functions 3

Functions – Part 3

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MATHEMATICS

Learner’s Study and

Revision Guide for

Grade 12

FUNCTIONS ‐ Part 3 TRIGONOMETRIC CURVES

Revision Notes, Exercises and Solution Hints by

Roseinnes Phahle

Examination Questions by the Department of Basic Education

Preparation for the Mathematics examination brought to you by Kagiso Trust

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Contents

Unit 7

Revision notes 3

Exercise 7.1 4

A review of parent functions in trigonometry 4

Answers to Exercise 7.1 7

Examination questions with solution hints and answers 8

More questions from past examination papers 12

Answers 18

How to use this revision and study guide

1. Study the revision notes given at the beginning. The notes are interactive in that in some parts you are required to make a response based on your prior learning of the topic from your teacher in class or from a textbook. Furthermore, the notes cover all the Mathematics from Grade 10 to Grade 12.

2. “Warm‐up” exercises follow the notes. Some exercises carry solution HINTS in the answer section. Do not read the answer or hints until you have tried to work out a question and are having difficulty.

3. The notes and exercises are followed by questions from past examination papers.

4. The examination questions are followed by blank spaces or boxes inside a table. Do the working out of the question inside these spaces or boxes.

5. Alongside the blank boxes are HINTS in case you have difficulty solving a part of the question. Do not read the hints until you have tried to work out the question and are having difficulty.

6. What follows next are more questions taken from past examination papers.

7. Answers to the extra past examination questions appear at the end. Some answers carry HINTS and notes to enrich your knowledge.

8. Finally, don’t be a loner. Work through this guide in a team with your classmates.


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REVISION UNIT 7: SKETCHING TRIGONOMETRIC GRAPHS

In the chapter on functions you studied the transformation of )(xfy = into qpxafy +−= )( . In

other words, saw the effect that the parameters qpa and , have on the graph of )(xf .

In Chapter 5 you looked at what effect these parameters have on the trigonometric graphs of sin, cos and tan . You saw there that the effects are the same on the trigonometric graphs as on graphs for algebraic functions – so you really have nothing new to learn so long as you can do sketches of the parent functions sin x , cos x and tan x .

In one respect, though, the graphs of the trigonometric functions are different from those of algebraic expressions. They are different in that they are periodic which means that their shapes repeat themselves over certain intervals.

What helps in determining the x ‐intercepts is the period of the function. For example, given

( ) kxxf sin= , how do you determine the period of ( )xf ?

Period =

The sine and cosine functions also have amplitudes which is given by:

Parent functions in trigonometry

In order to be able to sketch trigonometric functions you will need to know the basic shapes:

( ) xxfy sin==

( ) xxgy cos==

( ) xxhy tan==

and their characteristics or special features.

Sketching graphs of the trigonometric functions

What you must particularly note and mark on your sketches are the following:

• The x ‐intercepts, that is where the graphs cross the x ‐axis;

• The y ‐ intercepts, that is where the graphs cross the y ‐axis;

• The maximum value of the graph;

• The minimum value of the graph.


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Get yourself lots of graph paper on which to practice sketching the following trigonometric functions:

1. Start by making sketches of each of sin x , cos x and tan x over different intervals such as ( )oo 360 ;0 , ( )oo 360 ;360− , ( )oo 360 ;180− , ( )oo 270 ;90− or any interval of your choice.

2. Over the same intervals as above, now make sketches of a sin x , a cos x and a tan x where a

=21, 2, 3 or any value of your choice – but choose a value that does not make it awkward for

yourself!

3. Over the same intervals as above, make sketches of axsin , axcos and axtan .

4. Also over the same intervals make sketches of sin ( )px ± , cos ( )px ± and tan ( )px ± choosing

values of p such as 090± , 060± , 045± ; or 030± .

5. Repeat the above question for a sin ( )px ± , a cos ( )px ± and a tan ( )px ± where you choose

a and p again as above

EXERCISE 7.1

A. Determine the periods of the following functions and sketch them over the interval )360 ;360( oo− :

1. xy 2sin= 2. xy 4cos3= 3. xy21tan=

B. Draw sketches or use compound angle formulae (see Unit 16) to prove the following:

1. ( ) xx o cos90sin =+ 2. ( ) xx o sin90cos −=+

A REVIEW OF THE PARENT FUNCTIONS IN TRIGONOMETRY

A full review of the parent functions in trigonometry is given below.

In the review, general formulae are given for the location of the x ‐intercepts, the turning points and the asymptotes. In these formulae, n stands for the integers from ∞− to ∞+ .

That is n = . . . . . . ‐4; ‐3; ‐2; ‐1; 0; 1; 2; 3; . . . .

or simply, Zn∈

You will find these formulae useful for working out the x ‐coordinates of critical points within intervals in which you will be required to make sketches of trigonometric functions. You need not memorise the formulae as you can always draw very rough sketches of the shapes of the functions and from such sketches very easily derive them .


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Parent functions and their characteristics for values of ( )oo 360 ;360−∈x

Function Graph and characteristics xy sin=

-360-345-330-315-300-285-270-255-240-225-210-195-180-165-150-135-120-105 -90 -75 -60 -45 -30 -15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

-1

1

x

y

Period = o360 that is the shape repeats itself every o360 . −x intercepts at oo 180.0 kx += −y intercept at 0=y

Maximum value =+1 and occurs at oo 360.90 kx += Minimum value =‐1 and occurs at oo 360.90 kx +−= Domain (all values of x for which the function is defined): ( )∞+∞−∈ ;x

Range(all values of y for which the function is defined): [ ]1 ;1 +−∈y or 11 ≤≤− y Amplitude = (maximum value‐minimum value)/2 =1

xy cos=

-360-345-330-315-300-285-270-255-240-225-210-195-180-165-150-135-120-105 -90 -75 -60 -45 -30 -15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

-1

1

x

y

Period = o360 that is the shape repeats itself every o360 .

