Parabolas NCEA evel Achievement Standard 9157 (Mathematics and Statistics.) Appl graphical methods in solving problems Methods include: graphs at curriculum level 7, their features and their equations transformations of graphs. Parabolas Parabolas are the graphs of quadratic functions, which are functions of the tpe = a + b + c where a, b and c are constant numbers. Chapter 7 Graphs of quadratics = a + b + c where a = ±1 The graphs of functions of the tpe = a + b + c, where a = ±1, can be obtained b filling in a table of values (as in Chapter 6) and joining the resulting points b a smooth curve. Eample The graphs of = (graph I), = ( + 3) + (graph II) and = 4 ( ) (graph III) are shown. Each of the equations of these three graphs can be epanded and simplified to the form = a + b + c where a = ±1 (as shown below). I: = is alread in the form = a + b + c where a = 1, b = 0, c = 0 II I 6 Domain = Reals, Range = 0 II: = ( + 3) + 4 = + 6 + 9 + [epanding] III \ = + 6 + 11 and so is of the form = a + b + c where a = 1, b = 6 and c = 11 6 4 4 6 Domain = Reals, Range = III: = 4 ( ) 4 = 4 ( 4 + 4) [epanding] = 4 + 4 4 \ = + 4 and so is of the form = a + b + c where a = 1, b = 4 and c = 0 Domain = Reals, Range = 4 Note: If a (the coefficient of ) is negative, the graph is inverted. Graphs of = ±( a) + b The above eample illustrates that an graph of the tpe = ±( a) + b is congruent (identical in size and shape) to the graph of =, and has a verte (turning point) at (a,b). For eample, = ( + 3) + has a verte at (3,) which is a local minimum, and = 4 ( ) (which can be written as = ( ) + 4) has a verte at (,4), which is a local maimum. ESA Publications (NZ) td, Customer freephone: 080037 66
4 Achievement Standard 9157 (Mathematics and Statistics.) Graphs of quadratics in factorised form Quadratic functions which are factorised are easil sketched, b plotting intercepts and finding the verte using the ais of smmetr. Functions whose graphs are congruent to = have factorised form = ( a)( b) where a and b are real numbers. Eample The graph of = ( 1)( 3) is shown (the parabola) along with the ais of smmetr 6 (the dotted vertical line = ). 4 Note the following important points: The intercept is 3, which is found b setting 4 = 0 in the equation. This gives = (0 1)(0 3) = 3 The intercepts are 1 and 3 which are found b setting = 0 in the equation. The ais of smmetr is midwa between the intercepts. Its equation is = 1 + 3, which is = The coordinates of the verte are (,1). The verte is found b substituting = (the equation of the ais of smmetr) into the equation. This gives = ( 1)( 3) = 1 A smooth curve is drawn through the intercepts and verte, so that a smmetrical graph results. If a graph is congruent to =, its factorised equation will be of the form = ( a)( b) or = ( a)(b ). Eample Q. Write down the equation in factorised form of the parabola shown. A. The graph is congruent to = and has intercepts 3 and 1. The equation of the parabola is therefore: = ( + 3)( 1) Alternativel, since ( 1) = 1, the equation can be written: = ( + 3)(1 ) 4 4 4 Chapter 7 Activit 7B: Graphs of quadratics in factorised form Ans p. 357 1. a. Cop and complete in this table of values for the function = ( + 3)( 1) 4 3 1 0 1 b. Use our table to draw the graph of = ( + 3)( 1) c. What are the coordinates of the intercepts of the graph ou have drawn? d. What are the coordinate of intercept of the graph ou have drawn? ESA Publications (NZ) td, Customer freephone: 080037 66
e. What is the equation of the ais of smmetr of this graph? f. What are the coordinates of the verte of this graph? g. Write the equation of this parabola in the form = a + b + c.. a. Cop and complete the table and draw the graph of = ( + )( ) Chapter 7: Parabolas 43 3 1 0 1 3 5 0 b. What is the equation of the ais of smmetr of this graph? c. What are the coordinates of the verte of this graph? d. What are the coordinates of the intercepts of this graph? e. Write the equation of this parabola in the form = a + b + c. 3. Write the equation in factored form for each of the parabolas (I, II, III) shown. Each is congruent to =. 6 II I 4 6 4 4 6 4 III 6 Graphs of quadratics which can be factorised Quadratic functions which can be factorised are easil sketched, using the techniques discussed in the previous section. Eample = 6 + 5 can be factorised to give = ( 1) ( 5) 6 The intercept is found b substituting = 0: = (0 1)(0 5) = 5 4 The intercepts are found b substituting = 0: ( 1)( 5) = 0 = 1 or = 5 4 4 6 The ais of smmetr lies midwa between the intercepts and hence is the vertical line through = 3 4 The turning point will lie on the ais of smmetr and will have vertical coordinate: = (3 1) (3 5) = 4 [substituting = 3 into = ( 1)( 5)] \ the turning point is (3,4) Note: = 6 + 5 can be written as = ( 3) 4 which is in the form = a( b) + c and thus could be sketched using the method described in the first eample. Chapter 7 ESA Publications (NZ) td, Customer freephone: 080037 66
