Grade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations

Transcription

1 Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions in Verte Qudrtic Reltions in Verte P. 6 #1 - Form (Verticl Stretches) Form (Verticl Stretches) D Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsformtions) Form (Trnsformtions) D Grphing Qudrtic Reltions in Grphing Qudrtic Reltions P. 65 #7-9, 1,1,16 Verte Form in Verte Form D 5 Perfect Squre Trinomils nd Perfect Squre Trinomils nd P. 90 #1,,, 5, 6, 8, 9 Completing the Squres Completing the Squres D 6 Qudrtic Formul P. 0 #,5,8,9,11,1,17,0, Qudrtic Formul D 7 Optimiztion Prolems Optimiztion Prolems D 8 Optimiztion Prolems (More) P. 55 #1, P. 91 #11-18, 0, D 9 Discriminnts of Qudrtic Discriminnts of Qudrtic P. 0 #5 Functions & reltions Functions & reltions D 10 Work period D 11 Review Review: Qudrtic Functions P. 06 #6,9 P. 16 #8c Selective: Qudrtic Functions D 1 Review P. 8 #18 & Equtions P. 9 #1-1 D 1 Test

2 Mthemtics 10 Pge 1 of 7 The Qudrtic Function (Verte Form): Trnsltions Verte form of Qudrtic Reltions The epression p q defines qudrtic reltion clled the verte form with horizontl trnsltion of p units nd verticl trnsltion of q units. The verte of the qudrtic reltion is p, q nd is of smmetr is t p. A qudrtic reltion in verte form p q epnding nd collecting like terms. cn e converted to stndrd form c p q p p 0 & q 0 Verte : 0 p, q Ais of Smmetr : p Reflects out - is Concves down p, q p, q p 0 & q 0 Verte : 0 p, q q Ais of Smmetr : p intercepts p Concve up q intercept p q Bsic Points of Qudrtic Reltion (Prol) Step Ptterns: Emple 1: Prol with Verticl Trnsltion Given the qudrtic reltion, determine the intercepts, intercept, direction of opening, is of smmetr nd the verte. Determine mpping rule nd sketch of the reltion on the given grid. Descrie the trnsltion. ), RHHS Mthemtics Deprtment

3 Mthemtics 10 Pge of 7 The Qudrtic Function (Verte Form): Trnsltions ) Emple : Prol with Horizontl Trnsltion Given the qudrtic reltion, determine the intercepts, intercept, direction of opening, is of smmetr nd the verte. Determine mpping rule nd sketch of the reltion on the given grid. Descrie the trnsltion. ), , ), RHHS Mthemtics Deprtment

4 Mthemtics 10 Pge of 7 The Qudrtic Function (Verte Form): Trnsltions Emple : Prol with Trnsltions Given the qudrtic reltion, determine mpping rule nd sketch of the reltion on the given grid. Descrie the trnsltion. ), Eercise 1. Complete the chrt: ) Eqution ) c) 7 d) e) 5 f) 1 verte -intercepts (if n) -intercept direction tht curve opens eqution of is of smmetr. Grph the following equtions on the sme set of es using the nswers ou found in question 1. Lel ech prol with its corresponding eqution. ) ) c) d) e) f) RHHS Mthemtics Deprtment

5 Mthemtics 10 Pge of 7 The Qudrtic Function (Verte Form): Trnsltions. Descrie the effect of vrious vlues of "q" on the grph of q. Which grph est represents ech of the following. (Lel the grph with the pproprite letter) ) ) c) d) 1 5. Write n eqution tht could correspond to ech grph: c d ) ) c) d) 6. Fill in the following chrt: Eqution 5 direction of opening coordintes of verte -intercepts (if n) -intercept 7. For the generl qudrtic q : ) Wht re the co-ordintes of the verte? ) Wht restriction on the vlue of q eists in order for -intercepts to eist? RHHS Mthemtics Deprtment

6 Mthemtics 10 Pge 5 of 7 The Qudrtic Function (Verte Form): Trnsltions 8. Complete the chrt for ech eqution. ) ) c) d) e) f) g) Eqution ( ) ( ) ( ) ( 6) ( ) ( 6) verte eqution of is of smmetr -intercepts (if n) -intercept 9. Grph the following equtions on the sme set of es using the nswers ou found in question 8. Lel ech prol with its corresponding eqution. ) d) ( ) f) ( ) ) ( ) e) ( 6) g) ( 6) c) ( ) 10. Compre the grphs of nd ( p) when: ) p > 0 ) p < 0 RHHS Mthemtics Deprtment

