@ 2"! Edition ADDITIONAL 360 Yan Kow Cheong + Eric Chng Boon Keat * David Khor Nyak Hiong Consultant: Dr Toh Tin Lam Marshall Cavendish US) edication (taans 900_WoLA Fina 4 © anone 231 eM© 2013 Marshal Cavendish International (Singapore) Private Limited {© 2014, 2020 Marshall Cavendish Education Pte Ld Published by Marshall Cavendish Education Times Centre, 1 Now Industral Road, Singapore 536196 ‘customer Service Hotline: (65) 6213 9444 Email: mesales@mcedueation.com Website: www.mceducation com Fist published 2013, ‘Second edition 2020 All rights reserves,ACKNOWLEDGEMENTS The publisher would like to thank the following for granting permission to reproduce the copyrighted materials below: Front cover, back cover and title page into infinity geometry, abstract geometrical concentric swirl background, see shell ike structures, fractel swirl background, concentric wrapping geometry, stock illustration, © GarryKillian / iStock.com Chapter 1 p. 1, slingshot wood, ID 31004181 © Cunaplus / Dreamstime.com P. 14, a firebost putting out a shoreline fire, © Ketherine Welles / Shutterstock.com P. 15, Georgetown Bridge, landmark, highway, ID 31025832 © Mocut Dogan / Droamstime.com p. 15, swist and turne of a modern steel roller coaster, © Aneese / Shutterstock com Chapter 2 P. 27, Grugliasco, Italy - June 30 2074: car production line with unfinished cars in a row at Maserati factory, (© MikeDotta / Shutterstock.com Chapter 3 p. 50, Parthenon temple, Acropolis in Athens, Gre Dreamstime.com Hellenic, column, ID 88639249 © Kurylo54 / Chapter 4 p. 66, a roller coaster ride at amusement park in Vienna, ID 26377538 © Kewuwu / Dreamstime.com Chapter 5 p. 95, human pathogenic virus and bacter Yemelyanov / Dreamstime.com under microscope, viral dis, ID 109595847 © Makeym Chapter 6 P. 136, soccer football players compete at the stadium, two footballers running end kicking zoccer bell, frnthall mateh harlnrmind IM. 124098724 © Matimiv / Piraamatima namABOUT 360 ‘The books are specielly designed for students offering O and N(A}-level Additional Mathemetics syllabuses from the Ministry of Education, Singapore. Written besed on the belief that understanding mathematics goes beyond fects and procedural proficiency, ‘the books engage students in various tasks and learning experiences to build new knowledge, promote reasoning and problem solving, and thus ellowing students to see ‘the mathematical connections of ideas and make sense of their learning. The chapters have been organised into three strands - Algebra, Geometry and Trigonometry, and Calculus. The strands are indicated by the different colours of the chapter openers and chapter tabs. Chapters, sections and questions excluded from the Normal (Academic) Additional Mathematics syllabus are marked with a @) Key Features {Chapter pone << Emphasices the applications and relovance of mathematics to our everyday livos, and serves as a lead.in to the topic YY YX eae Beg $$ aaa Allows students to be aware of what they will encounter in each section. Activity Provides rich learning experiences and allows students to explore, investigate and communicate what they have discovered. tesa $$$ Iilusteates the application of concepts 2s well as the procedural steps to solve @ problem. Try Provides opportunities for students to assess their understanding immediately. Exercise Reinforces concepts and leads students to achieve mastery as they progress through the graded questions ‘natans 900_VoLA Prete 4 ® zone 231 Pu(plans 360_VOLA Pretns.is 5 Provides students with more challenging real-world problems or problems involving certain extensions of ‘concepts. It also provides coding opportunities for students Provides concise points to help students consolidate what they have learned. Invites students to reflect on their learning and develop awareness in their processes through ‘written communication, Allows students to consolidate and assess their learning, [7 In addition, the textbook contains these side features: Think Deerer, Enhances conceptual understanding and mathematical thinking through probing questions. Mate Connection Links concepts from different chapters/ sections, prosenting mathematics as a coherent and connected body of knowledge. Take rely Contains helpful pointers or tips and helps clarify possible misconceptions. Did You Know? Includes history of mathematics, information about real-life applications and ‘ther useful mathematical facts allowing for greater appreciation of mathematics Contains questions ov statements that allow for the smonitering of thinking and self-regulation of learning.The fight of 2 ball laund be modelled by a quedr ss ee2 Maximum and Minimum Values You Will Learn To : ‘© Find the maximum or minimum value of a quadratic function using the method of completing the square Notation f(x) na Cartesian plane, the relationship between the variables x and y can be represented graphically. We have graphed some relations previously in linear equations of the form y = mx + ¢ and quadratic equations of the form y * + bx + c. Observe that the graph of y = mx + c is a straight line and the graph of y= ax’ + by + cis a curve. stright line curve In both relationships, each input x has exactly one output y. Thine” We call this kind of relationship a furs Deeper We use the notation y = f(x) to mean that y is a function of x, Can two different inputs 5 - 3 "give the same output for 2 ‘or a quadratic function of x, we may write f(x)=ar'+bx+e. — gvennes Similarly, we may write a linear function of x as f(x) = mx + ¢. We can find a value of f(x) by substituting a value of x into the algebraic expression representing the function. For example, the value of [(x) =x + | when x = ~2 is given by ((2)=-2+1=-1. + Loan take on the values 24, 5.1, 10 and so on, We can represent all possible real values of f(x) for x > 0 using a single inequality f(x) > 1 If. x > 0, the function f(x)Quadiatic Functions 3 Example 1 Try 1 Given f(x) = 2x° - 4, find Given f(x) = -2x° + 4, find (its value when x= 0, @ fC, (ii) all possible real values of (x) 62 al posse rea vais of Solution Answers @) 2 () {@) = 209-4 {ii) Since x* = 0, 2° = 0 for all real values of x and so 2e-4>-4, Hence f(x) > -4. fay <4 For the function g(x) = -2y" - 4, how do | Find all pessibie real valves of (x)? Completed Square Form For the function f(x) 4, we can tell that f(x) > —4 by inspection, because x° > 0 for all real values of x. We can then deduce that the minimum value of {() is —4. What about f(x) = x7 + 6x — 4? How do we find all possible real values of this function? Furthermore, can we deduce whether this function yields a maximum or minimum value? To find all possible real values of f(x) = 1° + 6x — 4, we first use the method of completing the square to express the function in the form (x — h)° + k, where hand k are constants. Example 2. Try 2 Express f(x) = x7 + 6x — 4 in the form (x —h)? + k, where Given f(x) = x* + 3x — 6, find hand k are constants. Hence find (Wall possible real values of (i) all possible real values of f(x), ae aiiboarvalinist Gh fi). the minimum value of f(x), shavaliceteatehnh the value of x at which the minimum value of f(x) occurs. the minimum value of f(x) occurs. Answers @ w>4 3 w 2 34 [el Solution f(x) = 2+ Gx 4 The form x’ + br, where b = 6>0. =x +6r+($) -(3) =4 Add end subtract ( P+ 6r4+ 3°34 @+3 9-4 Use the first three terms 1B to form a perfect square. @) Since (x + 3720, (r+ 13 = -13 for all real values of x. Hence f(x) > -13. Gi), The minimum value of f(x) is -13. Gi) The minimum value of I(x) occurs when (x + 3)' = 0. 