Graphs of Trigonometric Functions - Open School...

Module 2

Module 2, Section 2

Graphs of Trigonometric Functions

IntroductionYou have studied trigonometric ratios since Grade 9Mathematics. In this module you will study the ratios asfunctions. You will look at tables of values of these functions,generate their graphs, and list their properties. It is importantfor you to do all the questions in the assignments because somenew concepts are developed within the assignments. Usingtechniques you developed in Module 1, you will studytransformations of these circular functions. In order to find thezeros of the functions, you will develop skills in circular functionequation solving and in relationships amongst these functionswhich are called identities.

Section 2 — Outline

Lesson 1 Sine and Cosine Graphs

Lesson 2 Transformations of the Sine and Cosine Functions

Lesson 3 Graphs of the Remaining Circular Functions

Review

Principles of Mathematics 12 Section 2, Introduction 65

Notes

Module 2

66 Section 2, Introduction Principles of Mathematics 12

Module 2

Lesson 1

Sine and Cosine Graphs

OutcomesUpon completing this lesson, you will be able to:• sketch the basic graphs of y = sin x and y = cos x• list the properties of each of the above two functions• sketch transformations of the above two functions and describe

how the basic properties have been altered• solve unrestricted sine and cosine equations

OverviewNow that you can find circular function values for any real numberyou are ready to sketch the graphs of these functions.

The Sine CurveIf you do not have a graphing calculator, use a pen or pencil to finishthe diagram below, recording your observations as the central angleθ moves from quadrant to quadrant. The idea is to get a feel for thegradual rise and fall of a y-value as it travels around a circle.

Keep in mind what you learned in the last section: on the unit circle,a y-value for a point is the same as a sin θ value for the central angle.

x

y

0

π2

π

3π2

θ

Principles of Mathematics 12 Section 2, Lesson 1 67

The result is a portion of the sine curve, y = sin θ. As the choice forthe independent variable is arbitrary, you will often be asked tosketch y = sin x instead of y = sin θ. It really does not matter; it’ssimply a choice of names.

θ

y

1

–1π

2

π

3

2

π

2π

as θθ increases y-values sin θθ curve

In Quadrant I,

from 0 to

are positive andincreasing from 0 to 1

starting at the origin,the curve rises slowly

from 0 to 1, at θ =

In Quadrant II,

from

are positive anddecreasing from 1 to 0

the curve declinesslowly from 1 to 0,at θ = π

In Quadrant III,

from

are negative anddecreasing from 0 to –1

the curve declinesslowly from 0 to –1,

In Quadrant IV,

from

are negative andincreasing from–1 to 0

the curve risesslowly from –1 to 0, at θ = 2π

π2 π

2

to2π π

π πto 32 at θ π= 3

2

32

2π πto

68 Section 2, Lesson 1 Principles of Mathematics 12

Module 2

On your graphing calculator, you can sketch the curve by followingthese steps:

1. Be sure your calculator is in radian mode.

2. Set your viewing window as follows:Xmin = –0.1Xmax = 7Xscl = 1Ymin = –1.2Ymax = 1.2Yscl = 1

3. Press Y= and enter sin(X) as Y1.

4. Press GRAPH.

You will get the following graph:

By zooming 7:ZTrig you will see the curve begin at θ = –6.152285 . . . and end at θ = 6.152285 (You can confirm thesevalues by checking the Xmin and Xmax values by pressing WINDOW).The curve will go through almost “two cycles.”

Extra for ExpertsYou can do a simulation to see how the sine curve evolves from theunit circle by plotting the unit circle and the sine curveparametrically. Follow these steps:

1. Press MODE and select Par instead of Func.

2. Press Y= and enter the unit circle as follows:X1T = cos(T)Y1T = sin(T)

These are the coordinates on the unit circle (cos θ, sin θ).


Module 2

3. On the same screen, enter:X2T = TY2T = sin(T)

Note: This is the equation y = sin θ.

4. Change your viewing window as follows:Tmin = 0Tmax = 7Tstep = 0.1Xmin = –1.2Xmax = 7Xscl = 1Ymin = –2.7Ymax = 2.7Yscl = 1

5. Press GRAPH and watch as the unit circle is drawn.

6. TRACE both curves and you will see how the y-values (i.e., sin θvalues) are related to the unit circle’s y-coordinates.

7. Try zooming ZSquare and ZTrig. TRACE both curves again to geta better feeling for the sine curve and its evolution.

Properties of Sine CurveYou should be able to draw a sketch of the sine curve without anyaids and list the properties stated below.