7 Mthemtics 10 Pge 6 of 7 The Qudrtic Function (Verte Form): Trnsltions 11. Which grph est represents ech of the following. Lel with pproprite letter. ) ( 1) ) ( ) c) ( ) d) ( ) 1. Write n eqution tht could correspond to ech grph: d c ) ) c) d) 1. Complete the following chrt: ) Eqution co-ordintes of verte eqution of is of smmetr direction of opening -intercept (if n) -intercept ) ( ) c) ( 8) d) ( ) e) ( ) RHHS Mthemtics Deprtment

8 Mthemtics 10 Pge 7 of 7 The Qudrtic Function (Verte Form): Trnsltions Answers 1) Verte: (0,0), -int: 0, -int: 0, opens up, = 0 ) Verte: (0,), -int: NA, -int:, opens up, = 0 c) Verte: (0,7), -int: NA, -int: 7, opens up, = 0 d) Verte: (0,-), -int:, -int: -, opens up, = 0 e) Verte: (0,-5), -int: 5 -int: -5, opens up, = 0 f) Verte: (0,1), -int: NA, -int: 1, opens up, = 0 ) Verticl trnsltions q units, d, c, ) 5) ) ) c) d) 5 1 6) Opens up, Verte: (0,5), -int: NA, -int: 5 7) (0, q) ) q < 0 ) Opens up, Verte: (0,-), -int:, -int: - c) Opens up, Verte: (0,), -int:na, -int: d) Opens up, Verte: (0,), -int:na, -int: 8) Verte: (0,0), = 0, -int: 0, -int: ) p > 0 Right p. ) Verte: (,0), =, -int:, -int: ) p < 0 Left p c) Verte: (-,0), = -, -int: -, -int: 16 d) Verte: (-,0), = -, -int: -, -int: 9 e) Verte: (6,0), = 6, -int: 6, -int: 6 f) Verte: (,0), =, -int:, -int: 16 g) Verte: (-6,0), = -6, -int: -6, -int: ) ) ( 5) ) ( 1) c) ( ) d) ( 6) c d 1) Verte: (0,0), = 0, opens up, -int: 0, -int: 0 ) Verte: (-,0), = -, opens up, -int: -, -int: 9 c) Verte: (8,0), = 8, opens down, -int: 8, -int: -6 d) Verte: (,0), =, opens up, -int:, -int: e) Verte: (-,0), = -, opens down, -int: -, -int: -16 RHHS Mthemtics Deprtment

9 Mthemtics 10 Pge 1 of The Qudrtic Function (Verte Form) Verticl Stretches Verticl stretches of Qudrtic Reltion (Prol) In generl, when the grph of qudrtic reltion is elongted in one direction, the word stretch (epnd) is used to descrie the trnsformtion. If it is shortened in one direction, the word compression is used to descrie the trnsformtion. In this course, we re going to focus onl on one tpe of stretches the verticl stretches of qudrtic reltions. Descriptions Emples > 1 Verticll epnd (stretch) fctor of 0 < < 1 Verticll compress fctor of Emple 1: Given the grph of, complete the grphs of on the sme is. Stte the mpping rule. ) ) c) 1 ) ) RHHS Mthemtics Deprtment

10 Mthemtics 10 Pge of The Qudrtic Function (Verte Form) Verticl Stretches c) Emple : Given prol in the form of, determine the vlue of if the prol psses the given points. Stte the mpping rule. 1 ), ),7.6 c), 5 RHHS Mthemtics Deprtment

11 Mthemtics 10 Pge of The Qudrtic Function (Verte Form) Verticl Stretches Eercise 1. Grph ech of the following on the sme set of es. Lel ech prol with its corresponding eqution. ) ) c) d) e) f) 1 1. Complete the following tle: ) ) Eqution co-ordintes of the verte direction of opening -intercepts -intercept c) d) e) 1 f) 1. Descrie the effect on the grph of s the vlue of "" vries.. All prols with the verte ( 0, 0) hve the generl eqution. If we re given point tht the prol psses through, we cn determine the vlue of. Determine the equtions of the prols with verte (0, 0) tht psses through the given points: Stte the mpping rule for ech of them. 1 ) (,18) ) (, 16) c ) (,) d ) (, 10) e ), RHHS Mthemtics Deprtment