44 Achievement Standard 9157 (Mathematics and Statistics.) Activit 7C: Graphs of quadratics which can be factorised Ans p. 358 Draw a neat graph for each of the following quadratic functions, showing clearl: i. intercept ii. intercepts iii. turning points 1. = 3. = 4 + 3 3. = ( 8) 4. = 6 + 8 5. = 3 4 6. = 4 + 5 Graphs of = a + b + c when a 1 When b, c are both zero, the graphs are of the tpe = a. Eample Sketches of =, =, = 1 3 and = 1 are drawn b plotting a few points and joining them. (Take care to square the values before multipling b the coefficient. For eample, if = 1 then = 1 =.) 4 3 1 1 1 = 3 = 1 3 1 1 1 = 4 3 1 1 1 = 1 1 1 1 1 Note: The graph of = is the graph of = stretched verticall. The graph of = 1 3 is the graph of = compressed verticall. Chapter 7 The graph of = 1 is the graph of = compressed verticall and inverted. Activit 7D: Graphs of the form = a Ans p. 358 Sketch the graphs of the following quadratic functions. abel the coordinates of a point on each graph. 1. = 3. = 1 3. = 4. = 3 5. = 4 ESA Publications (NZ) td, Customer freephone: 080037 66
Formulae NCEA evel Achievement Standard 9161 (Mathematics and Statistics.6) Appl algebraic methods in solving problems Methods include: manipulating algebraic epressions including rational epressions manipulating epressions with eponents. Chapter 0 Introduction The topics in this chapter provide a revision of the NCEA Mathematics evel 1 work on formulae. These topics are not assessed directl in evel, but are assumed knowledge throughout the evel course. Mathematical modelling Mathematical modelling involves epressing a relationship as a formula in which mathematical smbols and letters are used to represent variables. The abilit to do this accuratel and use the resulting formula is important in man situations. Eample Three quantities used in accounting are assets (A), proprietorship (P), and liabilities (). The relationship between these quantities is that assets are equal to the sum of proprietorship and liabilities. In mathematical form, the relationship can be epressed: A = P + Note: A, P and are variables, and the equation, A = P + is a formula for A in terms of the two variables, P and. Eample Q. The volume of a cone is one third of the product of its height and the area of its base. Epress this relationship in mathematical form. A. et the volume of the cone be V, the radius of the base be R, and the height be H. The base area is πr [formula for area of circle] the volume is V = 1 3 πr H [from information given] R H Eample Q. A trapezium is a quadrilateral with a pair of parallel sides. Find an epression for the area of the trapezium shown where one parallel side is one half the length of the other parallel side, and the distance between the two parallel 4 ESA Publications (NZ) td, Customer freephone: 080037 66
154 Achievement Standard 9161 (Mathematics and Statistics.6) sides is equal to one quarter of the length of the longer parallel side. Epress the area in terms of the length of the longer parallel side. A. et the length of the longer parallel side be and the area be A. The area of the trapezium is the sum of the areas of the two shaded triangles (as shown on the diagram, where the shorter parallel side is and the height is 4 ). Area of the smaller triangle is 1 4 = 16 1 Area of the larger triangle is 4 = 8 Area of the trapezium is A = 3 + = 8 16 16 [area is half base height] [adding fractions] Activit 0A: Writing formulae Ans p. 385 Chapter 0 1. A pair of socks costs $D. Write an epression for the cost of the socks in cents.. a. Epress the time of 3 minutes in hours. b. Epress this same time in seconds. 3. Peter weighs kilograms and grams. a. Epress Peter s weight in grams. b. Epress his weight in kilograms. 4. A rectangle has length metres and width W centimetres. Write an epression for: a. the area in cm b. the area in m c. the perimeter in mm 5. A square has area 4 in square metres. Write an epression for: a. the perimeter in cm b. the perimeter in mm 6. A rightangled triangle has base b and height a. Both have the same unit. a. What is the area of the triangle? b. What is the length of the hpotenuse (the third side)? 7. a. Find an epression for the volume of the clinder shown. b. Find an epression for the total surface area of the clinder. 8. a. Find an epression for the perimeter of the shape shown. b. Find a formula for the area of this shape. z 9. a. What is the perimeter of the shape shown? b. What is the volume of the bo which forms when the shape is folded along the dotted lines? M ESA Publications (NZ) td, Customer freephone: 080037 66