1. The domain is the set of all real numbers, since you can travelaround the unit circle as far as you wish, clockwise orcounterclockwise. Make sure you understand that the unit circleis used to generate the coordinates (cos θ, sin θ). However, thedomain of sin θ and cos θ is similar to a string wrapped aroundthe unit circle. This is the reason that these functions aresometimes referred to as “wrapping functions.”

2. The range is [–1, 1].

3. The y-intercept is 0.


Module 2

4. The zeros of the function are all integral multiples of π, thatis, . . . –2π, –π, 0, π, 2π, 3π. . . . A more concise expression ofthe function’s zeros is the notation {kπ | k ∈ I}. Notice thatthe solution is not restricted to an interval such as [0, 2π];therefore, the solution is infinite and we give the solution setas a formula to generate all the possible zeros.

Functions with a repeating pattern like this sine functionare called periodic functions. Examples of periodicfunctions are all around you: tides, the motion of a Ferriswheel, and so on. The time for the pattern to repeat itself iscalled the period. For patterns in wallpaper or in a cablesweater, the length between each pattern repetition is theperiod. Mathematically a period p can be a measure of lengthor time; after that period p, or interval p, the functionrepeats.

Definition: A function f(x) is a periodic function if thereexists a number p > 0, such that for all x in the domain of f , f(x + p) = f(x). The smallest such number p is called theperiod of f . (The period is the shortest distance you musttravel along the x-axis for the function to begin anothercycle.)

5. The period for the sine function is 2π.

θ

y

2π

2π

2π1

−1

θ

y

−3π −2π −π π 2π 3π 4π

1

−1


Module 2

6. The graph below shows that sin (x) is symmetric about theorigin. On any line drawn through the origin that intersects thegraph, the origin is the midpoint (the symmetry point) of thatstraight line. When that is true, it follows that sin (–x) = –sin(x)or sin(x) = -sin(–x). Symmetry about the origin is called “odd”symmetry. We'll explain why it's “odd” after you learn about theother kind (“even” symmetry) in the next Guided Practiceexercise.

7. Definition: Curves with wavelike forms, such as the sine curve,have a line midway between the high and low points of the curve.This line is called the axis of the curve. The distance from the axisto the maximum, or to the minimum, curve values is called theamplitude of the curve.

The amplitude of the sine curve is 1.

θ

y

1

−1

θ

y

1

−1


Module 2

If we restrict our attention to one cycle of the sine curve, from 0 to2π, and subdivide the curve into the four quadrants, two additionalproperties are apparent (see 8 and 9).

8. Since the curve is above the axis in Quadrants I and II, it followsthat sin θ > 0 in these two quadrants. Similarly, sin θ < 0 in Quadrants III and IV. This agrees with the CASTrule.

9. As you move along the curve from left to right the curve isincreasing (moving up) in Quadrants I and IV and decreasing inQuadrants II and III.

θ

y

I II III IV2ππ

1

−1


Module 2


Module 2

Guided Practice – The Cosine CurveSketch the curve y = cos θ using the unit circle or graphingcalculator. List the above nine properties of the cosine curve asindicated below.

1. Domain

2. Range

3. y-intercept

4. Zeros

5. Period

6. Symmetry: Does cos(x) = cos(–x)? or does cos(x) = –cos(–x)?

7. Amplitude

8. In one revolution, when is cos θ

a) positive? b) negative?

9. In one revolution, when is cos θ

a) increasing? b) decreasing?

Check your answers in the Module 2 Answer Key.

Lesson 2

Transformations of theSine and Cosine Functions

Now that you know how to draw the basic sine and cosine curves,you will turn your attention to some transformations of these basiccurves.

Example 1Sketch the graph of: a) y = 3 sin x and state its range and periodb) y = –cos x and state the values of its intercepts

Solutiona) Notice that you are graphing y against x. The choice of x or θ

letters is arbitrary. Just do not confuse this independent variablex with the dependent variable x on the unit circle. There is noconnection between them.

y = 3 sin x stretches the sin x graph vertically by a factor of 3.Range = [–3 , 3]. Period is still 2π.

x

y

−3

3

−π π 2π 3π


Module 2


Module 2

b)

Reflects the graph over the x-axis.

y-intercept = –1. x-intercepts are still

Example 2Sketch and state the period of:

(Recall: The period is the length of 1 cycle of the graph.)

