Play with Graphs PDF

Play with Graphs With Sessionwise Theory & Exercises Play with Graphs With Sessionwise Theory & Exercises Amit M. Agarwal ARIHANT PRAKASHAN (Series), MEERUT  ARIHANT PRAKASHAN (Series), MEERUT All Rights Reserved...

Play with Graphs With Sessionwise Theory & Exercises Play with Graphs With Sessionwise Theory & Exercises Amit M. Agarwal ARIHANT PRAKASHAN (Series), MEERUT  ARIHANT PRAKASHAN (Series), MEERUT All Rights Reserved © AUTHOR No part of this publication may be re-produced, stored in a retrieval system or by any means, electronic mechanical, photocopying, recording, scanning, web or otherwise without the written permission of the publisher. Arihant has obtained all the information in this book from the sources believed to be reliable and true. However, Arihant or its editors or authors or illustrators don’t take any responsibility for the absolute accuracy of any information published, and the damages or loss suffered thereupon. All disputes subject to Meerut (UP) jurisdiction only. Administrative & Production Offices Regd. Office ‘Ramchhaya’ 4577/15, Agarwal Road, Darya Ganj, New Delhi -110002 Tele: 011- 47630600, 43518550 Head Office Kalindi, TP Nagar, Meerut (UP) - 250002 Tel: 0121-7156203, 7156204 Sales & Support Offices Agra, Ahmedabad, Bengaluru, Bareilly, Chennai, Delhi, Guwahati, Hyderabad, Jaipur, Jhansi, Kolkata, Lucknow, Nagpur & Pune. ISBN : 978-93-25298-69-9 PO No : TXT-XX-XXXXXXX-X-XX Published by Arihant Publications (India) Ltd. For further information about the books published by Arihant, log on to www.arihantbooks.com or e-mail at [email protected] Follow us on  PREFACE It is a matter of great pleasure and pride for me to introduce to you this book “Play with Graphs”. As a teacher, guiding the Engineering aspirants for over a decade now, I have always been in the lookout for right approach to understand various mathematical problems. I had always felt the need of a book that can develop and sharpen the ideas of the students within a very short span of time. The book in your hands, aims to help you solve various mathematical problems by the use of graphs. Ways to draw different types of graphs are very easy and can be understood by even an average student. I feel glad to mention that the use of graphs in solving various mathematical problems has been well illustrated in this book. I would like to take this opportunity to thank M/s Arihant Prakashan for assigning this work to me. It is their inspiration that has encouraged me to bring this book in this present form. I would also like to thank Arihant DTP Unit for the nice laser typesetting. Valuable suggestions from the students and teachers are always welcome, and these will find due places in the ensuing editions. Amit M. Agarwal  CONTENTS 1. INTRODUCTION TO GRAPHS 1-50 1.1 Algebraic functions 1. Polynomial function 2. Rational function 3. Irrational function 4. Piecewise functions 1.2 Transcendental functions 1. Trigonometric function 2. Exponential function 3. Logarithmic function 4. Geometrical curves 1.3 Trigonometric inequalities 1.4 Solving equations graphically 2. CURVATURE AND TRANSFORMATIONS 51-137 2.1 Curvature 2.2 Concavity, convexity and points of inflexion 2.3 Plotting of algebraic curves using concavity 2.4 Graphical transformations 2.5 Sketching h(x)= maximum {f(x), g(x)} and h(x)= minimum {f (x), g(x)} 2.6 When f(x), g(x) — f(x) + g(x) = h(x) 2.7 When f(x), g(x) — f(x). g(x) = h(x) 3. ASYMPTOTES, SINGULAR POINTS AND CURVE TRACING 138-165 3.1 Asymptotes 