44350 When the coefficient of x is not equal to 1, we need to factorise the quadratic term and the linear term before we apply the method of completing the square. Example 3 Try 3. Express each of the following functions in the form Express each of the following a(x — hy’ + k, where a, h and k are constants, and a + 0. functions in completed square form. Hence find the maximum or minimum value of f(). Justify your answers. Hence find the minimum or maximum value of f(x). Justify your answers. (a) fx) = 3x" fa) {@) = 20 -3x-1 (b) (x) = 3x7 + 4v- 2 Answers (, 37 _17 fa) f6)=2(x-F) - Ze minimum 2 the brackets, where b= 2> 0. ' 2 . , ae (b) 10) =-3(x~2) -3, 230-24 1? 1)+2 Add and subtract (4) 37° ‘A maximum -= Use the first three terms within the = brackets to form a perfect square. =3@-1)P-1 Since (x - 1)? > 0, 3(r— 1)? — 1 > 1 for all real values of x. ele Hence f(x) has the minimum value -1. ore. fa) = ax’ +b has the minimum value when a > 0.Quadkatic Functions 5 (b) 3 Z>0. within the brackets, where b= 5 ay ° -(3) | +3. Add and subtract (4) Use the first three terms within the brackets to form a perfect square. 3 e on wh Take cf = ah jeverse the sign when 0, a ae 3) <0 multiplying by a negative Note ay 3 a fQ) = ar + bv +e ee has the maximum value when a <0. Hence f(x) has the maximum value -. How do | check that the two forms ave equivalent? C9 ay 3 Fev example, i 2x4) + equivalent to Ziy= See Re From Example 3, we see how a quadratic function in general form ax” + bx + ¢ can be transformed to completed square form a(x — hy? + k. Activity 14: Use the method of completing the square to transform the quadratic function f(x) = ax? + bx + € to the form a(x — hy +k. (i) Show that A = [Derive the quedtatc formule] for f(x) = 0.Example 4 Show that x° - 2x + 6 > 5 for all real values of x. Solution Since (x — 1)? = 0, (x- 1° + 5 = 5 for all real values of x. Try 4 Show that —1° ~ 4x — 10 is negative for all real values of x. Take ely Expressing in the form a(x — hi + k helps you find all the possible real values ‘that the function can take on. Given f(x) = 2x" + 1, find @) 13), Gi) (2), Giii) all possible real values of f(x). Express each of the following in the form a(x = hy + k. fa) + 12 (b) +1 (© 3 +3x-2 (d) 3x°-6x +2 (e) 32° + 6x — () sO -4x41 I _3r45 + 5x43 State the minimum or maximum value of I(x) and the corresponding value of x Explain why (x + 3)" + 1 is positive for all real values of x. Find the minimum or maximum value of f(x) and the corresponding value of x. fa) f(x) = 2v -4x-5 (b) f(x) =? 243 () fa) =2-3x430r (d) f(x) =3x+4-2 Given f(x) = x° — 2x + 3, show that f(x) = 2 for all real values of x. Show that 4x — negative. Tis always Explain why 4° — 12x | 10 is positive for all real values of x. Solve 3x*— 10x — 8 = 0 using the following methods. © Factorisation * Completing the square * Quadratic formula Compare the three methods and write down the reason(s) for your preferred method.Quadsatic Functions 7. Graphical Representation of 1 Quadratic Functions You Will Learn To ‘Use the discriminant to determine the number of x-intercepts of quadratic grephs ‘© Use the discriminant to determine whether ax” + bx + c is always positive or (always) negative he graph of a quadratic function changes direction at @ point called the ‘turning point or the vertex of the graph. turning peint ‘turing point’ The general form ax’ + bx + ¢ is not very useful in sketching quadratic graphs, as it does not show features like intercepts and turning points. As such, we would convert quadratic functions from the general form to either the factorised form or the completed square form. © The factorised form a(x ~ p(x — 4), where p and g are the x-intercepts * The completed square form a(x — hy? + k, where (i, k) ate the coordinates of the tuning point Do the following activity to explore the graphs of quadratic functions in various forms. Activity 1B [Explore the graphs of quackatic functions in the forms y= at + br + c = aGx~ pXx—Q) and y= atx =} +} Use a graphing software to draw the graphs of quadratic functions in various forms and ‘observe their shapes. () (a) Display on the same screen some graphs of quadratic functions of the form y= ar’ + bx +c, where a> 0. What do the shapes look like? (b) Display on the same screen some graphs of quadratic functions of the form y = ax" + bx + ¢, where a <0. What do the shapes look like? (ii) Draw graphs of quadratic functions of the form y = a(x —p)(x — q). (a) How does the sign of a affect the shape of the graphs? {b) At which points does each of the graphs intersect the x-axis? How are these points related to p and q? (@) If the line of symmetry has the equation « = h, how is / related to p and q? (iti) Draw graphs of quadratic functions of the form y = a(x — hy? + k. {a) How does the sign of a affect the shape of the graphs? (b) What are the coordinates of the turning point for each graph plotted? How do you determine whether a turning point is a maximum or minimum point? What conclusion can you make? (&) What is the equation of the line of symmetry for each graph plotted? What conclusion can you make?8 [el From the activity, we have the following results. ‘The coefficient, a, of x° determines the shape of a quadratic graph and the nature of the turning point. maximum point yea? thes ca<0 beteaso ‘minimum point The factorised form gives the x-intercepts p and q and enables us to find the line of syminetry. yEaee= pye-q),a>0 yaale~pXe=9) a<0 +a pea (line of symmetry) (line of symmetry) The completed square form gives the coordinates of the turing point and enables us to find the line of symmetry. (ine of symmetry) i (1,1), meximum point ate hard yootehf + ka<0 1), minimum point line of symmetry)Quadtatic Functions Example 5 (i) Find the coordinates of the turning point and the (ii) Find the exact values of x for which y=0. Solution @ 2s4are4 Pj+4 S21 +244 -1%+6 4] The turning point is (1, 6). ao When x =0,y =4. - The y-intercept is 4. The graph is shown (i) From part i as an illustration. 200 = 1)? +6= Try 5 (Find the coordinates of the turning point and the y-intercept on the graph of y=4u? - 124+ 9, Solve the equation y = 0. (i Answers Letting 4 = 0 enables us to find the y-intercept. Letting = 0 enables us to find intercey Solving ax’ + br + ¢= 0 for x gives the x-interceptts) of the graph of y= ax" + br + c The values that satisfy an equation are called the roots or solutions of the equation. In Example 5, we say that the equation ~2x° + 4r + 4 = 0 has two different (or distinct) real roots, 1 + J3 and 1-3. Example 6 (i) Find the x-intercepts and the coordinates of the turning point on the graph of y = 2x - 6x8. (i) Hence explain why 2° — 6x — 8 = 0 has two distinct real roots. Solution () 22 - @c-8 = 207 - 3x- 4) =2(e+ D@-4) The x.intercepts are -1 and 4. inates of the turing point on the graph of y=-2r 4 2k 44. explain why 2x 4= 0 has two distinct real roots. Answers @ -1.2,(05,4.5) Finding the coordinates of the turning point?