Solution

a)

Period is because the curve is compressed by a factor of 4.24 2π π=

x

y

π 2π

−1

1

3π2

π2

−π2

a)

b)

y x

y

=

= −FHGIKJ

sin

cos

4

2θ π

( )k k2 1 I .2

odd multiples of 2

π

π

+ ∈

x

y

−π π 2π

−1

1


b)

Shifts the graph units to the right.

To illustrate the use of the hint, determine the starting andending value of one cycle.

Start End

22

42 2

52

πθ π

π πθ

πθ

− = = +

=

θ π

θ π

−FHGIKJ =

=

20

2

2π

x

y

2ππ2

π

−1

1

A useful hint: Many students find it easier to sketchcircular function curves by using the normal period of thefunction to determine where the transformed curve beginsand ends one cycle. For example, the period of a basic sinecurve is 2π; therefore, find the starting and ending point ofone cycle as follows:

Curve Start one cycle at End one cycle at Period

basic y = sin x 0 2π 2π

transformedfunction

y = sin 4x

basic y = cos x

transformedfunction

= 1cos

2y x

When does 4x = 0?When x = 0

0

When does

When x = 0

=10?

2x

When does 4x = 2π?

2π

When does

When x = 4π

12 ?

2x π=

2π

4π

π2When x = π

2


Module 2

Begin drawing the basic cycle of the cosine curve at

and finish the cycle at

The horizontal translation of to the right is also called a phase

shift of

Note: We’ll define phase shift more fully in Section 4 Lesson 1.

Period is still

Note: With experience you will notice that the period is only affected ifyou change the numerical coefficient before the x (or θ).

Example 3The graph below is a representation of the function y = a sin b(x – c). Find a, b, and c.

x

y

−5

5

ππ2

−

52 2

2π π π− = .

x

y

π2

5π2

start finish

1

−1

.2πθ =

π2

θ π= 52

.

θ π=2

Solution

Example 4Sketch at least one cycle of the graph of:

a)

Compare to the general form y = A sin [B(x – C)] + D

From A = –2, the base graph of y = sin x is reflected in the x-axis and the amplitude is changed to 2.

From B = 3, the period of the graph is

From D = 0, there is no vertical displacement.

The graph is most easily sketched in stages:

1. Vertical shift is zero, so the graph lies along the x-axis.2. The amplitude is 2, so lightly sketch a straight line 2 units up

from the axis and 2 units down, i.e., through y = 2 and y = –2.

3. The phase shift is , so the graph will begin units to

the left. Mark an X at .

4. The period is , so mark points on the x-axis in

multiples of 2 from .

3 2π π−

23π

– ,02π

2π–

2π

leftFrom C , the phase shift is or to the .2 2 2π π π− −=

2 2.

B 3π π=

π−A = –2; B = 3; C = ; D = 0

2

π = − + 2sin 3

2y x

By inspection = 52

period 2

–phase shift

6 6

a

bb

c

ππ

π π

= = → =

−= → =


Module 2


5. Each period has four major points, always in the same order, foreither a sine or cosine curve: zero, maximum, zero, minimum.In the case of a sine curve with a negative amplitude, the order iszero, minimum, zero, maximum. Divide each periodinto four equal sections, starting at , and place an x at

the four major points of each period.Check that there are x’s in each of the zeroes you calculated instep 4.

y

π6

1

5π

6−1

2−

x−π

9π62

2

−7π6

∗ ∗∗ ∗

∗

∗ ∗

∗ ∗∗ ∗

∗

∗ ∗

∗ ∗

∗

∗

–2π

y

π6

1

5π6

−1

2−

x−π

9π62

2

−7π6

∗

2e.g. –2 3 6

2 56 3 65 2 9 36 3 6 2

And to the left.2 7– –

2 3 67 2 11– –6 3 6

So far your set-up should look like this :

π π π

π π π

π π π π

π π π

π π π

+ =

+ =

+ = =

− =

− =


Module 2

6. Join the points with a smooth curve.

b)

Note that this function is NOT in the form y = A cos [B(x – C)] + DFirst, we have to remove the coefficient of x from inside thebrackets by taking out a factor of 2.

1. Vertical shift is –1, so the axis lies on the line y = –1.2. The amplitude is 1, so the maximum and minimum points go

through lines one unit up and down, respectively, from –1,i.e., y = 0 and y = –2.