Chapter 0: Formulae 155 10. The length of the closed bo shown is 4. The height and width of the bo are both half the length. a. Write down an epression for the volume of the bo. b. Write down an epression for the surface area of the bo. 4 Changing the subject of a formula The subject of a formula is a variable which appears alone on one side (usuall the left side) of the equals sign. The subject is epressed in terms of the other variables. Changing the subject of a formula means rearranging the formula (using inverse operations) so that another variable becomes the subject. Eample Q. Make the subject of = 6 A. = 6 + 6 = [adding 6 to both sides] = + 6 [swapping sides] = + 6 [dividing both sides b ] Sometimes the variable to be made subject appears more than once in the formula. You will need to collect terms in this variable on one side and then factorise. Chapter 0 Eample Q. Make T the subject of AT + D = E T AT D A. = E + T AT + D = ET [multipling both sides b T] ET = AT + D [swapping sides] ET AT = D [collecting terms in T on one side] T(E A) = D [factorising ET AT ] T = D E A [dividing b E A (providing E A 0)] Note: D = 0 if E A = 0 [substituting E A = 0 in T(E A) = D] Squaring and taking the square root are inverses. Take care with signs when taking a square root (if = then = ± ). ESA Publications (NZ) td, Customer freephone: 080037 66
answers Chapter 1: Coordinate geometr Activit 1A: Distance between points (page 4) 1. a. 3 b. 10 c. 13 d. 40 e. 41. a.. km b. 6.1 km c. 3.6 km d. 8.6 km e. 5.8 km 3. 4.166 km 4. 7.691 km 5. b. AB = 6, AC = 6, BC = 3 6. Side lengths are 40, 40, 3 7. (1,) 8. a. 5 cm b.. (1 dp) 9. = or 6 10. a = 1 or 5 Activit 1B: Midpoints of line segments (page 6) 1. a. (3.5,6) b. (,) c. (4,3) d. (1.5,0) e. (0.5,4) f. (a + 1,a 1). (4,1) 3. (7,7) 4. 1 km east and km north of camp 5. a. 40 km west and 0 km north of island b. 60 km west and 40 km north of port 6. 13 7. a = 4, b = 9 8. midpoint AC = midpoint BD = ( 1, 1 ) 9. a. 4.7 b. QR = 8.544 which is double 4.7 10. Student proof Chapter : Gradient of a straight line Activit : The gradient of a straight line (page 9) 1. a. m = 1 3 b. m = 3 c. m = 1 d. m = e. m is undefined f. m = 0 4. Both lines have gradient 3. Both lines have gradient 3 4. Gradients are 1 and 3 5. Not parallel 6. Parallel 7. k = 3 8. k = 4 9. k = 3 10. k = 11 11. = 1 1. = 13 13. = 13 14. 4 15. 4 16. Proof (grad AB = grad BC = ) 17. Proof (grad AB = grad BC = 3) 18. a = 7 19. a = 11 3 0. a. 3 b. 3 c. The two lines are parallel 8 8 d. Student proof ESA Publications (NZ) td, Customer freephone: 080037 66
Formulae and tables Eternall assessed standards Quadratics If a + b + c = 0 b ± b 4 ac then = a and = b 4ac ogarithms If = b then = log b log b () = log b + log b log b = log b log b log b ( n ) = n log b If = e then = log e (= In ) Calculus d d (n ) = n n 1 n d = Probabilit Z = µ σ n + 1 + c n+ 1 Internall assessed standards Coordinate geometr 1 = m( 1 ) Sequences and series t n = a + (n 1)d S n = n [a + (n 1)d] t n = a r n 1 S n = a (1 r n ), r 1 1 r a S = for r < 1 1 r Statistics ( ) s = n IQR median ± 1.5 n Trigonometr a b c = = sina sinb sinc a = b + c bc cos A b + c a cos A = bc Area of triangle = 1 bc sin A Arc length = r θ Area of sector = 1 r θ Area of segment = 1 r (θ sin θ) ESA Publications (NZ) td, Customer freephone: 080037 66
answers Inde absolute value 3, 83 absolute value function 87, 9 acceleration 1, 3 actual probabilit 75 algebraic equation 133 algebraic fractions 147 altitude 3 amplitude 75 antiderivative 19 antidifferentiation 19 arbitrar constant 19, 0 arc 15 arc length 15 area of sector 15 area rule (triangle) 119 arithmetic sequence 99 asmptote 34, 57, 69 average 41 ais of smmetr 4, 58 backtoback stemandleaf diagram 48 base (indices) 169 basechanging rule 69 bearings 10 bimodal 49 boundar line 339 boandwhisker plot 5, 58 calculus 197 Cartesian notation 198 categor variable 90 census 35 change of base rule 69 cluster sampling 37 clusters 47, 50 coefficient 144 collinear 9, 10 common denominator 149 common difference 99 common factor 143 common ratio 105 complement of event 79 compress 80 conclusion (inference) 65 concurrent 5 conditional probabilit 85 confidence interval 57 congruent 39, 4 constant term 161, 0 continuous population 95, 308 convention for labelling triangles 115 cosine (cos) 75 cosine rule 115 cube root 173 cubic functions 51 deca curve 65 decreasing function 3, 05 degree (polnomial) 51, 159, 161 denominator 146, 147 derivative 197 derived function 197 difference of two squares 144 differentiable 197 differentiating 197 discontinuit 58 discriminant 166 distance of point from line 3 distribution 47 distributive law 141 divisor 148 domain 30, 57, 60, 75 dot plot 47, 50 enlarge 80 equall likel outcomes 75, 76 equation of straight line 11 equation of tangent 03 event 75, 76 epanding 141 epected number or value 89 eperiment 75 eperimental distributions 307 eperimental probabilit 75, 3 ESA Publications (NZ) td, Customer freephone: 080037 66