3.2 Singular points 3.3 Remember for tracing cartesian equation HINTS & SOLUTIONS 166-182  er Introduction of Graphs t ap INTRODUCTION OF 1 C h GRAPHS ➥ In this section, we shall revise some basic curves which are given as. Polynomial Rational Algebraic Modulus Irrational Signum Piecewise Greatest integer function Fractional part function FUNCTIONS Least integer function Trigonometric Exponential Logarithmic/Inverse of exponential Transcendental Geometrical curves Inverse trigonometric curves 1.1 ALGEBRAIC FUNCTIONS 1. Polynomial Function A function of the form: f ( x ) = a 0 + a1 x + a 2 x 2 + … + a n x n ; where n ∈ N and a 0, a1 , a 2,…, a n ∈ R. Then, f is called a polynomial function. “f ( x) is also called polynomial in x”. 1  Some of basic polynomial functions are y (i) Identity function/Graph of f (x) = x y=x A function f defined by f ( x) = x for all x ∈ R, is called the Play with Graphs identity function. 45° x Here, y = x clearly represents a straight line passing through O the origin and inclined at an angle of 45° with x-axis shown as: The domain and range of identity functions are both equal to R. Fig. 1.1 (ii) Graph of f (x) = x 2 y y=x2 A function given by f ( x) = x 2 is called the square function. The domain of square function is R and its range is R + ∪ { 0} or [0, ∞) Clearly y = x 2 , is a parabola. Since y = x 2 is an even x O function, so its graph is symmetrical about y-axis, shown as: (iii) Graph of f (x) = x 3 Fig. 1.2 A function given by f ( x) = x 3 is called the cube function. y y = x3 The domain and range of cube are both equal to R. Since, y = x 3 is an odd function, so its graph is symmetrical about opposite quadrant, i.e., “origin”, shown as: x O (iv) Graph of f (x) = x 2 n ; n ∈ N If n ∈ N, then function f given by f ( x) = x 2n is an even function. So, its graph is always symmetrical about y-axis. Fig. 1.3 Also, x 2 > x 4 > x 6 > x 8 > … for all x ∈ ( −1, 1) and x 2 < x 4 < x 6 < x 8 < … for all x ∈ ( −∞, − 1) ∪ (1, ∞ ) Graphs of y = x 2 , y = x 4 , y = x 6 ,…, etc. are shown as: y=x4 y=x6 y y = x6 y = x2 y = –x y=x x –1 O 1 Fig. 1.4 (v) Graph of f (x) = x 2 n −1; n ∈ N If n ∈ N, then the function f given by f ( x) = x 2n −1 is an odd function. So, its graph is symmetrical about origin or opposite quadrants. 2  Here, comparison of values of x, x 3 , x 5 ,… y=x 5 y=x 3 Introduction of Graphs y y=x for x ∈ (1, ∞ ) x < x3 < x5 < … x ∈ ( 0, 1) x > x3 > x5 > … x –1 O 1 x ∈ ( −1, 0) x< x < x x3 > x5 > … Graphs of f ( x) = x, f ( x) = x 3 , f ( x) = x 5 ,… are shown as in Fig. 1.5 Fig. 1.5 2. Rational Expression A function obtained by dividing a polynomial by another polynomial is called a rational function. P ( x) ⇒ f ( x) = Q ( x) Domain ∈ R − { x | Q( x) = 0} i.e., domain ∈R except those points for which denominator = 0. Graphs of some Simple Rational Functions 1 (i) Graph of f (x) = y x 1 A function defined by f ( x) = is called the reciprocal (1,1) x 1 function or rectangular hyperbola, with coordinate axis as x 1 –1 O 1 asymptotes. The domain and range of f ( x) = is R – { 0}. (–1,–1) –1 x Since, f ( x) is odd function, so its graph is symmetrical about opposite quadrants. Also, we observe lim f ( x) = + ∞ and lim f ( x) = − ∞. Fig. 1.6 x → 0+ x → 0– and as x → ± ∞ ⇒ f ( x) → 0 1 Thus, f ( x) = could be shown as in Fig. 1.6. x 1 (ii) Graph of f (x) = 2 x 1 Here, f ( x) = 2 is an even function, so its graph is symmetrical about y-axis. x Domain of f ( x) is R − { 0} and range is (0, ∞). y Also, as y→∞ as lim f ( x) or lim f ( x). x → 0+ x → 0− and y→0 as lim f ( x). x→ ±∞ x 1 O Thus, f ( x) = could be shown as in Fig. 1.7. x2 Fig. 1.7 3  1 (iii) Graph of f (x) = 2n − 1 ; n∈N 1 x y= x y =1 1 y x3 Here, f ( x) = is an odd function, so its graph is x 2n – 1 Play with Graphs symmetrical in opposite quadrants. 