[ ]

Function is

cos 2 – 2 16

or cos2 – 16

Now A 1 amplitude = 1 no change

2 B 2 period = 2

C phase shift is to the right6 6

D 1 vertical displacement is 1i.e., mid-line or ax

y x

y x

π

π

π π

π π

= − = −

= ⇒

= ⇒ =

= ⇒

= − ⇒ −is for the graph is 1y = −

2We do this by rewriting as or 2 to create a common3 6 6

factor of 2.

π π π

π = − cos 2 – 1

3y x

y

π6

1

5π

6−1

2−

x−π

9π62

2

−7π6

∗ ∗∗ ∗

∗

∗ ∗

∗ ∗∗ ∗

∗

∗ ∗

∗ ∗

∗

∗

3. The phase shift is , so the graph will begin units tothe right.

4. The period is π, so mark points on the graph in multiples

of π, starting at

5. The cosine pattern is max, zero, min, zero. Divide each periodinto four equal sections and mark in the major points.

6. Join the points with a smooth curve.y

π6

17π

6

−1

2−

x

13π6

−5π6

2

−11π6

y

π6

17π

6

−1

2−

x

13π6

−5π6

2

−11π6

∗ ∗ ∗∗ ∗

∗∗

∗

∗∗ ∗ ∗

∗∗∗∗

∗

7 13 5 11i.e., , , and left at – , – .

6 6 6 6 6 6π π π π π π

6π

6π


Module 2

Guided Practice

1. Sketch each of the following. State the domain, range, amplitude,y-intercept, and period.

2. For each of the following graphs of y = a sin b(x + c), find thevalues of a, b, and c.

a)

b)

x

y

−3

3

−5π π2

3π2

π2

−2

y

π

−2

2

2π x

( )a) 2 sin b) sin 2

c) sin d) 2 cos

e) cos 2 f) cos 2 14

g) sin h) 3sin 23

i) sin 3 j) 2cos 12

1k) cos l) 2 sin2

y x y x

y x y x

y x y x

y x y x

y x y x

y x y x

π

π

π

π

π

= == − =

= − = + − = = −

= − = − = = −


Module 2

3. Use the same graphs as in Question 2, but change the function toy = a cos b(x + c) and find the values of a, b, and c.

4. Find the period of each function.

5. Find the x-intercepts of each circular function.

Extra for Experts6. Sketch y = cos x – |cos x|.

7. Sketch y = sin x + |sin x|.


( )

a) sin 3 b) cos 2

c) sin d) sin2

e) cos f) 3 sin 2

y x y x

y x y x

xy y x

ππ

ππ

= =

= − =

= =− −

a) b)

c) d)

e) f)

y x y x

y x y x

y x y x

= = −

= − = − −

= + = −FHGIKJ

3 2

3 4

412

sin sin

cos sin

cos sin

π π

π

π

b gb g b gb g


Module 2

Lesson 3

Graphs of the Remaining Circular Functions

OutcomesUpon completing this lesson, you will be able to:• sketch the graph of y = tan θ• list the properties of the function y = tan x• sketch the basic graphs of the reciprocal functions: y = sec x, y =

csc x, and y = cot x• list the properties of each of the above reciprocal functions

Reciprocal Functions

Example 1: Sketching y = tan θFor angles terminating in Quadrant I, tan θ begins very small(infinitely close to 0) and becomes infinitely large as θ gets

closer and closer to Of course, since

which is undefined. Therefore, we see an asymptote at

θ

y

π2

θ π=2

.

π

ππ π= = =2

2

sin 1cos 0, tan

2 2 0cosπ

.2


Module 2

In Quadrant II, for values of θ very close to but greater than like 1.58r, 1.59r, etc., value of tan θ start as very largenegatives (tan 1.58r = –108.65, tan 1.59r = –52.067). As θ gets closerto π, the values for tan θ approach zero (tan 3.11 = –0.032, tan 3.13 =–0.012, tan π = 0).

In Quadrants III and IV, the curve repeats its performances inQuadrants I and II, respectively.

⇒

Extra for ExpertsYou can plot the tangent curve by following these steps:

1. Press MODE and select Func.

2. Enter Y1 = tan(X).

3. Press GRAPH, then Zoom ZTrig.

4. TRACE the branches of the curve.

θ

y

π2

3π2

θ

y

π2

π 2π

3π2

θ

y

π2

π

π,

2


Module 2

We can observe and record the following properties of the function y= tan x:

Domain:

Range: ℜ

y-intercept: 0

Zeros occur at: 0, π, –π, 2π, –2π, etc. . . .

Equations of asymptotes:

Period: π

Symmetry: tan(–x) = –tan(x), or symmetric with respect to (0,0).