−x intercepts at oo 90).12(0 ++= kx −y intercept at 1+=y

Maximum value =+1 and occurs at oo 360.0 kx +=

Minimum value =‐1 and occurs at oo 360.90 kx +±= Domain (all values of x for which the function is defined): ( )∞+∞−∈ ;x

Range(all values of y for which the function is defined): [ ]1 ;1 +−∈y or 11 ≤≤− y Amplitude = (maximum value‐minimum value)/2 =1


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Function Graph and characteristics xy tan=

-360-345-330-315-300-285-270-255-240-225-210-195-180-165-150-135-120-105 -90 -75 -60 -45 -30 -15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 30

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

y

Period = o180 that is the shape repeats itself every o180 . −x intercepts at oo 180.0 kx += −y intercept at 0=y

Asymptotes at oo 90).12(0 ++= kx No maximum value No minimum vale

Domain: ( ) ( ) o90.12 ; +−∞+∞−∈ kx or all values of x between ∞− and ∞+ but

excluding o90).12( += kx Range: ℜ∈y or all real values of y between ∞− and ∞+ or +∞<<∞− y

Although working with the formulae of general solutions that appear in the table above is very useful, as advised earlier, it will be just as good enough to always draw on rough paper a little sketch of the basic shape in which you mark in the intercepts, turning points and asymptotes and from which you will be able to deduce the characteristics of whatever function you are required to sketch.

You will find a fuller treatment of the sketching of trigonometric functions in UNIT 17. You would do well to study this section now because it ties in very well with this unit.


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ANSWERS

EXERCISE 7.1

A . 1. Period = o180 ; 2. Period = o90 ; 3. Period = o360

B. If you use sketches to prove these identities then the graph of the LHS must coincide with the

graph of the RHS.

Take question 1:

1. Solving graphically: Draw the graphs of ( )o90sin += xy and xy cos= on the same set of

axes. If the graphs coincide then you have proved the identity.

2. Using compound angle formulae:

LHS = ( )o90sin +x

= xx cos90sin90cossin oo +

= xx cos10sin ⋅+⋅

= xcos

= RHS

and so the identity is proved.

3. You could also use your knowledge of angles greater than o90 and the property of complementary angles as follows:

( ) ( )( )ooo 90180sin90sin +−=+ xx

= ( )oo 90180sin −− x

= ( )x−o90sin

= xcos

Now question 2

Repeat any of the above procedures.


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PAPER 1 QUESTIONS 7 DoE/ADDITIONAL EXEMPLAR 2008


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PAPER 1 QUESTION 7 DoE/ADDITIONAL EXEMPLAR 2008

Number Hints and answers Work out the solutions in the boxes below 7.1 Revision question:

How do you work out the period of the sin or the cosine of kx ? And what about tan kx ? Now write down the period of g . Answer:

o180

WARNING: all of Question 7 involves very little or no calculation to be worked out to get an answer. In cases such as these where you can see the answer make doubly sure that you understand the answer and that you can find it without looking at the answer.

7.2 Revision question: At what values of x does tan xhave asymptotes? Now write down the values of x at which f will have asymptotes in the given interval. Answer:

ox 45−=

WARNING: see above.

7.3 Revision question: What do you do to reflect a function in the x ‐axis? Now write down the equation of k . Answer: ( ) xxk 2sin−=

WARNING: see above.


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PAPER 1 QUESTION 6 DoE/NOVEMBER 2008


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PAPER 1 QUESTION 6 DoE/NOVEMBER 2008

Number Hints and answers Work out the solutions in the boxes below 6.1 What are the asymptotes of the parent

function tan x ? How is the graph of tan x transformed when x is replaced by ( )o45−x ? So what are the asymptotes of tan ( )o45−x ? The answer you give must lie strictly within the required interval for

[ ]oo 180;90−∈x . Answer: You write down the answer.

6.2 Use tan ( )θ− = − tanθ and then describe the transformation in words. Answer: You write down the answer.

6.3 What is the period of sin x ? What is the period of sin kx ? So what is the period of sin 2 x ? Write down the equation of the resulting function in the form =y Answer: You write down the answer.


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MORE QUESTIONS FROM PAST EXAMINATION PAPERS

Exemplar 2008


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Preparatory Examination 2008


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Feb – March 2009

DIAGRAM SHEET 1


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November 2009 (Unused paper)


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November 2009 (1)


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Feb – March 2010

DIAGRAM SHEET 2


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ANSWERS

Exemplar 2008 8.1 Period = o360 8.2 The shift changes the range of g and will Become [‐1; 3]. 8.3 ( ) xxh cos= Preparatory Examination 2008 8.1 2 8.2 oo 180180 ≤<− x or ( ]oo 180;180−

8.3 ( ) ( )o60sin2 −= xxg

8.4 ;0o ;90o o180 Feb/March 2009 8.1 Sketch:

8.2 [ ]4;4−∈y

8.3 o720 8.4 o90=θ or oo 36090 n+=θ

November 2009 (Unused papers) 6.1 2=p and 1=q

6.2 [ ]2;0∈y or 20 ≤≤ y 6.3.1 By finding the x ‐values of the points of Intersection of the graphs of fand g .

6.3.2 o180 or o180− or about o5,112−

November 2009(1) 7.1 [ ]3;3−∈y or 33 ≤≤− y

7.2 ( )oo 38,0;63,82B

7.3 Period = o120 7.4 o180−=x Feb/March 2010 8.1 Sketch:

8.2 o60−=x 8.3 Reflection about the x ‐axis.

functions 3

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