1 (1,1) A Also, y → ∞ when lim f ( x) and x→ 0+ x O y→−∞ when lim f ( x). –1 (–1,–1) 1 x → 0− –1 B 1 1 Thus, the graph for f ( x) = ; f ( x) = 5 , …, etc. will x3 x 1 be similar to the graph of f ( x) = which has asymptotes x as coordinate axes, shown as in Fig. 1.8 Fig. 1.8 1 (iv) Graph of f (x) = 2 n ; n ∈ N x 1 1 We observe that the function f ( x) = 2n is an even y y = x2 x y = 14 x function, so its graph is symmetrical about y-axis. Also, y → ∞ as lim f ( x) or lim f ( x) x → 0+ x → 0− (–1,–1) (1,1) B 1 A and y → 0 as lim f ( x) or lim f ( x). x → −∞ x→ + ∞ x –1 O 1 The values of y decrease as the values of x increase. 1 1 Thus, the graph of f ( x) = 2 ; f ( x) = 4 , … , etc. will be Fig. 1.9 x x 1 similar as the graph of f ( x) = 2 , which has asymptotes as coordinate axis. Shown as in Fig. 1.9. x 3. Irrational Function The algebraic function containing terms having non-integral rational powers of x are called irrational functions. Graphs of Some Simple Irrational Functions (i) Graph of f (x) = x1/ 2 y Here; f ( x) = x is the portion of the parabola y 2 = x, which y=x lies above x-axis. y= x Domain of f ( x) ∈ R + ∪ { 0} or [ 0, ∞ ) (1, 1) 1 and range of f ( x) ∈ R + ∪ { 0} or [ 0, ∞ ). Thus, the graph of f ( x) = x1/2 is shown as; O 1 x Note If f (x) = x n and g (x) = x1/n , then f (x) and g (x) are inverse of Fig. 1.10 each other. ∴ f (x) = x n and g (x) = x1/n is the mirror image about y = x. 4 (ii) Graph of f (x) = x1/ 3 y=x3 Introduction of Graphs y y=x As discussed above, if g( x) = x 3. Then f ( x) = x1/3 y =x1/3 is image of g( x) about y = x. 1 where domain f ( x) ∈ R. x –1 O 1 and range of f ( x) ∈ R. –1 Thus, the graph of f ( x) = x1/ 3 is shown in Fig. 1.11; Fig. 1.11 (iii) Graph of f (x) = x1/ 2 n ; n ∈ N y y=x4 y=x2 y=x Here, f ( x) = x1/ 2n is defined for all x ∈ [ 0, ∞ ) and the values taken by f ( x) are positive. So, domain and range of f ( x) are [ 0, ∞ ). y = x1/2 1 y = x1/4 Here, the graph of f ( x) = x1/ 2n is the mirror image of the graph of f ( x) = x 2n about the line y = x, when x ∈ [ 0, ∞ ). Thus, f ( x) = x1/ 2 , f ( x) = x1/ 4 , … are shown as; O 1 x Fig. 1.12 (iv) Graph of f (x) = x1/ 2 n −1 , when n ∈ N Here, f ( x) = x1/ 2n −1 is defined for all x ∈ R. So, y y = x5 y=x3 y=x domain of f ( x) ∈ R, and range of f ( x) ∈ R. Also the graph of f ( x) = x1/ 2n −1 is the mirror image of the graph y =x1/3 y = x1/5 of f ( x) = x 2n −1 about the line y = x when x ∈ R. 1 Thus, f ( x) = x1/ 3 , f ( x) = x1/ 5 , …, are shown as; O x –1 1 Note We have discussed some of the simple curves –1 for Polynomial, Rational and Irrational functions. Graphs of the some more difficult rational functions will be discussed in chapter 3. Such as; Fig. 1.13 x 1 x2 + x + 1 y= y= 2 , y= 2 ,…. x+1 x −1 x −x+1 4. Piecewise Functions As discussed piecewise functions are: (a) Absolute value function (or modulus function), (b) Signum function. (c) Greatest integer function. (d) Fractional part function. (e) Least integer function. (a) Absolute value function (or modulus function)  x, x ≥ 0 y = |x|=  − x , x < 0 5  y = –x y y=x “It is the numerical value of x”. “It is symmetric about y-axis” where domain ∈R and range ∈[0, ∞). 