Behaviour: Increasing in all quadrantsPositive in Quadrants I and III.Negative in Quadrants II and IV

Example 2: Reciprocal Functions—Sketching y = sec xWe are going to graph y = sec x by using the skills acquired inModule 1, Section 2, Lesson 5.

To sketch

Step 1: Sketch y = cos x

x

y

1

–1

1sec

cosy x

x= =

( )π π π⎧ ⎫= = = + ∈ Ι⎨ ⎬⎩ ⎭

3, , ...., 2 1 ,

2 2 2x x x k k

{ }π∈ℜ = ∈Ι/, ,x x k k

πThe set of all real numbers which

are not odd multiples of .2

( )π⎧ ⎫∈ℜ ≠ + ∈Ι⎨ ⎬⎩ ⎭

, 2 1 ,2

x x k k


Module 2

Step 2: We now need to graph the reciprocal of this function. Sinceany x intercept (y = 0) has no reciprocal, it follows that the verticalasymptotes are at:

where n are integers

Step 3: Plot the invariant points which occur when y = 1 and y = –1

( ) 1, ,

,0 ( , )2 23 3,0 ( , )2 2

x y xy

undefined

undefined

π π

π π

→

→ →

( )2 13 5, , ,.....

2 2 2 2n

xπ π π +

= ± ± ± =


Module 2

Step 4: Plot a few points by taking the reciprocal of the y values.

Notice that very small positive and negative y values, found oneither side of the asymptotes, become very large positive and verylarge negative values respectively.

Step 5: Complete the sketch by joining these points with smooth curves.

Step 6: The characteristics of the graph y = sec x are

( )

( )

2 1Domain: ,

2Range: 1 or 1

2 1Asymptotes: where are integers

2

nx x

y y

nx n

π

π

+∈ℜ ≠ ±

≥ ≤ −+

=

( )

±→

±

±=

±→

±

±=

±→

±

→

2,32

1,3

414.1,41

2,42

1,4

155.1,63

2,62

3,6

1,,

ππ

πππ

πππ

yxyx


Module 2

Example 3: Graph y = 2 csc x

Step 1: To graph , graph y = sin x first.

Step 2: We now need to graph the reciprocal of this function. Sinceany x intercept (y = 0) has no reciprocal, it follows that the verticalasymptotes are at:

where n ∈ are integers

Step 3: Find the invariant points which occur when y = ±1

, 2 , 3 ,.....x nπ π π π= ± ± ± =

xx

xysin

2)sin

1(2csc2 ===


Module 2

Step 4: Plot a few points by taking the reciprocal of the y values andsketch a smooth curve.

Step 5: Vertically expand the graph by a factor of 2.

Step 6: Notice the characteristics.

Domain: , where are integersRange: 2 or 2Asymptotes: where are integers

x x n ny yx n n

π

π

∈ℜ ≠≥ ≤ −=

( ) 1, ,x y xy

→


Module 2


Module 2

Guided Practice1. Each of the remaining functions is a reciprocal of one of the basic

three circular functions. Use the graphs of the basic functions tosketch their reciprocal functions: y = csc x, y = sec x, and y = cot x.

2. For each of the three functions in Question 1, list all theproperties of each function:

• Domain• Range• y-intercept• Zeros• Equations of asymptotes• Periods• Symmetry (odd or even)• Behaviour

3. Sketch each of the following transformations:

Note: You may wish to draw the reciprocal functions on thecalculator by sketching:


Review Section 2 before attempting the review questions followingthe summary on the next page. These questions should help youconsolidate your knowledge as you prepare Section Assignment 2.2.

yxy

xy

x= = =1 1 1

sin,

cos,

tanand

a) tan b) sec 1

c) csc d) cot2

e) tan 1 f) 2sec

y x y x

y x y x

y x y x

π

= = +

= − =

= − =

Principles of Mathematics 12 Section 2, Summary 93

Module 2

Summary of Section 2

Know the basic shapes of all trigonometric functions.