12 Mthemtics 10 Pge of The Qudrtic Function (Verte Form) Verticl Stretches Answers 1) c f d ) ) c) d) e) f) 1 1 e ) Verte: (0,0), opens up, -int: 0, -int: 0 ) Verte: (0,0), opens up, -int: 0, -int: 0 c) Verte: (0,0), opens up, -int: 0, -int: 0 d) Verte: (0,0), opens down, -int: 0, -int: 0 e) Verte: (0,0), opens down, -int: 0, -int: 0 f) Verte: (0,0), opens down, -int: 0, -int: 0 ) > 1, Verticll epnd (stretch) fctor of 0 < < 1, Verticll compress fctor of ) ) c) d) e),,,, 6,, 6 5 7, 5,, 7, RHHS Mthemtics Deprtment

13 Mthemtics 10 Pge 1 of 8 The Qudrtic Reltion (Verte Form) Trnsformtions Qudrtic Reltions in Verte Form The epression p q defines qudrtic reltion in verte form. The coordintes of the verte of the corresponding prol re p, q. If > 0, the prol opens upwrd. If < 0, the prol opens downwrd. A qudrtic reltion in verte form p q cn e converted to stndrd form c epnding nd collecting like terms. A qudrtic reltion in stndrd form c cn e converted to verte form p q completing the squres which will e discussed in this unit. p q p p 0 & q 0 Verte : 0 q p p Emple 1: Find the verte, the is of smmetr, the direction of opening, -intercept(s) nd the -intercept for the grph of the qudrtic reltion. Stte the mpping rule. 1 : ) 5 p, q Ais of Smmetr : p Reflects out - is Concves down p, q p, q Mpping rule p 0 & q 0 Verte : p, q 0 Concve up q Ais of Smmetr : p intercepts intercept q RHHS Mthemtics Deprtment

14 Mthemtics 10 Pge of 8 The Qudrtic Reltion (Verte Form) Trnsformtions ) Mpping rule c) 1 Mpping rule RHHS Mthemtics Deprtment

15 Mthemtics 10 Pge of 8 The Qudrtic Reltion (Verte Form) Trnsformtions Emple : Determine qudrtic reltion in verte form which contins verte, 5nd psses through the point, 6. Determine the mpping rule for the trnsformtions. Eercise 1. Complete the following tle: ) ) Eqution ( 5) c) ( 5) 5 verte is of smmetr Opening -intercepts -intercept d) ( ) e) ( ). Use the tle ove to sketch the grph of the qudrtic reltion. Stte the mpping rule. ( 5) ( 5) 5 ( ) ( ) RHHS Mthemtics Deprtment

16 Mthemtics 10 Pge of 8 The Qudrtic Reltion (Verte Form) Trnsformtions. Complete the following tle: Eqution verte is of smmetr Opening -intercepts -intercept ) ( ) ) ( ) c) ( ) d) ( ) 5. Complete the following tle: Eqution verte is of smmetr Opening -intercepts -intercept ) ( ) ) ( 1) c) ( ) d) ( 5) 6. Use the tles in #,5 ove to sketch the grphs of the qudrtic reltion. Stte the mpping rules. RHHS Mthemtics Deprtment

17 Mthemtics 10 Pge 5 of 8 The Qudrtic Reltion (Verte Form) Trnsformtions 7. Complete the following tle: Eqution verte is of smmetr Opening -intercepts -intercept ) ( ) ) ( ) c) ( ) d) ( ) 8. Complete the following tle: Eqution ) ( ) verte is of smmetr Opening -intercepts -intercept ) 1 ( ) c) 1 ( ) 9. Use the tles in #7,8 ove to sketch the grphs of the qudrtic reltion. Stte the mpping rules. RHHS Mthemtics Deprtment

18 Mthemtics 10 Pge 6 of 8 The Qudrtic Reltion (Verte Form) Trnsformtions 10. In compring the grph of ( ) 5 to the grph of ) wht chnge is cused the 5?, eplin: ) wht chnge is cused the? c) wht chnge is cused the? 11. Given the reltion ( ) 5 ) wht is the verte of the prol? ) does the grph open upwrds or downwrds? c) is the verte mimum or minimum point? The eqution ( p) q is clled the verte form of the prol. 1. Given the reltion ( p) q ) wht chnge is cused the q? ) wht chnge is cused the p? c) wht chnge is cused the? RHHS Mthemtics Deprtment