135° 45° Play with Graphs Properties of modulus functions x O (i) |x | ≤ a ⇒ − a ≤ x ≤ a ; ( a ≥ 0) (ii) |x | ≥ a ⇒ x ≤ − a or x ≥ a ; ( a ≥ 0) Fig. 1.14 (iii) |x ± y| ≤ |x| + |y| (iv) |x ± y| ≥ |x| − |y |. (b) Signum function; y = Sgn(x) y-axis It is defined by; 1 |x| x  +1, if x> 0  or ; x≠0  y = Sgn( x) =  x |x| =  −1, if x< 0 x-axis O  0 ; x = 0.  0, if x=0 –1 Here, Domain of f ( x) ∈ R. and Range of f ( x) ∈ { −1, 0, 1}. Fig. 1.15 (c) Greatest integer function [x] = n [x] indicates the integral part of x which is nearest and smaller integer to x. It is also known as floor of x. Thus, [ 2.3] = 2, [ 0.23] = 0, [ 2] = 2, [ −8.0725] = − 9, …. n n+1 In general; x n ≤ x < n + 1 ( n ∈ Integer) ⇒ [ x] = n. Fig. 1.16 y-axis Here, f ( x) = [ x] could be expressed graphically as; 3 x [x] 2 0≤ x< 1 0 1 1≤ x< 2 1 2≤ x< 3 2 –3 –2 –1 O 1 2 3 4 x-axis –1 Thus, f ( x) = [ x] could be shown as; –2 Properties of greatest integer function –3 (i) [ x] = x holds, if x is integer. Fig. 1.17 (ii) [ x + I] = [ x] + I, if I is integer. (iii) [ x + y ] ≥ [ x] + [ y ]. (iv) If [ φ ( x)] ≥ I, then φ ( x) ≥ I. (v) If [ φ ( x)] ≤ I, then φ ( x) < I + 1. (vi) [ − x] = − [ x], if x ∈ integer. (vii) [ − x] = − [ x] − 1, if x ∉integer. “It is also known as stepwise function/floor of x.” 6  Introduction of Graphs (d) Fractional part of function Here, {.} denotes the fractional part of x. Thus, in y = { x}. x = [ x] + { x} = I + f ; where I = [ x] and f = { x} ∴ y = { x} = x − [ x] , where 0 ≤ { x} < 1; shown as: y x {x} 0≤ x< 1 x 1 1≤ x< 2 x−1 3 2 1 1 2 3 x+ x+ x+ x– x– x– x 2≤ x< 3 x−2 x –3 –2 –1 O 1 2 3 −1 ≤ x < 0 x+1 −2 ≤ x < − 1 x+2 Fig. 1.18 Properties of fractional part of x (i) { x} = x ; if 0≤ x< 1 (ii) { x} = 0 ; if x ∈ integer. (iii) { − x} = 1 − { x} ; if x ∈ integer. (e) Least integer function y = ( x) = x , ( x) or x indicates the integral part of x which is nearest and greatest integer to x. It is known as ceiling of x. [x ] = n (x) = x = n +1 Thus, 2.3023 = 3, ( 0.23) = 1, ( −8.0725) = − 8, ( −0.6) = 0 In general, n < x ≤ n + 1 ( n ∈ integer)) i. e., x or ( x) = n + 1 n x n+1 shown as; Fig. 1.19 Here, f ( x) = ( x) = x, can be expressed graphically as: y-axis x x = ( x) 3 −1 < x ≤ 0 0 2 0< x≤ 1 1 1 x-axis 1< x≤ 2 2 –4 –3 –2 –1 O 1 2 3 4 –1 −2 < x ≤ − 1 −1 –2 −3 < x ≤ − 2 −2 –3 Properties of least integer function Fig. 1.20 (i) ( x) = x = x, if x is integer. (ii) ( x + I) = x + I = ( x) + I ; if I ∈ integer. (iii) Greatest integer converts x = I + f to [ x] = I while x converts to ( I + 1). Note We shall discuss the curves: y = {sin x} , y = {x3 } , y = {sin−1 (sin x)} y = [sin x] , etc. in chapter 2. (Curvature and Transformations). 7  1.2 TRANSCENDENTAL FUNCTIONS 1. Trigonometric Function Play with Graphs (a) Sine function Here, f ( x) = sin x can be discussed in two ways i.e., Graph diagram and Circle diagram where Domain of sine function is “R” and range is [–1, 1]. Graph diagram y (On x-axis and y-axis) 3π π, 1 f ( x) = sin x, increases – ,1 2 1 2 y =1 strictly from –1 to 1 as x increases B D π π from − to , decreases strictly h1 h1 2 2 x π 2π 3π π π OA π π 3π 2π C from 1 to –1 as x increases from 2 2 2 2 2 3π y = –1 to and so on. We have graph π , –1 –1 3π, –1 2 2 2 as; Here, the height is same after Fig. 1.21 every interval of 2π. (i.e., In above figure, AB = CD after every interval of 2π). ∴ sin x is called periodic function with period 2π. Circle diagram π/2 π ,1 5π , 1 … = (On trigonometric plane or using 2 2 quadrants). Let a circle of radius ‘1’, y=1 γ i.e., unit circle. a β Then, sin α = , 1 b 1 1 a b α sin β = , 1 …, 5π, 3π, π y = 0 3π O 0=y x = 0, 2π, 4π ,… c c 2 sin γ = − , d 1 d 2π–δ sin δ = − , … , shown as. 1 y = –1 ∴ sin x generates a circle of radius ‘1’. 