{ }

period = asymptotes at , domain = | ,

range =

k k I

x x k k I

ππ

π∈

≠ ∈

ℜ

y

x

y = cot x

( )

( )

period = asymptotes at

2 1 ,

22 1

domain = | ,2

range =

kk I

kx x k I

π

π

π

+∈

+ ≠ ∈

ℜ

y

x

y = tan x

[ ]

period = 2amplitude = 1domain

range 1,1

π

= ℜ= −

y

2π

x

y = cos x

[ ]

period = 2amplitude = 1domain

range 1,1

π

= ℜ= −

y

2π

x

y = sin x

Transformations of Sine and Cosine

( ) ( )

[ ]

sin cos

domain =

range = ,

amplitude =

2period =

phase = (units left)axis at

y A B x C D or y A B x C D

A D A D

A

BCy D

π

= − + = − +

ℜ− + +

=

( )

( )

( )

period = 2

2 1asymptotes at ,

22 1

domain = |2

range = , 1] [1,

kk

kx x

ππ

π

+

+ ≠−∞ − ∪ ∞

y

x

y = sec x

{ }( )

period = 2asymptotes at ,

domain = | ,

range = , 1] [1,

k k I

x x k k I

ππ

π∈

≠ ∈

−∞ − ∪ ∞

y

x

y = csc x

94 Section 2, Summary Principles of Mathematics 12

Module 2

Module 2, Section 2Review

1. Find the period of each function.

2. Sketch each of the following. State the domain, range, amplitude,y-intercept, and period.

3. Sketch at least one period of:

For each graph determine:

i) the domain

ii) the range

iii) the period

iv) the amplitude

v) the phase shift

vi) the y-intercept

( )( )

4

12

a) 3sin 2 1

b) 5cos 3

y x

y x

π

π

= − +

=− + +

( )

( )

y x y x

y x y x

y x y x

y x y x

y x y x

y x y x

a) sin b) cos 2

c) sin d) 2 sin

e) sin 2 f) cos 2

g) sin h) sin

1i) cos j) tan2

k) sec 1 l) csc

π

=− =

= + =

=− = −

= =

= =−

= − =

( )

( )

a) 4 sin 2

b) 2 sin

c) cos 3

y x

y x

y x

π π

π

=−

= −

= −

Principles of Mathematics 12 Section 2, Review 95

Module 2

4. Sketch the graphs of the following. Do your graphs look familiar?


Now do the section assignment which follows this section. When it iscomplete, send it in for marking.

a) sin2

b) cos2

y x

y x

π

π

= −

= −

Rewrite as 2 2

Hint : x xπ π − − −

96 Section 2, Review Principles of Mathematics 12

Module 2

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PRINCIPLES OF MATHEMATICS 12

Section Assignment 2.2

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Version 05 Module 2


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Version 05 Module 2

98 Section Assignment 2.2 Principles of Mathematics 12

Module 2


Graphs of Trigonometric Functions

1. Fill in the blanks with the correct response.

a) The figure below shows a sketch of the graph ofy = sin 3x. The x-coordinate of the point P in the diagram is: _____

b) The period of function f is 5. If f (1) = 4, f (2) = 5, and f (4) = –2, the value of f (7) is: _____

c) The phase of the function f(x) = 5 cos (2x + 6) is: _____

d) The range of the function y = 3 sec x – 2 is: _____

e) The y-intercept of the function g(x) = 2 sin 2π(x – 1) + 1 is: _____

f) The range of the function _____

g) The domain of the function y = tan 2θ is: _____

h) Find the equations of the asymptotes for y = csc(3θ): _____

( ) –4sin 2 is:3

f x xπ = − −

P x

y


Module 2

Total Value: 40 marks(Mark values in margins)

(8 marks)

2. Sketch the graph of f(x) = –3 sin x and state the period andrange of this function.

Period: ________________ Range: ___________________

3. Sketch the graph of g(x) = |cos πx|and state the period andzeros of this function.

Period:

_____________

Zeros:

_____________

x

g(x)

x

f(x)


Module 2

(4)

(5)

4. Graphically solve the equation

[Hint: Sketch two graphs and find their points of intersection.]

5. If y = p cos(qx + r) + s, identify the meaning of each of the fourconstants p, q, r, and s. Describe in a sentence how they affect thegraph of y = cos x.

x

y

sin cos where 0 2π= ≤ <x x x


Module 2

(3)

(6)

6. Sketch at least one period of the graph of

State the domain, range, period, and phase.

7. Identify the equation of the graph below in the formy = A sin B(x – C) + D

x

y

π

−2

2

3π2

π2

−π2

−6

2cos4 .2

y xπ = −


Module 2

(6)

(4)

8. Find the area of rectangle ABCD if the equation of the curveis y = tan x.

Send in this work as soon as you complete this section.

x

y

B

π4

π2−

CD

A


Module 2

(4)

Total: 40 marks


Module 2

Graphs of Trigonometric Functions - Open School...

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