— (i) The real roots of 2x° — 6x -8 = 0 are the x-intercepts of the graph of yo2v-6r-8 The graph has two «-intercepts, J and 4. Hence 2x” - 6x — distinct real roots. Make Conneti We can also solve 2° -6r-8=0 by factorisation to check that the equation has two distinct real roots = 0 has two Discriminant, Roots and x-intercepts When the value of the quadratic function f(x) = ax? + br + ¢ is 0, we will obtain its corresponding quadratic equation ax” + bx + ¢ = 0. We have learned previously that we may solve ¢ quadratic equation by the quadratic formula x = . The expression b? ~ 4ac is called 2a the discriminant. Without working out the roots, we can use the sign of the discriminant to determine the number of real roots of the quadratic equation and the number of x-intercepts of the corresponding quadratic graph. Do the following activity to explore the relationships between the discriminant, roots and x-intercepts. Activity 1C-_ am pani nd en on eee sign ofthe cectninart and erplin how he relates tothe nate of roots ofthe eqitin| (i) Copy and complete the table. Face Note When we describe the nature of roots an equation has, ‘we mean whether the equation has real roots; if there Value of disctiminant (= dae) are real rots, whether they are distinct or repeated. (a) When does the quadratic equation ax” + bx + ¢ = 0 have * teal roots, * teal and distinct roots, * real and repeated roots, * no real roots? (b) How does the sign of the discriminant affect the nature of roots? (ii) Use @ graphing software to draw the graphs mentioned in the table. (a) Take note of the sign of the discriminant and the number of x-intercepts of the graphs. What can you conclude? (b) How does the sign of the discriminant affect the position of the graphs?Quadratic Functions 111 From the activity, we summarise as follows. Mate Comme! We will use the relationship The curve intersects the x-axis at 2 points. between the discriminant oz and the neture Ae fa 0, * below the x-axis when a < 0. symbol. A> B Hence we have the following results. means A and B are equivale Discriminant <0 and a> 0 4 y = ax’ + br + ¢ > 0 for all real values of x, Discriminant <0 and a<0 & y =ar + br + ¢ <0 for all real values of x. Example 7 Try 7 Use the discriminant to determine the number of x-intercepts | Use the discriminant to for the graph of each given quadratic function. seers: a fuse! af ‘ y=? cea acintercepts for the graph of a) ae ec) ) y=Q-ad ‘each given quadratic function. Solution (b) y=(1-ay 42 (a) Discriminant = 6 - 4(1)(9) = 0 Answers Hence the graph of y =x" + 6x + 9 has one x-intercept. (a) 2 (b) 0 5 es (b) y x41 av+5 Discriminant = (-4)" - 4(1)3 Hence the graph of y = (2 — x)’ + | has no x-intercepts.. Example 8 Try 8 Find the non-zero values of p for which the graph of Find the values of p for which y= px — 2x + p touches the x-axis. the graph of y= 3x" + 12 - px meets the a-axis at one point only. Solution Answers As the graph touches the x-axis, there is one x-intercept =12, 12 only. Discriminant = 0 (2) = 40K) = 0 4 4p Example 9 Try 9 Explain why the graph of y = x° + (1 — p)x —p intersects Explain why the graph of the x-axis. Y 2v+2-p, where p> 1, intersects the x-axis at ‘two points. Solution Discriminant = (1 — p)’ — 4p) Thine 1-2p + p+ 4p Deeper, How many points of intersection are there between the graph of yx’ + (1—p)x-p and the waxi Hence the graph intersects the x-axis, Example 10 Try 10: (a) Show that the graph of x —2kv + (+ Lis entirely | (a) Show that the graph of above the x-axis. ) P+ ke - FF + 1) (b) Show that -2x* + 4kx — 2° — 1 is negative for all real ies Lumpletely below the values of x. avaxis, ns / {b) Show that px" + kv 4 T Solution is always positive. Since the discriminant is negative and the coefficient of x* is positive, the graph is entirely above the x-axis. How caw \ aiso use the method of completing the square to prove the statements? (b) Since the discriminant is negative and the coefficient of x* is negative, ~2x” + dv — 2k — 1 is negative for all real values of x.Exercise eal Quadratic Functions 13 to) For each of the following, {)_ state the coordinates of the turning point, (ii) find the axial intercepts (if any). (a) Ax + 2-8 ) © @) ) @) y= 5 () y=—4(e4 27 49 Bp For each of the following, i) state the x-intercepts, (id) find the coordinates of the turning point. @) y=2@-Dw+2 (b) y=-30@x- DGr-2) (©) y= + 202r-3) (@) y=3@—4)2x-5) () y=20r+ 9G-¥) @ y=2Gr-45-2) Bp For what value(s) or range of values of k does y = 3x" - 2x + k have {no xintercepts, (ii) one x-intercept, (iii) two x-intercepts? Is it true that y = 2x" + bx +c has the same number of x-intercepts as y = 2x" — by + c? Justify your answer. Given that y = ax” + bx + c has two wintercepts, show that ac < a : 2 (i) Find the coordinates of the turning point and the equation of the line of symmetry on the graph of y= 2+ 2r-1, (ii) Find the exact values of x for which 2x” + 2x -1=0. (i) Find the intercepts and the coordinates of the turning point on the graph of y = -2x7 + 4x +3. Gi) Explain why -2x* + 4r two distinct real roots. Find the value(s) or range of values of p for which the graph of (a)_y = px’ — 6x + p touches the x-axis, y = 3x" + 2x — p intersects the x-axis at two distinct points, (ce) y = 2x7 + 3x + 2p intersects the x-axis, d) y=pe the x-axis. ¢—4 does not intersect Find the range of values of k for which (a) the graph of y=x7 + 2x+k41 lies completely above the x-axis, (b) 2° + 4x + kis always negative. Find the range of values of k for which + 2x + k is always positive, + 6x + k= 1 is always negative, (6) the graph of y = 2x" + x~ 2k lies entirely above the x-axis. Find the range of values of ¢ for which y= 3x" -2x + ¢~ 1 intersects the axis. Explain why the graph of ¢ +4v—k+ | has no x-intercepts when k = 5, () Prove that 2x° + 2x >—4 for all real values of x. (ii) Hence explain why the graph of y=2v + 2r+4has no wrintercepts.. Find the range of values of p for which (a) Find the range of values of m for the graph of which x — 10x + 4—m> 0 for all (@) y=@+ DQr-D-@-2 real values of x. intersects the x-axis at two (b) Find the range of values of k distinct points, for which (b) y= plix+ DO -3)-x+4p+2 av + dkx — Hk - DK + 2)<0 does not intersect the x-axis. for all real values of x. Find the least value of k for which the graph of y =x -2kx + -G +0) intersects the x-axis. Applications of 1: Quadratic Functions You Will Learn To + Use quadratic functions as models Qoizitssferetion can be ute to model 9 wide range ad You of practical problems. We can form relationships Krow? between quantities with quadratic equations and use 7 quadratic graphs to represent the trajectory of objects, Transport engineers use physics phenomena or parabolic structures. parabolic curves for design of railway tracks and roads. Let us look at some examples of quadratic functions. The trajectory of water sprayed onto the sea can be modelled by a quadratic function.Quadratic Functions 15 Parabolic arches support the bridge such that the weight of the deck is evenly distributed on the columns. Did You Know? A parabola is a member of the family of conic sections. This section of roller coaster is in the shape of a parabola, which gives riders a sensation of weightlessness as they