19 Mthemtics 10 Pge 7 of 8 The Qudrtic Reltion (Verte Form) Trnsformtions 1. Given the reltion ( p) q ) wht is the verte of the prol? ) does the grph open upwrds or downwrds? c) is the verte mimum or minimum point? 1. Write the corresponding eqution in verte form for the qudrtic reltion with the given vlues. verte eqution ) ( 1,) ) ( 1, ) c) ( 1, ) d) ( 1, ) e) ( 1,) 1 f) (,7) 15. For ech of the following: ) crete the generl eqution in verte form. ) use the given point to determine the vlue of. c) write the defining eqution. (i) verte (,5) (ii) verte ( 1, 7) point on prol ( 1, ) point on prol ( 0, ) (iii) verte (, 5) (iv) verte (,1) point on prol (, 7) point on prol ( 1,7) RHHS Mthemtics Deprtment

20 Mthemtics 10 Pge 8 of 8 The Qudrtic Reltion (Verte Form) Trnsformtions Answers: 1) Verte: (0, 0); Ais of smmetr: = 0; Opens up; -int: = 0; int: = 0; ) Verte: (5, 0); Ais of smmetr: = 5; Opens up; -int: = 5; int: = 5; 5, c) Verte: (5, 5); Ais of smmetr: = 5; Opens up; -int: NA; int: = 0; 5, 5 d) Verte: (-, 0); Ais of smmetr: = -; Opens up; -int: = -; int: = ;, e) Verte: (-, -); Ais of smmetr: = -; Opens up; -int: = - & 0; int: = 0;, ) Verte: (, 0); Ais of smmetr: = ; Opens up; -int: = ; int: = 9;, ) Verte: (, -); Ais of smmetr: = ; Opens up; -int: = 1.6 &.; int: = 7;, c) Verte: (, ); Ais of smmetr: = ; Opens up; -int: NA; int: = 1;, d) Verte: (, -); Ais of smmetr: = ; Opens up; -int: = 1 & 5; int: = 5;, 5) Verte: (, ); Ais of smmetr: = ; Opens up; -int: NA; int: = 1;, ) Verte: (1, ); Ais of smmetr: = 1; Opens up; -int: NA; int: = 5; 1, c) Verte: (-, ); Ais of smmetr: = -; Opens up; -int: NA; int: = 8;, d) Verte: (-5, ); Ais of smmetr: = -5; Opens up; -int: NA; int: = 9; 5, 7) Verte: (, ); Ais of smmetr: = ; Opens up; -int: NA; int: = 8;, ) Verte: (, ); Ais of smmetr: = ; Opens up; -int: NA; int: = 16;, c) Verte: (, ); Ais of smmetr: = ; Opens down; -int: = 0 & ; int: = 0;, d) Verte: (, ); Ais of smmetr: = ; Opens down; -int: = 0.8 &.; int: = -8;, 8) Verte: (-, -); Ais of smmetr: = -; Opens up; -int: = -.7 & -0.; int: = 1;, ) Verte: (-, -); Ais of smmetr: = -; Opens up; -int: = -. & 0.; int: = -1; 1, c) Verte: (-, -); Ais of smmetr: = -; Opens down; -int: NA; int: = -5; 1, 10) Verticl trnsltion 5 units up ) Horizontl trnsltion units right c) Verticl epnsion fctor of 11) (, 5) ) upwrd c) minimum 1) Verticl trnsltion q units (if q > 0, up. q < 0, down) ) Horizontl trnsltion p units (if p > 0, Right p units, p < 0, Left p units) c) Verticl stretch fctor of (if > 1, epnd, 0 < < 1, compress) (if > 0, opens up, < 0, opens down) 1) p, q ) if > 0, opens up, < 0, opens down c) if > 0, min, < 0, m 1) 1 ) 1 c) 1 d) 1 e) 1 f) i) ) 5 ) - c) 5 ii) ) 1 7 ) c) 1 7 iii) ) 5 ) 1 1 c) 5 v) ) 1 ) c) 1 RHHS Mthemtics Deprtment

21 Mthemtics 10 Pge 1 of Grphing Qudrtic Reltions in Verte Form RHHS Mthemtics Deprtment Qudrtic Reltions in Verte Form The epression q p defines qudrtic reltion in verte form. The coordintes of the verte of the corresponding prol re q p,. If > 0, the prol opens upwrd. If < 0, the prol opens downwrd. Eercise: 1) Mtch the reltion with the proper digrm 1 ) ) 5 ) 1 ) d c ) 1 ) 1 ) 1 ) h g f e 5 ) ) 1 ) ) l k j i

22 Mthemtics 10 Pge of Grphing Qudrtic Reltions in Verte Form ) Grph ech of the following using the ke fetures. (Don t forget to lel!!) ) ) 5 1 c) 6 d) 7 e) 6 7 f) 1 6 c d - f Answers 1) i k f d c j l e h g ) e f c d RHHS Mthemtics Deprtment