3π/2, 7π/2, … Fig. 1.22 (b) Cosine function Here, f ( x) = cos x The domain of cosine function is R and the range is [–1, 1]. Graph diagram (on x-axis and y-axis) y As discussed, cos x decreases strictly (–2π, 1) (0, 1) (2π, 1) from 1 to –1 as x increases from 0 to π, increases strictly from –1 to 1 as x increases from π to 2 π and so on. Also, x cos x is periodic with period 2 π. –2π –3π/2 –π –π/2 O π/2 π 3π/2 2π (–π, –1) (π, –1) 8 Fig. 1.23  y Introduction of Graphs Circle diagram y=1 Let a circle of radius ‘1’, i.e., a unit circle. a b Then, cos α = , cos β = − , b 1 β1 1 1 a δ α c d y=0 y=0 x cos γ = , cos δ = − O 1 1 d 1 γ c 1 ∴ cos x generates a circle of radius ‘1’. y = –1 (c) Tangent function Fig. 1.24 f ( x) = tan x y The domain of the function y = tan x is;  π 3π 5π  R − ± , ± ,± ,…  2 2 2   π  1 i. e., R − ( 2n + 1)   2 x and Range ∈ R or ( −∞, ∞ ). –π π O π π – 4 4 The function y = tan x increases strictly –1 from − ∞ to + ∞ as x increases from π π π 3π 3π 5π − to , to , to , … and so on. 2 2 2 2 2 2 x = –3π/2 x = –π/2 x = π/2 x = 3π/2 The graph is shown as : Fig. 1.25 π 3π 5π Note Here, the curve tends to meet at x = ± ,± ± , … but never meets or tends to 2 2 2 infinity. π 3π 5π ∴ x=± , ± , ± … are asymptotes to y = tan x. 2 2 2 (d) Cosecant function f ( x) = cosec x y y = cosec x y = cosec x y = cosec x π ,1 2 1 y = sin x x –2π –3π/2 –π –π/2 O π/2 π 3π/2 2π –1 – π ,–1 3π ,–1 2 2 y = cosec x y = cosec x y = cosec x Fig. 1.26 9  Here, domain of y = cosec x is, R − { 0, ± π, ± 2 π, ± 3 π, …} i. e., R − { nπ| n ∈ z} and range ∈ R − ( − 1, 1). Play with Graphs as shown in Fig. 1.26. The function y = cosec x is periodic with period 2 π. (e) Secant function f ( x) = sec x  π  Here, domain ∈ R − ( 2n + 1) n ∈ z  2  Range ∈ R − ( −1 , 1) Shown as: y y = sec x y = sec x (–2π, 1) (0, 1 ) (2π, 1 ) 1 x –2π – 3π –π π O π π 3π 2π – 2 2 2 2 y = cos x –1 (–π, –1 ) (π, –1 ) Fig. 1.27 The function y = sec x is periodic with period 2 π. Note (i) The curve y = cosec x tends to meet at x = 0 , ± π , ± 2 π , … at infinity. ∴ x = 0, ± π , ± 2π , … or x = nπ , n ∈ integer are asymptote to y = cosec x. π 3π (ii) The curve y = sec x tends to meet at x = ± ± , … at infinity. 2 2 π 3π 5π π ∴x = ± , ± , ± , … or x = (2 n + 1) , n ∈ integer are asymptote to y = cosec x. 2 2 2 2 Here, we have used the notation of asymptotes of a curve in the context of special curves, but we would have a detailed discussion in chapter 3. (f) Cotangent function f ( x) = cot x Here, domain ∈R − { nπ| n ∈ z} Range ∈ R. which is periodic with period π, and has x = nπ, n ∈ z as asymptotes. As shown in Fig. 1.28; 10  y y = cot x Introduction of Graphs x –2π 3π –π π O π π 3π 2π – – 2 2 2 2 asymptotes Fig. 1.28 2. Exponential Function Here, f ( x) = a x, a > 0, a ≠ 1, and x ∈ R, where domain ∈R, Range ∈ ( 0, ∞ ). Case I. a > 1 Here, f ( x) = y = a x increase with the increase in x, i.e., f ( x) is increasing function on R. y y = a x, a > 1 (0,1) x O Fig. 1.29 4x y-axis 3x For example; 2x y = 2 x , y = 3 x , y = 4 x ,… have; 2x < 3x < 4x < … for x > 1 (0,1) and 2 > 3 > 4 > … for 0 < x < 1. x x x O x-axis and they can be shown as; Fig. 1.30 y=ax y-axis Case II. 0 < a < 1 Here, f ( x) = a x decrease with the increase in x, i.e., f ( x) is 0 0) and a ≠ 1 is a logarithmic function. Thus, the domain of logarithmic function is all real positive numbers and their range is the set R of all real numbers. We have seen that y = a x is strictly increasing when a > 1 and strictly decreasing when 0 < a < 1. 