descend. Example 11 Projectile Motion. A ball was launched from a slingshot. Its height, fm, above the ground is given by h =-2x° + 8x + 1, where x m is the horizontal distance from the slingshot. Find the height of the ball above the ground when it just left the slingshot. Find the greatest height of the ball after it was launched from the slingshot. (iil) IF a toy is 3 m horizontally from the slingshot and 7 m above the ground, justify if the ball will hit the toy directly. Solution () When the ball just left the slingshot, it was 0 m horizontally from the slingshot. Thus x = 0. When x = 0, h = ~2(0) + 8(0) + 1 The ball was | m above the ground when it just left the slingshot20 = 4x42 2-2)+1 8+1 9 The greatest height is9 m. (2, 9) is the maximum point on the graph of k= -2x* + Bv+ 1 Gil) When x = 3, = -28 - 2y'+9 The flight of the ball passes through (3, 7) and so the ball will hit the toy directly. Try 11 Projectile Motion. A projectie was launched from a catapult to smash a defence structure on 2 fot. Its height, /t m, above the ground is given by ht = —saq5x" + 5 z +3, where x m is the horizontal distance from the catapult. (i) Find the height of the projectile when it just loft the catapult. {ii) Find the greatest height of the projectile after it was launched from the catapult. {iil If the defence structure is 150 m horizontally from the catapult and 5 m above the ground, justify if the projectile will smash the structure. a0 Answers: (i) 3 m (ii) 7 1 (iii) No Example 12 Architectural Design. A curved arch that supports a bridge can be modelled by a quadratic function with its graph shown. In this model, x m is the horizontal distance from O and y m is the height of the arch above the surface of water. The arch is 120 m wide at its base and 50 m high in the middle. (i) Write 2 quadratic function in the form y = a(x — p)(x — q) to represent the arch. (i) A point on the arch is 10 m horizontally from O. What is the height of the arch above the surface of water at this point? (iii) Another point on the arch is 20 m vertically from the surface of water. What is the width of the arch at this point?Quadratic Functions 17 Solution (i) Let the function representing the arch be y= ax(x ~ 120), where a is a constant. The arch is 120 m wide, When x = 60, y = 50. 50 = a(60)(60 — 120) 30 =-3600a 0 3600 1 72 a= The function representing the arch is y = “+ x(x — 120). Try 12 Architectural Design. The opening of a tunnel can be y modelled by a quadratic function with its graph shown. In this model, x m is the horizontal distance from one end of the tunnel and y m is the height of the tunnel. The tunnel is LO m wide at its base and 5 m high in the middle. (Write a quadratic function in the form y = atx p)(x— 9) to represent the opening of the tunnel. i) A point on the opening of the tunnel is 2m horizontally from one end. What is the height of the tunnel at this point? Another point on the opening of the tunnel is o x 4.2 m vertically from the base. What is the width iim of the tunnel at this point? Answers: (i) y= x(e— 10) Gi) 3m Gi) 4m— Example 43-——--—__________—_ Architectural Design. An arched underpass has the shape of a parabola as shown. In the diagram, x m is the horizontal distance from one end of the arch and y m is the height of the arch. A one-way road passing under the arch is 6 m wide and the maximum height of the arch is 5 m. (i) Write 2 quadratic function in the form y = a(x —h)’ + k to represent the arch. Find the height of the arch when its width is 4m Decide whether it is possible for a truck that is 4 m wide and 2.5 m tall to navigate through the underpass. Explain your answer. Solution (i) Let the function representing the arch be y = a(x —3)° + 5, where a is aconstant. The turning point of the quadratic graph is (3, 5). When x =0, y= 0. (0-3) +5 The graph passes through the origin (0, 0). The function representing the arch is y = 3 -3F +5. 3a -3) +5 When the width of the arch is 4m, a value : of vis (6-4) =2=1. 2 The height of the arch is 22 m when its wieth is 4 m, Think Deeper, Gi) From part (i), the maximum possible height for a vehicle faiths " ili 51 Would you set 22m as of width 4 m to navigate through the underpass is 25 ™. tno maximum hoight that a Since 2.5 m < 22 m, itis possible for the tuck to navigate Yeni can nevasts ste through the underpass. Explain.Quadratic Functions 19 Try 13 Architectural Design. An arched underpass has the shape of a parabola as shown. In the diagram, x m is the horizontal distance from one end of the arch and y mis the height of z the arch. A river passing under the arch is 3 m wide, and the maximum height of the arch is 2 m. () Write @ quadratic function in the form y represent the arch. i) Find the height of the arch when its width is I m. (ii) Decide whether it is possible for a boat that is 1 m wide and 18 m tall to navigate through the underpass. Explain 7 om your answer. Answers: (i) y = GZ m Car manufacturers use quadratic functions to determine Did You what types of brakes and tyres are needed to stop cars Know? moving at various speeds During the investigation of car accidents, police used Let us investigate how 2 quadratic function can be used to SN ASS PE model such a situation. thosneads cf the cars.when the car collisions occurred. Activity, Digi teva qua ncaa ox boa nnd ts oat npaiatl daa ‘Automobile Engineering. The table shows the approximate rapid braking distances, d m, in a testing workshop for different speeds, v km/h, of a car. v(kmih) | 20 | 30 | 40 | 50 | 6 | 70 | 80 | 9 | 100 aim | 10 | 15 | 20 | 35 | 45 | oo | 75 | 95 | 110 (1) Use a graphing software to draw a scatter plot of the data with v as the horizontal axis and das the vertical axis. Why is a quadratic function suitable to model these data values? Discuss. (ii) Display on the same screen a curve of best fit for the given data. (ili) The data can be modelled by d= av" + bv + c, where a, b and c are constants. Find this function. (iv) Use the function obtained in part (iii) to. (a) estimate the braking distance for a speed of 65 km/h, (b) find the speed of the car for a braking distance of 70 m. (v) State two factors that may affect the braking distance besides the speed of the car.