave-like appearance (also referred to as oscillating)

Amplitude = 1 (height)

Period = (length of one cycle)

Frequency = 1 (# of cycles in radians)

Domain =

Range =

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:___________________________________________________ DATE:____________

Algebra 2/Trig – Graphing Sine and Cosine HOMEWORK PERIOD:__________

1. Graph y = sin x from

2. Sketch the graph y = cos x from

3. Sketch the graph y = 2 sin x from

REGENTS REVIEW

3. Solve the equation algebraically for all values of x.

5. Solve 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 . State the domain and range, in interval notation, of this

function.

9. The inverse of the function is

1)

2)

3)

4)

10. Write a quadratic equation that has as one of its roots.

NAME:___________________________________________________ DATE:____________

Algebra 2/Trig – Graphing with Properties of Trigonometry Graphs CLASSWORK 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 y = a sin bx or y = a cos bx

Frequency y = a sin bx or y = a cos bx

Period y = a sin bx or y = a cos bx

State the amplitude, period, and frequency of each graph below:

xy sin3

xy

2

1sin

xy 2cos2

1

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:___________________________________________________ DATE:____________

Algebra 2/Trig – Graphing with Properties of Sine and Cosine Graphs HOMEWORK PERIOD:__________

1. Graph xy 2sin3 from

2. Sketch the graph xy2

1cos from

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: 14352 x

9. Solve the inequality 1563 x for x. Graph the solution on the line below.

10. Solve algebraically for x:

NAME:___________________________________________________ DATE:____________

Algebra 2/Trig – Properties of Trig Graphs and System of Trig Graphs CLASSWORK 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 between0 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

20 x .

At 2

x

, find the difference: 2 sin x – cos 2x

PRACTICE: Graph the functions y = cos2

1x and y = -3 sin x from on the same set of axis.

Then identify the number of points that satisfy the equation cos2

1x = -3 sin x between .

NAME:___________________________________________________ DATE:____________

Algebra 2/Trig – Properties of Graphs and Systems of Graphs HOMEWORK PERIOD:__________

1. Sketch the graph of on the interval 0 ≤ x ≤ . State the amplitude, frequency and the period.

2. Sketch the graph of

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?

1)

2)

3)

4)

6. Write 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)

2)

3)

4)

8. What is the period of the function

?

1)

3)

2)

4)

9. What is the minimum value of in the equation

?

1)

2)

3)

4)

NAME:______________________________________________ DATE:____________

Algebra 2/Trig – Graphing with Vertical Shifts CLASSWORK PERIOD:__________

AIM: Graphing Trigonometric Graphs with Vertical Shifts

Vertical shifts can move your sinusoid curve either up or down!

or

Amplitude = Frequency=

Period = Vertical Shift =

The green graph moved up by 2

The green graph moved down by 1

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

on the interval 0 ≤ x ≤ 2. State the amplitude, frequency,

period, and vertical shift.

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 where a, b, and c are

real numbers.

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:______________________________________________ DATE:____________

Algebra 2/Trig – Graphing with Vertical Shifts HOMEWORK 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 20 x .

Then identify the number of points that satisfy the equation 2 sin x = cos 2x between0 x 2 . 4. The path traveled by a roller coaster is modeled by the equation .3013sin27 xy

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 . What is the period of this wave?

(1) 5 (3)

(2) 2 (4)

9. The graph below incorrectly represents the equation . Write a mathematical explanation of why this

graph is incorrect.

10. Which graph represents a sound wave that follows a curve whose period is π and that is in the form y a bx sin ?

NAME:______________________________________________ DATE:____________

Algebra 2/Trig – Graphing with Horizontal Shifts and Tangent Graphs CLASSWORK 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 or 𝒚 = 𝒂 𝐜𝐨𝐬 𝒃(𝒙 − 𝒄) + 𝒅

Amplitude = Frequency= Period = Vertical Shift =

Phase Shift =

1. State the amplitude, frequency, period, vertical shift and horizontal shift of the following functions.

2x

2

1sin2y

π + 3

f(x) = - sin 4(x − 2) – 1

2. Write a trigonometric equation with the following properties:

amplitude of 3, frequency of 1

2, shifted down 4 units and

shifted to the right 5𝜋 units

amplitude of 2, period of 𝜋, shifted up 1 unit and shifted to the left 3 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: ____________________________________________ DATE:_________ Algebra 2/Trig – Horizontal Shift and Tangent Graphs HOMEWORK PERIOD:_______

1. Which type of symmetry does the equation y = cos x have? (1) line symmetry with respect to the y-axis (2) line symmetry with respect to y = x (3) point symmetry with respect to the origin (4) point symmetry with respect to (π,0)

2. Write the equation of the graph of y = 3 sin x translated 2 units up and right units. 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

5. As x increases from 2

to

2

3,

the value of sinx (1) decreases, then increases (2) increases, then decreases (3) increases, only (4) decreases, only

6. The graph below incorrectly represents the

equation y = 1.5sin2

1x. Write a mathematical

explanation of why this graph is incorrect.

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

the equation m(x) = 4cos2

1x , and the female skater’s path is represented by the equation f(x) = −sinx.

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:____________________________________________________ DATE:_________

Algebra 2/Trig – Graphing Reciprocal Trig Functions CLASSWORK 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 repeats every units. The x-intercepts of y = sin x are the asymptotes for y = csc x.

Secant Function:

period: amplitude: none, graphs go on forever in vertical directions.

There are vertical asymptotes. The asymptote that occurs at repeats every units. 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 .


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.


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:____________________________________________________ DATE:_________

Algebra 2/Trig – Graphing Reciprocal Trig Functions HOMEWORK PERIOD:_______

1. Sketch the graph of in the interval .

2. Sketch the graph from .

3. Sketch the graph of in the interval .

4. Which graph represents one complete cycle of the equation ?

1)

2)

3)

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 increases from

to

, the value of

(A)

(B)

(C)

(D)

(E)

9. Sketch the graph

from . State the amplitude, frequency, and period.

10. On the same set of axes, sketch the graphs of and for the values of x in the interval

0 ≤ x ≤ 2.

State the number of values of x in the interval 0 ≤ x ≤ 2 that satisfy the equation .

NAME:_____________________________________________ DATE:_____________

Algebra 2/Trig – Graphs of Inverse Trig Functions CLASSWORK 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 , what is the value of ?

1) 1

2)

3)

4)

2. What is the value of ?

1)

2)

3)

4)

3. If , the value of y is

1)

2)

3)

4)

4. If , the value of y is

1)

2)

3)

4)

5. What is the principal value of ( √

)?

1) 3)

2) 4)

6. If , 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. is a function. The domain of Cos x could be

(1) { (3) {

(2) { (4) {

3. The inverse of Tan x is a function. The domain of Tan x could be

(1) {

(3) {

(2) {

(4) {

NAME:____________________________________________________ DATE:_________

Algebra 2/Trig – Unit 15: Trigonometry V REVIEW SHEET PERIOD:_______

1. The graph of which equation has amplitude 2 and period ?

1) 3)

2)

4)

2. A monitor displays the graph . 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 is:

1)

2)

3)

4)

5. As x increases from

to

, the value of

(A) (B) (C) (D) (E)

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 and for the values of x in the interval

-2 ≤ x ≤ 2.

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 where a, b, and c

are real numbers.

State the values of a, b, and c, and write an equation for the graph.

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 ?

Name: Teacher: Pd:€¦ · Day 3: Graphing with Properties of Trig Graphs and System of Trig Graphs...

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