11  Thus, the function is invertible. The If 0 < a < 1 If a > 1 y-axis y-axis inverse of this function is denoted by log a x, we write y = a x ⇒ x = log a y ; Play with Graphs where x ∈ R and y ∈( 0, ∞ ) (1,0) x-axis x-axis O O (1,0) writing y = log a x in place of x = log a y , we have the graph of y = log a x. Thus, logarithmic function is also known as inverse of exponential function. Fig. 1.32 Properties of logarithmic function 1. log e ( ab) = log e a + log e b { a, b > 0}  a 2. log e   = log e a − log e b { a, b > 0}  b 3. log e a m = m log e a {a > 0 and m ∈ R} 4. log a a = 1 {a > 0 and a ≠ 1} 1 5. log bm a= log b a { a, b > 0, b ≠ 1 and m ∈ R} m 1 6. log b a = { a, b > 0 and a, b ≠ 1} log a b log m a 7. log b a = { a, b > 0 ≠ {1} and m > 0} log m b 8. a loga m =m { a, m > 0 and a ≠ 1} 9. a logc b = b logc a { a, b, c > 0 and c ≠ 1}  x > y , if m>1 { m, x, y , > 0 and m ≠ 1} 10. If log m x > log m y ⇒  x < y , if 0< m < 1 which could be graphically shown as; If m > 1 (Graph of log m a) Again if 0 < m < 1. (Graph of log m a) logm x logm a logm x logm y logm y O 1 y x O x y1 Fig. 1.33 Fig. 1.34 ⇒ log m x > log m y when x > y and m > 1. ⇒ log m x > log m y ; when x < y and 0 < m < 1. 11. log m a = b ⇒ a=m b { a, m > 0 ; m ≠ 1 ; b ∈ R}  a > m ; if m > 1 b 12. log m a > b ⇒   a < m ; if 0 < m < 1 b  a < m b ; if m > 1 13. log m a < b ⇒ .  a > m ; if 0 < m < 1 b 12  Introduction of Graphs 4. Geometrical Curves y (a) Straight line ax + by + c = 0 (represents general equation of straight line). We 0, – c know, b c c y =– when x = 0 – ,0 a b x c O and x=− when y = 0 a joining above points we get required straight line. Fig. 1.35 (b) Circle We know, (i) x 2 + y 2 = a 2 is circle with centre ( 0, 0) (ii) ( x − a) 2 + ( y − b) 2 = r 2 , circle with and radius r. centre (a, b) and radius r. y y (a, b) x C r C (0, 0) (r, 0) x Fig. 1.36 Fig. 1.37 (iii) x 2 + y 2 + 2gx + 2fy + c = 0 ; (iv) ( x − x1 ) ( x − x 2 ) + ( y − y 1 ) ( y − y 2 ) = 0; centre ( −g, − f ); radius g 2 + f 2 − c. End points of diameter are ( x1 , y 1) and y ( x 2 , y 2). O r A B C (x1, y1) (x2, y2) (–g, –f ) x O Fig. 1.39 Fig. 1.38 (c) Parabola (i) y 2 = 4 ax (ii) y 2 = – 4 ax Vertex : (0, 0) Vertex : (0, 0) Focus : (a, 0) Focus : (– a, 0) Axis : x-axis or y = 0 Axis : x-axis or y = 0 Directrix : x=−a Directrix : x=a 13  y y y 2= – 4ax F Play with Graphs x x V (a, 0) Focus V (0, 0) Directrix (–a, 0) Vertex y 2 = 4ax x=a x = –a Fig. 1.40 Fig. 1.41 (iii) x 2 = 4 ay (iv) x 2 = − 4 ay Vextex : ( 0, 0) Vertex : ( 0, 0) Focus : ( 0, a) Focus : ( 0, − a) Axis : y-axis or x = 0 Axis : y-axis or x = 0 Directrix : y =−a Directrix : y =a y x 2 = 4ay y y=a F (0, a) V (0, 0) x x V (0, 0) y = –a F (0, –a) Directrix x 2 = –4ay Fig. 1.42 Fig. 1.43 (v) (y − k) 2 = 4a (x − h) Vertex : ( h, k ) y=k Focus : ( h + a, k ) V (h, k) F (h + a, k) Axis : x=h Directrix : x=h−a x = h–a x = h directrix (d) Ellipse Fig. 1.44 x2 y2 (i) 2 + 2 = 1 (a 2 > b 2 ) a b a 2> b 2 Centre : (0, 0) b Focus : ( ±ae, 0) Vertex : ( ±a, 0) –a O a b2 Eccentricity : e = 1 − –b a2 a x = –a/e x = a/e Directrix : x=± e Fig. 1.45 14  Introduction of Graphs x2 y2 (x − h) 2 (y − k) 2 (ii) + 2 =1 (a 2 < b 2 ) (iii) + = 1 (a 2 > b 2 ) a2 b a2 b2 y y b2> a2 y = b/e (0, b) (h, k +b) y=k A′ (a+h, k) (a–h, k) (h,k) A x (–a, 0) O (a, 0) (h, k–b) O x (0,–b) x=h y = –b/e directrix Fig. 1.46 Fig. 1.47 (e) Hyperbola y x2 y2 b (i) 2 − 2 = 1 