[41> Architectural Design. A section of cable is suspended between two towers of a bridge. The height, y m, of the cable above the roadway is given 1 2 by y= Tog9 300)" + 10, where x m is the horizontal distance from tower P. @) Find the height of each tower above the roadway. (ii) Find the distance between the two towers, Gil) A car on the road is 20 m beneath the cable. How far from tower P could the car be? Roller Coaster Engineering. The height, y m, of a rider above the ground in a section of roller coaster ride is given by y= 4x°-2r +8, where x mis the rider’s horizontal distance from the start of the ride. (i) Express the function in the form yatx—hy +k. Gi) Find the rider's minimum height above the ground. (iil) If the rider is 8 m above the ground after the ride starts, find the rider's horizontal distance from the start of the ride. Projectile Motion. Two shells are fired from a battleship at two enemy frigates Their fights can be represented on a Cartesian plane as shown The vertical height, y m, of the first shell from the gun on the battleship is aiven by y=—soa +S, whore m is the horizontal distance travelled. ‘Assume this shell hits the enemy frigate. (How far is the enemy frigate from the battleship? (ii) What is the maximum vertical height that the shell can reach? The vertical height of the second shell is given by y =-aig ee 1500)" + 450. (ii) If the second enemy frigate is 3 km from the battleship, justify whether the second shell will hit it. Architectural Design. A section of curved cable that hangs between two towers of a suspension bridge can be modelled by a quadratic function with its graph shown.In the model, x m is the horizontal distance from the left tower and y m is the height of the cable above the road. The cable touches the surface of the road halfway between the two towers, which are 20 m above the roadway and 120 m apart. (i) Find a quadratic function to model this situation, (ii) Find the height of the cable above the roadway at a point that is 20 m away from the left tower. Architectural Design. A section of curved cable of » suspension bridge can be modelled by a quadratic function with its graph shown. 200m) In the model, x m is the horizontal distance and y m is the height of the cable above the bridge. Vertical supporting wires are spread out in equal intervals of 1 m apart hanging from the cable. The longest length of these wires is 40 m and the shortest length of these wires is 10 m. The two longest wires are 200 m apart. (i) Find a quadratic function in the form y= a(x — hy’ + k to model this situation. (ii) Find the length of the vertical supporting wire that is 30 m horizontally from the origin. Quadratic Functions 27 Architectural Design. Suppose the cross section of a skateboard ramp is parabolic in shape and can be represented by y = 0.25x° — x + 2, where x m is the horizontal distance and y m is the height of the ramp above the ground. (i) Find the y-intercept and the coordinates of the turning point on the graph of y against x. i) Find the maximum depth of the ramp. (ii) For safety reasons, the maximum height of the ramp above the ground is limited to 2 m. Find the maximum possible width of the curved part of the ramp. Projectile Motion. The height, y m, of a shot above the ground after it has been thrown is given by + C+ get 5, where xmis the horizontal distance travelled. (i) Express the function in the form y=a(x—hy +k. (i) Find the intercepts and the coordinates of the turning point ‘on the graph of y against x. (iii) Find the maximum height that the shot can reach. (iv) Find the horizontal distance that the shot has travelled before it lands.. Projectile Motion. A soccer ball is kicked from the ground. The height, hm, of the ball above the ground at time ¢ seconds is given by h = 8t—4P. (@) Express the function in the form h= alt - pylt—q. Find the intercepts and the coordinates of the turning point on the graph of ft against 1 Find the maximum height that the ball can reach. Find the time it takes for the ball to retum to the ground, A player whose height is 1.8 wishes to head the ball as it is falling. Find the earliest time he can do so with his feet on the ground. Gii) Gili) Ww) w) A wire that is 40 em long is bent into the shape of a rectangle whose width isxem (i) Find an expression, in terms of x, for the area, A cm’, of the rectangle. Find the x-intercepts on the graph of A against x. il) Find the maximum area that can be formed. (iv) Show that this maximum area is only possible if the shape formed is a square. ii) * Projectile Motion. The height, y m, of a baseball above the ground x seconds after it has been hit is given by y=—Sr' + 20x+ c, where c isa constant. (i) Use the discriminant to find the range of values of c if the baseball did not reach a height of 50 m. ii) If ¢ = 20, express y in the form y=a(x—h) + k Hence find the coordinates of the maximum paint. (iii) Relate the answer in part (j) to the answer in part (i). Projectile Motion. The following functions model the paths of three water jets. Water jet 4: y = -0.3. Water jet B: y= 0.20 + 18 Water jet C: y = -0.08x° + 2.4 Which water jet will (i) send water the farthest, (ii) send water the highest, (il) produce the narrowest path? Explain. Water jet D can be modelled by y= D.Sc + 2x4 k, where k is a constant and all distances are measured in metres. (iv) Find the range of values of & if this water jet will send water to a maximum height of at least 10 m.Quadratic Functions 23 @ In mathematics, a zero of a function f(x) is a value of x such Did ‘Tou that f(x) = 0. Know? The following algorithm can be used to obtain approximate ahaa ib roots of f(x) = 0. A eoteieprsice Step 1: White f(x) = 0 in the form x= (3), where a(0) Se iaence is a function of x. find out mora applications For example, 2x° + 3x—4 of this algorithm. Of =A 3 a= 3 where g(x) = Step 2: Let x,,=2 ee of x after x. where x, is an input value of x and x, is the next input value Step 3: Choose an input value, x), as the initial approximation of x. Step 4: Substitute x, into the formula in Step 2 to obtain the next approximation of x. Repeat the process of substitution to obtain successive approximations until = x, correct to 2 decimal places. For example, by choosing x, = —1, i tle whe wl elu ele Since x,= x, correct to 2 decimal places, ~2.35 is an approximate root of the equation. Using the above algorithm, write a computer program that can be used to find the approximate values of the zeros of a quadratic function.Maximum and minimum values Any quadratic function in general form ax” + br + ¢ can be transformed to completed square form a(x — h)° + k. es {line of symmetry) {U8 maximum point atx hy +k a>0 yeacr— ay +k <0 %h.0, minimum point x =h line of symmetry) Discriminant, roots and x-intercepts Diseriminant | Nature of roots of | Number of Graph of (dae) | at tbr intercepts af +be+e ER ‘The curve intersects the x-axis al 2 points, “oO (repeated) roots 2 real and ee distinct roots The curve intersects the a-axis at 1 point (or touches the x-axis), oof A fot <0 Neratines f ‘The curve does not intersect the x-axis. ax + bx + ¢ > 0 for all real values of x. ax’ + bx + ¢ <0 for all real values of x. When discriminant < 0 and a > 0 <9 y When discriminant <0 and a <0 © y‘Quadratic Functions 25 ATE Al) Automobile Engineering. A car travels down a straight section of a test track. The distance, d m, it takes for the car to stop completely when a red light flashes can be modelled by @ quadratic function of v, where v km/h is the speed of the car. It is given that the car will stop in 20 m if it travels at 40 kmih, and it will stop in 42 m if travels at 60 km/h. Why is a quadratic function a suitable model for this case? Discuss with your classmates. Find a quadratic function to model the motion of the car in the given context. Write your answer in the form d= av" + by +c. Your classmates claim that poor braking systems would increase the value of a, and slippery track surface would increase the value of c. Comment on their statements. Are they correct? Why or why not? CAVES (6) 4) a 1 | SetA A1 Physics. The amount of deflection, A3_ The equation of a curve is dmm, of a plastic rod supported at =x + 2v+8~-p, where p is both ends under a load in the middle is constant, given by d = 0,002m° + 0.02m + 