Functions – Part 3
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
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Contents Unit 7 Revision notes
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Exercise 7.1
4
A review of parent functions in trigonometry
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Answers to Exercise 7.1
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Examination questions with solution hints and answers
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More questions from past examination papers
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Answers
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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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Functions – Part 3
REVISION UNIT 7: SKETCHING TRIGONOMETRIC GRAPHS In the chapter on functions you studied the transformation of y = f (x ) into y = af ( x − p ) + q . In other words, saw the effect that the parameters a, p and q have on the graph of f (x ) . 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
f ( x ) = sin kx , how do you determine the period of f ( x ) ? 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:
y = f ( x ) = sin x
y = g ( x ) = cos x
y = h( x ) = tan x
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. 3
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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
(0 ; 360 ), (− 360 ; 360 ), (− 180 ; 360 ), (− 90 ; 270 ) or any interval of your choice. o
o
o
o
o
o
o
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2. Over the same intervals as above, now make sketches of a sin x , a cos x and a tan x where a =
1 , 2, 3 or any value of your choice – but choose a value that does not make it awkward for 2
yourself! 3. Over the same intervals as above, make sketches of sin ax , cos ax and tan ax .
4. Also over the same intervals make sketches of sin (x ± p ) , cos (x ± p ) and tan (x ± p ) choosing values of p such as ± 90 0 , ± 60 0 , ± 45 0 ; or ± 30 0 .
5. Repeat the above question for a sin (x ± p ) , a cos (x ± p ) and a tan (x ± p ) 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 o ; 360 o ) : 1.
y = sin 2 x
2.
y = 3 cos 4 x
3.
y = tan
1 x 2
B. Draw sketches or use compound angle formulae (see Unit 16) to prove the following: 1. sin x + 90 o = cos x 2. cos x + 90 o = − sin x
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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, n ∈ Z 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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Functions – Part 3
Parent functions and their characteristics for values of x ∈ (− 360 o ; 360 o ) Function
Graph and characteristics
y = sin x
y
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x -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
Period = 360 o that is the shape repeats itself every 360 o . x − intercepts at x = 0 o + k .180 o y − intercept at y = 0 Maximum value =+1 and occurs at x = 90 o + k .360 o Minimum value =‐1 and occurs at x = −90 o + k .360 o Domain (all values of x for which the function is defined): x ∈ (− ∞; + ∞ )
Range(all values of y for which the function is defined): y ∈ [− 1; + 1] or − 1 ≤ y ≤ 1 Amplitude = (maximum value‐minimum value)/2 =1 y = cos x y
1
x -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
o
o
Period = 360 that is the shape repeats itself every 360 . x − intercepts at x = 0 o + (2k + 1).90 o y − intercept at y = +1 Maximum value =+1 and occurs at x = 0 o + k .360 o Minimum value =‐1 and occurs at x = ±90 o + k .360 o Domain (all values of x for which the function is defined): x ∈ (− ∞; + ∞ )
Range(all values of y for which the function is defined): y ∈ [− 1; + 1] or − 1 ≤ y ≤ 1 Amplitude = (maximum value‐minimum value)/2 =1 5
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Function
y = tan x
Graph and characteristics y
9 8 7 6 5 4 3 2 1
-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
-1 -2 -3 -4 -5 -6 -7 -8 -9
Period = 180 o that is the shape repeats itself every 180 o . x − intercepts at x = 0 o + k .180 o y − intercept at y = 0 Asymptotes at x = 0 o + (2k + 1).90 o No maximum value No minimum vale Domain: x ∈ (− ∞; + ∞ ) − (2k + 1).90 o or all values of x between − ∞ and + ∞ but excluding x = (2k + 1).90 o Range: y ∈ ℜ 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. 6
Functions – Part 3
ANSWERS EXERCISE 7.1 A . 1. Period = 180 o ; 2. Period = 90 o ; 3. Period = 360 o 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:
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1. Solving graphically: Draw the graphs of y = sin x + 90 o and y = cos x on the same set of axes. If the graphs coincide then you have proved the identity. 2. Using compound angle formulae:
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LHS = sin x + 90 o = sin x cos 90 o + sin 90 o cos x = sin x ⋅ 0 + 1 ⋅ cos x = cos x = RHS and so the identity is proved. 3. You could also use your knowledge of angles greater than 90 o and the property of complementary angles as follows:
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sin x + 90 o = sin 180 o − x + 90 o
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= sin 180 o − x − 90 o
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= sin 90 o − x = cos x 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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Functions – Part 3
PAPER 1 QUESTION 7 Number 7.1
7.2
7.3
Hints and answers 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: 180 o Revision question: At what values of x does tan x have asymptotes? Now write down the values of x at which f will have asymptotes in the given interval. Answer: x = −45 o Revision question: What do you do to reflect a function in the x ‐axis? Now write down the equation of k . Answer: k (x ) = − sin 2 x
Work out the solutions in the boxes below 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.
WARNING: see above.
WARNING: see above.
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DoE/ADDITIONAL EXEMPLAR 2008
Preparation for the Mathematics examination brought to you by Kagiso Trust
PAPER 1 QUESTION 6
DoE/NOVEMBER 2008
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Functions – Part 3
PAPER 1 QUESTION 6
Number Hints and answers 6.1 What are the asymptotes of the parent function tan x ?
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So what are the asymptotes of tan x − 45 o ?
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The answer you give must lie strictly within the required interval for x ∈ − 90 o ;180 o .
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6.2
6.3
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Answer: You write down the answer. Use tan (− θ ) = − tan θ and then describe the transformation in words.
Answer: You write down the answer. 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. 11
DoE/NOVEMBER 2008
Work out the solutions in the boxes below
How is the graph of tan x transformed when x is replaced by x − 45 o ?
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MORE QUESTIONS FROM PAST EXAMINATION PAPERS Exemplar 2008
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Functions – Part 3
Preparatory Examination 2008
13
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Feb – March 2009
DIAGRAM SHEET 1
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Functions – Part 3
November 2009 (Unused paper)
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November 2009 (1)
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Functions – Part 3
Feb – March 2010
DIAGRAM SHEET 2
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ANSWERS Exemplar 2008 8.1 Period = 360 o 8.2 The shift changes the range of g and will Become [‐1; 3]. 8.3 h( x ) = cos x Preparatory Examination 2008 8.1 2 8.2 − 180 o < x ≤ 180 o or − 180 o ;180 o 8.3 g ( x ) = 2 sin x − 60 o 8.4 0 o ; 90 o ; 180 o Feb/March 2009 8.1 Sketch:
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8.2 y ∈ [− 4;4] 8.3 720 o 8.4 θ = 90 o or θ = 90 o + n360 o
November 2009 (Unused papers) 6.1 p = 2 and q = 1 6.2 y ∈ [0;2] or 0 ≤ y ≤ 2 6.3.1 By finding the x ‐values of the points of Intersection of the graphs of f and g . 6.3.2 180 o or − 180 o or about − 112,5 o November 2009(1) 7.1 y ∈ [− 3;3] or − 3 ≤ y ≤ 3 7.2 B 82,63o ;0,38 o 7.3 Period = 120 o 7.4 x = −180 o Feb/March 2010 8.1 Sketch:
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8.2 x = −60 o 8.3 Reflection about the x ‐axis.
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