y =– x a te b y= x a a b pto ym Centre : (0, 0) as Focus : ( ±ae, 0) Vertices : ( ±a, 0) x (–ae,0) (–a, 0) O (a, 0) (ae, 0) 2 b Eccentricity : e = 1 + a2 as ym a p tot Directrix : x= ± x = –a/e x = a/e e e b Fig. 1.48 In above figure asymptotes are y = ± x. a x2 y2 (x − h) 2 (y − k) 2 (ii) − + 2 =1 (iii) − =1 a2 b a2 b2 y y te pto ym (0, b) as y=k (h, k) O x x as (0, –b) ym pt ot e x=h Fig. 1.49 Fig. 1.50 15  (iv) x 2 − y 2 = a 2 (Rectangular hyperbola) (v) xy = c 2 As asymptotes are perpendicular. Therefore, Here, the asymptotes are x-axis and y-axis. called rectangular hyperbola. Play with Graphs y asymptote y=x to te ymp as (c, c) x O O asymptote as (–c, –c) ym pto te y = –x Fig. 1.51 Fig. 1.52 Note In above curves we have used the name asymptotes for its complete definition see chapter 3. Inverse Trigonometric Curves As we know trigonometric functions are many one in their domain, hence, they are not invertible. But their inverse can be obtained by restricting the domain so as to make invertible. Note Every inverse trigonometric is been converted to a function by shortening the domain. For example: Let f ( x) = sin x We know, sin x is not invertible for x ∈ R. In order to get the inverse we have to define domain as:  π π x ∈ − ,   2 2  π π ∴ If f : − ,  → [ −1, 1] defined by f ( x) = sin x is invertible and inverse can be represented  2 2 by:  π −1 π y = sin −1 x.  − ≤ sin x ≤   2 2 Similarly, y = cos x becomes invertible when f : [ 0, π] → [ −1, 1]  π π y = tan x ; becomes invertible when f :  − ,  → ( − ∞, ∞ )  2 2 y = cot x ; becomes invertible when f : ( 0, π ) → ( − ∞, ∞ )  π y = sec x; becomes invertible when f : [ 0, π] –   → R − ( −1, 1)  2  π π y = cosec x; becomes invertible when f : – ,  – { 0} → R – (–1, 1)  2 2 16  y (i) Graph of y = sin −1 x ; Introduction of Graphs y = sin–1x y=x where, π/2 1 f (x) = sin x x ∈ [ −1, 1]  π π x and y ∈ − ,  (–π, 0) (–π/2,0) (–1,0) O (1,0) (π/2,0) (π, 0)  2 2 –1 As the graph of f −1 ( x) is mirror image of f ( x) about y = x. –π/2 Fig. 1.53 y (ii) Graph of y = cos −1 x ; y = cos–1x π y=x Here, π/2 domain ∈ [ −1, 1] 1 π/2 Range ∈ [ 0, π] –1 O 1 π x f (x) = cos x (iii) Graph of y = tan −1 x ; Fig. 1.54  π π Here, domain ∈R, Range ∈  − , .  2 2 y y = tan x y y = π/2 y= tan–1x x x O O y = –π/2 x = –π/2 x = π/2 Fig. 1.55 As we have discussed earlier, “graph of inverse function is image of f ( x) about y = x” or “by interchanging the coordinate axes”. (iv) Graph of y = cot −1 x ; We know that the function f : ( 0, π ) → R, given by f(θ) = cot θ is invertible. ∴ Thus, domain of cot –1 x ∈ R and Range ∈ ( 0, π ). y y = cot x y (0, π) y=π π/2 O (π,0) x y = cot –1x (π/2,0) x O x=π Fig. 1.56 17  (v) Graph for y = sec −1 x;  π The function f : [ 0, π] −   → ( −∞, − 1] ∪ [1, ∞ ) given by f(θ) = sec θ is invertible.  2 −1  π Play with Graphs ∴ y = sec x, has domain ∈ R − ( −1, 1) and range ∈[ 0, π] −   : shown as  2 y y y=π 1 –1 y = sec x (0, 1) x-axis y = π/2 O π/2 π –1 (π,–1) x –1 O 1 x = π/2 x = π y = sec x y = sec–1 x Fig. 1.57 (vi) Graph for y = cosec −1x;  π π As we know, f : − ,  − { 0} → R − ( −1, 1) is invertible given by f(θ) = cos θ.  2 2 ∴ y = cosec −1 x ; domain ∈ R − ( −1, 1)  π π Range ∈ − ,  − { 0}.  