0.005, (i) Find the range of values of p for where m g is the mass of the load. which the curve lies completely (i) Find the deflection when a load of above the x-axis. 200 g is added. (ii) In the case where p = 2, find the (ii) Explain if this model is suitable for coordinates of the turning point a load of 100 kg. and the y-intercept on the graph of y=x° 4 px+8—p. A2 A curve has the equation yor + 4043, A4 (i) Given that ax” — 4x ~c is always (i) Show that the lowest point on the Positive, what conditions must curve has coordinates (1, 5). apply to the constants a and c? (ii) Find the x-coordinates of the (ii) Give a pair of values of a and c points at which the curve intersects. that satisfy the conditions found in the xaxis, pert (i). AS. Roller Coaster Engineering. A section of roller coaster track is parabolic in shape. The height, ht m, of the first capsule above the ground at time t seconds is given by h = — 12 + 40. (@ Explain why this capsule cannot reach a height of 2 m. (ii) Find the time taken for the capsule to reach the lowest point of this section of the ride. A6 A set of experimental data is tabulated below. Pao) a as t [3s [7 [as[ 17 [235 (i) Suggest a quadratic function to model the relationship between xv and y. Explain how you found the function. Gi) Deduce the range of values of xx for which y < 0. SetB B1 (i) Express y = 2x7 — 10x + m in the form a(x —h)’ + k. 19 (ii) if the minimum value of y is — 2, find the value of m. Gil) Find the range of values of m for which y = 2x" - 10x + m has two distinct intercepts. B2_ A quadratic curve has the equation y= ax —x+c. The curve passes through the points (-2, -8) and (1, 3). ()_ Find the value of a and of c. Gi) Find the coordinates of the turning point. Hence explain why the value of y can never exceed 8. Projectile Motion. A ball is thrown from the top of a building, The height, him, of the ball above the ground at time 1 seconds is given by haar + 12. (i) Find the height of the ball 3 seconds later. (i) Find the maximum height that the ball can reach. i) Find the values of t when hi = 0. (iv) Interpret your answer from part (i in this context. Arectanyle isa cm long and y cur wide. (i) Given that its perimeter is 16 cm, express y in terms of x. (il) Express its area, A cm’, as a function of x. (iii) Find the coordinates of the turning point on the graph of A against x. Hence explain why the greatest. area of the rectangle occurs when x=y. Architectural Design. A cable in the shape of a parabola is suspended at the ends of a bridge. * The minimum height of the cable above the roadway is 30 m. * The horizontal span of the bridge is 180 m. * The cable starts at (0, ends at (180, 75) when its shape is drawn on a Cartesian plane Suggest a quadratic function to model this situation, assuming that the x-axis is the roadway. Explain how you found the answer. (a) Find the smallest value of the integer a for which ax” + 4x + 6 is positive for all real values of x. (b) Given that the graph of y=-6x" + by ~3 lies completely below the x-axis, show that Ben,eS Se . — a ag | 2 Sn 1 . a] ees, cs eee Utes ta | — iS| ‘ < Simultaneous Equations : You Will Learn To '* Solve simultaneous equations in two variables by substitution, with ‘one of the equations being linear * Apply simultaneous equations in real-world contexts, Yy” have learned to solve @ pair of simultaneous linear equations by various methods. For example, solve for x and y: (1) (2 Solving by elimination Solving by substitution pad graphically (1) + (2): Se From (1), y=7—x (3) a2 ‘Substitute (3) into (2). When x =2, 2+ y 4v-(7-x) =3 y Sx = 10 Hence x =2 andy x=2 When x= 2, ya7-2= How do | solve the non-linear equation y* + (2x + 3)* = and the linear equation 2x + y = I simultaneously? Are a the three methods mentioned above still applicable? Which method should | use? To solve a linear equation and non-linear equation simultaneously, we can solve for ‘one variable in the linear equation and then substitute the resulting expression into the non-linear equation. Example 1 Try 1 Solve, for x and y, the simultaneous equations Solve, for x and y, the y+ Qx+3)'= 10 and2e+y=1 simultaneous equations 2x-y=3 and ay=x+2y. Answers v= 1,y=-LorEquations and inequalities Solution Take ir y+ Qr+3)=10 (1) Note ae+y=l We need to pair up the values yel-2e (2) of x and y and write them in order as the final answer. For Substitute (2) into (1). Sry He Pett en (= 20° + 2x43 = 10 ee oe 1—4x + 4x’ + 4x" + Lx + 9 = 10 because each pair of values 8x + 8x =0 ‘of x and y must satisfy the Brix + 1) =0 simultaneous equations -lorx=0 together. When x =-1, y= 1 —2(-1) =3. ave: y= Connetion When solving two linear equations in two unknowns simultaneously, we generally get ‘one solution. When solving a non-linear equation and a linear equation simultaneously, we could get two solutions. Do you know why? Explain. Lot us now investigate the relationship between the graphical and algebraic solutions of a pair of simultaneous equations. Activity 2A: Use a graphing software to draw on the same screen the graphs of the equations in Example 1. {i) How mony points of intersection are there? {ii) What are the coordinates of the points of intersection? (iii) How are the points of intersection related to the solutions of the simultaneous equations? [Relate points of intersection to solutions of simultaneous equations] From the activity, we know that (1, 3) and (0, 1) are the points of intersection between the curve y° + (2x + 3)° = 10 and the line 2v + In general, we have this result. Thine Deerer, The points of intersection between the graphs What are the maximum representing a pair of simultaneous equations give number and the minimum ‘the graphical solutions of the simultaneous equations. number of solutions that pair of simultaneous (one Finear, one non-linear) ‘equations has? 29Example 2 Try 2 Find the coordinates of the points of intersection between Find the coordinates of the the curve y =x = 2x+2 and the linex + y=4. points of intersection of the line y —x +3 = 0 and the curve y -drek Solution a Answers xe c ‘ 2 (1, 2) and (4, 1) Substitute (2) into (1). Make Connection v-x-2=0 The line intersects the curve G+ Die 5) and (2, 2). The coordinates of the points of intersection are (—1, 5) and (2, 2). We can follow these steps to solve problems in real-world contexts involving simultaneous equations. Step 1: Read the problem carefully and highlight important information. Step 2: Define the unknowns, say x and y. Step 3: Formulate two equations (usually one linear equation and one non-linear equation) in x and y from the given information. Step 4: Solve the simultaneous equations for the unknowns. Step 5: Check whether the answers are reasonable and correct. Example 3 Try 3 Measurement. A rectangular car license plate has an area of Measurement. A rectangular 600 em*, Its perimeter is 4 cm more than 10 times its length. | picture frame has an area of 154 cm? and a perimeter of 50 em. Find the dimensions of this picture frame. Find the dimensions of this license plate. Solution Answer Let em be the length and w em be the width of the license | 14 ¢m by 11 om plate. dw = 600 (1) The area is G00 cm’, 2w + = 101+ 4 The perimeter is 4 cm more than 10 times its length.Equstions and inequalities 31 Substitute (2) into (1) HAL + 2) = 600 4F + 21-600 = 0 2F + 1-300 (Qt + 25yt- 12) Since length cannot have a negative value, / = 12. When I = 12, w = 4(12) + 2 = 50. The dimensions of the license plate are 50 cm by 12 em How do | check that the. avswer i covrect? S Give two methods of solving the simultaneous equations y = x” ~ 4 and y = 2x —4, Explain how you worked iagram shows the curve x + Land the line y= x out the answers. Ee eee Solve the following simultaneous EEE feel svat apAray equations. (a) (b) (©) ae+hy Find the coordinates of the points of intersection between the curves and the lines. 