2 2 y y y = π/2 1 x –1 O 1 x –1 y = – π/2 x = – π/2 x = π/2 x = –1 x=1 y = cosec x y = cosec –1x Fig. 1.58 Note If no branch of an inverse trigonometric function is mentioned, then it means the principal value branch of that function. In case no branch of an inverse trigonometric function is mentioned, it will mean the principal value branch of that function. (i.e.,) 18  Introduction of Graphs Function Domain Range Principal value branch 1. sin −1 x [ −1, 1]  π π π π − ≤ y ≤ , where y = sin −1 x − 2 , 2 2 2 2. cos −1 x [ −1, – 1] [ 0, π] 0 ≤ y ≤ π, where y = cos −1 x 3. tan −1 x R  π π π π − ,  − < y < , where y = tan −1 x  2 2 2 2 4. cosec –1 x ( −∞, − 1] ∪ [1, ∞ ) − π , π  – {0} π π − ≤ y ≤ ; y ≠ 0, where y = cosec −1 x  2 2 2 2 5. sec –1 x ( −∞, − 1] ∪ [1, ∞ ) [ 0, π] −  π   π 0 ≤ y ≤ π ; y ≠   , where y = sec −1 x    2  2 6. cot −1 x R ( 0, π ) 0 < y < π ; where y = cot −1 x. 1.3 TRIGONOMETRIC INEQUALITIES To solve trigonometric inequalities including trigonometric functions, it is good to practice periodicity and monotonicity of functions. Thus, first solve the inequality for the periodicity and then get the set of all solutions by adding numbers of the form 2nπ ; n ∈ z, to each of the solutions obtained on that interval. 1 EXAMPLE 1 Solve the inequality; sin x > −. 2 SOLUTION As the function sin x has least positive period 2 π. {That is why it is sufficient to solve inequality of the form sin x > a, sin x ≥ a, sin x < a, sin x ≤ a first on the interval of length 2 π, and then get the solution set by adding numbers of the form 2 πn, n ∈ z, to each of the solutions  π 3π  obtained on that interval}. Thus, let us solve this inequality on the interval − , , where  2 2  1 graph of y = sin x and y = − are taken two curves on x-y plane. 2 y sin x > 1 1 2 π π π x O π 7π 3π 2 6 2 6 2 –1/2 1 y=– –1 2 y = sin x 2π Fig. 1.59 1 y = sin x and y =− 2 19  1 π 7π From above figure, sin x > − when − < x <. 2 6 6 Thus, on generalising above solution; π 7π 2nπ − < x < 2nπ + ; n ∈ z. Play with Graphs 6 6 which implies that those and only those values of x each of which satisfies these two inequalities for a certain n ∈z can serve as solutions to the original inequality. 1 EXAMPLE 2 Solve the inequality:cos x ≤ −. 2 SOLUTION As discussed in previous y example, cos x is periodic with period 2 π. 1 So, to check the solution in [ 0 , 2 π]. 1 It is clear from figure, cos x ≤ − when; 1/2 2 2π 4π π O π 2π π 4π 3π ≤ x≤. 2 2 3 3 2 2π x 3 3 –1/2 y = –1/2 On generalising above solution; –1 1 2π 4π cos x – 2nπ + ≤ x ≤ 2nπ + ; n ∈z 2 3 3 Fig. 1.60 1 ∴ Solution of cos x ≤ − 2  2π 4 π ⇒ x ∈ 2nπ + , 2nπ + ; n ∈ z.  3 3  EXAMPLE 3 Solve the inequality:tan x < 2. SOLUTION We know tan x is periodic with period π. y  π π So, to check the solution on the interval  − , .  2 2 y=2 π (t an–1 2, 2) It is clear from figure, tan x < 2 when; 1, 4 1 π π − < x < tan −1 2 or − < x < arc tan 2 x 2 2 –π/2 –π/4 O π/4 π/2 ⇒ General solution –1 π 2nπ − < x < 2nπ + tan −1 2 tan x < 2 2  π  ⇒ n ∈  2nπ − , 2nπ + arc tan 2 x = –π/2 x = π/2  2  x = tan–12 = arc tan 2 Fig. 1.61  3x π 1 EXAMPLE 4 sin  Solve the inequality: + <.  2 12 2  3x π 1 3x π SOLUTION Here, sin  + < ; put + =t  2 12 2 2 12 20  1 y Introduction of Graphs ∴ sin t < , now sin t is periodic 2 y = sin t with period 2π, thus to check on 1 1 y= 1  π 5π   π 3π   2 , 2  or − , 2 2  2 2  O π 3π 3π 2π 9π 5π t π =t 2 4 2 4 2 From figure, –1 1 3π 9π sin t < 1 sin t < , when < t<. 2 2 4 4 t = 3π/4 t = 9π/4 ∴ generalsolution 3π 9π Fig. 1.62 2nπ + < t < 2nπ + ; n ∈z 4 4 3x π Substituting t = + 2 12 3π 3x π 9π 2nπ + < + < 2nπ + 4 2 12 4 4 4 13 4 ⇒ π + πn < x < π + nπ ; n ∈ z. 9 3 9 3 EXAMPLE 5 Solve the inequality : cos 2x − sin 2x ≥ 0. SOLUTION Here, cos 2x − sin 2x can be reduced to,  1 1   π π  2 cos 2x − sin 2x ⇒ 2  cos cos 2x − sin sin 2x  2 2   4 4  π  ⇒ 2 cos  + 2x ?