2-52 +97 () Explain why the equation x° + | =x has no solutions. (ii) Draw on the same axes a line that is parallel to y =x and intersects the curve at two distinct points. State clearly the equation of the line drawn. Find the coordinates of the points of paneer Pelween the-curveriend (iil) Hence write down a quadratic ‘equation that has two solutions in ) pre the forma? + 1=x + &, where k is (b) 2x-y=4, 2744 2 : (0) 3x+y= 1, (c+ et a constant.. It is given that the following pair of simultaneous equations has one solution. y= 20" + bx + ¢, where b,c>0 y=-4 0 ear equation were replaced 4+ d, where d > 0, how many solutions would the pair of simultaneous equations have? Gi) if the linear equation were replaced by y = -4—d, how many solutions would the pair of simultaneous equations have? Enclosure Design. A rectangular enclosure is made with 18 m of fencing on three sides as shown. If the area of wall the enclosure is 40m’, calculate its possible dimensions. ym xm) Estate Planning. A town council decides to set aside a rectangular piece of land for a community garden. The piece of land is x m long and y m wide @) Find the expressions, in terms of x and y, for the area and perimeter of the land. (ii) If its area is 216 m’ and it is enclosed by 60 m of fencing, calculate the dimensions of the piece of land. Ladder Problem. A ladder that is 5m long is placed on a horizontal floor and is leaning against a vertical wall. The bbase of the ladder is x m from the wall and the top of the ladder is y m from the floor. Ei \Gse ym Ee oxm (i) Form a nonlinear equation in x and y. (ii) Solve your non-linear equation with the linear equation y (il) Explain why only one of the solutions is acceptable. A cylindrical wooden block with base radius r cm and height /r cm has a total surface area of 327 cm’. (i) Show that 7° + hr = 16. (ii) Given that its height is 4 cm more than its base radius, find the value of rand of h. The curve 12x° — Sy = 7 intersects the line 2px — at the point (1, p). (i) Find the value of p. (ii) Find the coordinates of the other point of intersection. y 28 [BD sove +2 =Sand2+ simultaneously for x and y. ain Bp A pair of simultaneous equations is given. s=xr+y t=xy Find x and y in terms of s and 1 if x>y>0. [BD 4 pair of simultaneous equations is given, 21x + 6y = k, where k is 8 real number (x — 20) + (Gy + 10" = 200 (If k= 10, find the exact solutions of the simultaneous equ: (ii) If &= 1, explain how you could tell that there are no solutions without actually solving the simultaneous equations. (iii) Find the values of k for which the simultaneous equations have one solution only. (iv) Explain why there cannot be more than two solutions for all values of k. ns.Equations and inequalities 33 Nature of Roots of a 1 Quadratic Equation You Will Learn To * Use the discriminant to determine whether @ quadratic equation has {i distinct real roots, (i) repeated real roots or (il) no real roots * Use the discriminant to determine whether @ given line (i) intersects o given curve, (i) is a tangent to a given curve or (ii) does not intersect a given curve ecall that the discriminant of the quadr: * _ 4ac, which is the expression inside the radical sign (J) in the b equation ax’ + bx+¢=0 quadratic formule x = ae 2a The discriminant is named as such, because it can ‘discriminate’ between the possible types of the solutions that the equation has—whether it has real (and distinct) roots, equal roots or no real roots. In Chapter 1, we learned the connections between the discriminant, the nature of roots of the quadratic equation ax’ + by + c = 0 and the number of x-intercepts of the quadratic function y = ax" + bx +c. We can now extend the concept of the nature of roots of the quadratic equation ax’ + bx + ¢ = 0 to the intersection between a curve and a straight line. Intersection Between a Curve and a Straight Line Recall that when we solve ax + br + c = 0, we are actually finding the number of points of intersection between © the quadratic graph of y = ax" + br + ¢ (the curve), and © the linear graph of y = 0 (the x-axis) By simply replacing the right side of the quadratic equation ax” + bx + ¢ = 0 with mx + k, we can find the number of points of intersection between * the quadratic graph of y = ax’ + br + ¢ (the curve), and © the linear graph of y = mx + k (the line). Let us examine some cases where the curve y =x" + 6x + k and the line y = 2x + 1 are drawn on the same axes. To determine the relationship between the two graphs, we need to equate the two given equations to obtain a new quadratic equation in x. V4 Gr ace k= 1 24d The discriminant of the above equation is given by — AU - 1) = 16 4k + 4 = 20 4k. Different values of k in the discriminant, 20 - 4k, determine the nature of the roots of the equation x° + 4x + (k - 1) = 0, and hence the number of points of intersection between the curve y =x" + 6x + kand the line y= 2x + 1 Number of Quadratic points eqcaton formed | Dissiminant ot | Nature ofroois of | ofittersaion | "atonhip tormen + de+ (E-1) | P+ 4ee(R-1)=0| 24 4e4+(k-1)=0 between a ee A =0 yoreerek) Meyer and y=2v4+1 2 real and peo Set ad distinct roots (+38 ad 2 real and equal 20-45) =0 (repeated) roots 1 (2 and -2) pes) Pte) Yar k-1) bos |F2 20-4K<0 No eal rots 6 Y Now let us use a graphing software to explore the relationship between a curve and « line. Activity 2B [Explore the relationship between the discriminant and the points of intersection Between ncurve and line} (i) Use a graphing software to draw the line y= 2x + 1. On the same axes, draw the curves pox + r+ k for k= 1, 2,3, 4,5, 6 and 7. (a) What values of k make the curve and the line intersect at two points? (b) What values of & make the curve and the line not intersect? (¢) What value of k makes the line a tangent to the curve? Gi) For each value of k in part (i), substitute the linear equation into the quadratic equation and find the discriminant of the new equation formed. How is the sign of the discriminant related to the number of points of intersection between the curve and the line?Equations and inequalities 35 We can find the points of intersection between y = ax" + bx+c Did You and y = mux + k by solving the two equations simultaneously, which results in solving ax” + bx + ¢ = mx + k, or ax’ + (b—m)x + (c — k) = 0. Thus, we have the following results. Know? The square root of 2 negative number is called an imaginary number, because it is not roal. When ‘the discriminant is less than 0, the quadratic equation has no real roots. We also ‘say that the equation has ‘two complex roots. Each complex root could contain an imaginary number, i, and areal number. An example of such a root is 3 + 2i, where represents «/=t. Example 4- (a) Explain why the line y = 2x + 3 does not intersect the curve y = 4x" +7. (b) If the line y = 2x +3 intersects the curve x° + xy = kat two distinct points, find the range of values of k. (©) If the line y = xm is 2 tangent to the curve 3x° + 3x, find the value of m. Solution (0) Substitute y Try 4 (a) Explain why the line year + intersects the curve y = 2x" + Sy at two distinct points. {b) IF the line y= x4 kisa tangent to the curve y = 2x" + Sr, find the value of k. . (€) If the curve y = 3x7 + m does not intersect the line y =~2r, find the range of values of m. Ancwore. (0) Substitute y x-m 3x + Qe + m= 0 x + 3x. 2+ 3x Since the line is a tangent to the curve, discriminant When a line is a tangent to a quadratic curve, the line meats the curve at one point only. Example 5: Find the value(s) or range of values of p for which (a) 4x’ - 2x + p—2 =0 has two equal real roots, (b) px? - 4x + 1 = 2x —6 has real roots, (©) + +.x-2 +p =0 has two real and distinct roots, (d) °-x>9+p for all real values of x. Solution (2) Since the equation has two equal real roots, iscriminant = 0 ? — HAY ~ 2 4 16p +32 36 - 16p =0 9 Pa Since the equation has real roots, discriminant > 0 Try 5: Find the value(s) or range of valuce of p for which (a) px’ — 2x -2= has two equal real roots, (b) 4x°- 2 + 1 = 2v—phas ‘two real and distinct roots, (0) = + 3r-2+p=Ohas real roots, {d) 17+ 2x < 5 + p for all real values of x. Answers () p= (bh) p<0 ead, (@ p=-t vie (@) p>—4(d) V-x>9+p ¥-x-O+p)>0 Since x*— x — (9 + p) > 0 for all real values of x, the curve y x — (9 + p) does not intersect the x-axis. Hence the equation y = 0 has no real roots and the discriminant is less than 0. C1)? 4()[-O + pl <0 1+36+4p <0 dp <-37 37 ps Example 6 Equations and Inequalities 37 If the line y = mr —8 intersects the curve y show that m° + 6m—7 > 0. Solution Substitute y = mx — 8 into y me-8 v= (5+ m)x + (m+8)=0 —Se+m. —Srtm Since the line intersects the curve, discriminant 2 0 [HS + m)F - 40m + 8) > 0 25+ 10m + mi — 4m —32>0 m+6m-720 Example 7. Business. Tom owns a factory that produces bicycle helmets. The profit per year, SP, from producing x number of helmets is given by P = -0.003x° + 12x + 27 760. Can Tom make a profit of $40 000 per year? Explain. Solution Substitute P ~ 40 000 into P = -0,003x7 + 12x + 27 760. 40 000 = -0,003x° + 12x + 27 760 0,008x7 - 12x + 12 240 Discriminant = (—12)° - 4(0.003)(12 240) = -2.88 <0 no real roots to the equation. is not possible for Tom to make a profit of $40 000 per year. Try 6 ifthe line y not intersect the curve v4 x43, show that =2m-3<0. Try 7. Business. Jerry owns a business that sells tires. The revenue, SR, from selling xx number of tires in July is given by R = x(200- 0.4n). Can Jerry's business generate 2 revenue of $20 000 in July? Justify your answer. Answer: Yes Maks Comme’ ‘tion We can also solve Example 7 using the method of completing the square that ras introduced in Chapter 1.ed 2.2 | ())_ State the condition for which the quadratic equation 20 + br + 3b =0 has real and distinct roots. (ii) Suggest a value of b that satisfies the condition in part Find the value(s) or range of values of p for which (a) px = 6x + p = 0 has real and equal roots, (b) 3x7 + 2x —p = 0 has real and distinct roots, (e) 2x° + 3x + 2p = 0 has real roots, (d) px -x-4= 0 has no real roots. Use the discriminant to determine whether each pair of the curve and the line intersects. if they intersect, state intersects the curve y =m where m < 1. Find the value(3) or range of values of p for which the equation (@) 6 +(~42x+4~—DH=0 has equal real roots, (b) 3x° = 2r +p — L has real and distinct roots, (©) x + p? = 3px — Shas real and repeated roots, (d) (+ 1)Qr- 1) = p—2 has real and unequal roots, (e) pox + 1-3) =x~4p—2has no real roots. Show that the roots of the equation x4 (p+ Dx = 5 — 2p are real for all real values of p Dp Find the range of values of p for which the curve y +x42p-Land the line y + 2x = p intersect at two distinct points. Find the value(s) or range of values of & for which (a) the line y = kx — 5 iis a tangent to the curve x = 2y + 1, (b) the line x + 3y =k— 1 intersect ar+5, kx + 2 intersects 8r—x° at two the curve y (c) the line the curve y distinet points, (d) the line y=x+k—1 does not intersect the curve (y- 1)? = 4x. Both the curve y = x" and the line = mx — m+ I pass through the point (a,b. (@) For what value of m is there only ‘one point of intersection? (ii) What can you say about the line with regard to its position to the curve? , show that the equation x — 1) has no real roots. (i) Are there values of k such that the curve y = (k + 3)x° — 3x lies completely above the line y= x4 K? Justify your answer. (ii) Explain the relationship between the curve and the line in part (i). ) Suggest a quadratic function f(x) that has the following properties: * There are no real solutions to the equation f(x) = 10. * The graph of the quadratic function has the maximum point. Explain your choice briefly.Projectile Motion. Mary is outside her apartment building. She needs to pass her key to John, who is 6 m above the ground floor of the building. Mary has no time to run upstairs, so she throws the key towards John. Suppose the height, / m, of the key above the ground floor at time t seconds is given by h=-lIP + 201+ L. |s it possible for John to catch the key? Justify your answer. Business. The cost in thousands of dollars, C, of producing n hundred pairs of running shoes is given by C= Ln? 14.4n + 53.7. (i) Explain why it is impossible to have a cost of production of 10 thousand dollars. (ii) Explain why it is possible for the cost of production to reach 80 thousand dollars. (iii) Find the range of values of x if the cost of production is always more than « thousand dollars. number line Equetions and inequalities 39 Projectile Motion. The height, y m, of a ball above the ground at time t seconds after it has been thrown is given by the equation y= 0.51 + 7t+ k, where k is a constant. The ball is 26 m above the ground after 10 seconds. (i) Find the initial height of the ball. (ii) Find the range of values of r when the height of the ball is above 30 m. (iii) Deduce the maximum height of the ball. (iv) Using your result in part (ii) or otherwise, express y in the form a(t — p) + y, where a, p and g are the real constants to be found, (v) A curve has the equation y= 05x? + 7x +c, where x and ¢ are real. Is this curve the same as the curve that represents the motion of the ball? Justify your answer. Quadratic Inequalities You Will Learn To 1 1 Sah qudeic equine a opteantth elton nthe * Apply quadratic inequalities in real-world contexts ecall that solving 2 linear inequality is similar to solving a linear equation, except that you need to reverse the inequality sign whenever you multiply or divide both sides of the inequality by a negative number. The following properties are useful in solving inequalities Ifa>b, thenat+c>b+e. ifa>b and c > 0, then ac > be. ifa> b and c <0, then ac < be. Solving Quadratic Inequalities A quadratic inequality in one variable is an inequality involving quadratic expressions. For example, x° - 5x +4 <0 or (x= 1)(r—4) <0) is a quadratic inequality in x