23 Mthemtics 10 Pge 1 of Perfect Squre Trinomils nd Completing The Squre A qudrtic reltion in stndrd form c cn e rewritten in verte form p q creting perfect squre in the epression, then fctoring the squre. This technique is clled completing the squre. Completing the squre involves the following steps: 1) Remove the common constnt fctor from oth the nd term. ) Find the constnt tht must e dded nd sutrcted to crete perfect squre. This vlue equls the squre of hlf of the coefficient of the term in step 1. Rewrite the epression dding, then sutrcting, this vlue fter the term inside the rckets. ) Group the three terms tht form the perfect squre. Move the sutrcted vlue outside the rckets multipling it the common constnt fctor. ) Fctor the perfect squre nd collect like terms. Completing the squre cn e used to find the verte of qudrtic in stndrd form without finding the zeros of the reltion or two points equidistnt from the is of summetr. Completing the squre llows ou to find the mimum or minimum vlue of qudrtic reltion lgericll, without using grph. Emple 1: Convert ech of the following into verte form completing the squre. ) 6 ) 8 c) 5 Emple : Convert ech of the following into verte form completing the squre. Determine the verte, is of smmetr, opening, intercepts nd the mpping rule. ) 5 ) 1 5 RHHS Mthemtics Deprtment

24 Mthemtics 10 Pge of Perfect Squre Trinomils nd Completing The Squre c) 5 1 TRY: c Eercise 1. Epnd ech of the following, giving finl nswers onl. Do not show middle step. ) ( 5) ) ( ) c ) ( 1) d ) ( 7) e ) ( 1 ) f ) ( Ech of the nswers in #1 is clled perfect squre trinomil ecuse it is the result of squring inomil. Epnd ( 10) nd ( 10)( 10). In wht w re the nswers different? In wht ws re the nswers the sme? ). Which of the following trinomils re perfect squre trinomils? (circle the letter) (tr to discover pttern from the nswers ove) ) 6 9 ) 6 9 c ) 6 9 d ) 6 9 e ) 8 6 f ) 6 6 g ) 16 6 h ) 16 6 i ) 9 81 ) 9 16 RHHS Mthemtics Deprtment j k ) 9 l ) 9 m ) 5 10 n ) 1. Fill in the lnk with the pproprite numer to mke ech perfect squre trinomil. ) 1 ) 0 c) 10 d) e ) 9 f ) g ) 1 h ) 11

25 Mthemtics 10 Pge of Perfect Squre Trinomils nd Completing The Squre. Beside ech of the trinomils in #, write its fctored form. Epress our nswer s ( p) or ( p). 5. Convert ech of the following into verte form completing the squre. ) 8 ) 10 c) 18 d ) 16 e ) 1 5 f ) 10 1 g ) 7 h ) 7 1 i ) 5 6. Without grphing ech function, stte whether it hs mimum or minimum vlue. Stte the mimum or minimum vlue of the function. Stte the vlue of when it occurs. ) 6 ) 6 8 c ) 5 d) 1 e) 6 f ) g ) 8 h) 16 i) 8 10 j ) Complete the chrt. Stndrd form Verte form Direction of opening Eqution of is of smmetr Verte Is verte minimum or mimum point? Minimum or mimu m vlue -intercept ( 1) ( ) 7 5 RHHS Mthemtics Deprtment

26 Mthemtics 10 Pge of Perfect Squre Trinomils nd Completing The Squre Answers 1 ) 10 5 ) 8 16 c ) 1 d ) 1 9 e ) 1 9 f ) 9 16 ) c g h k 1 c 0, c ) 6 ) 100 c) 5 d) e) 6 f) g) h) ) 6 ) 10 c) 5 d) e) f) g) 1 h) 11 5) 16 ) 5 5 c) 9 81 d) 8 68 e) 6 1 f) g) h) i) 6) Min vlue = -7 when = - ) Min vlue = -11 when = -1 c) Min vlue = when = 1 15 d) M vlue = 18 when = - e) M vlue = 1 when = - f) Min vlue = when = 8 g) M vlue = 0 when = 1 h) Min Vlue = -16 when = i) M vlue = - when = 5 j) Min vlue = 0 when = 6 7) Stndrd form Verte form Direction of opening Eqution of is of smmetr Verte Is verte minimum or mimum point? Minimum or Mimum vlue -intercept 8 ( ) 1 Up = (, -1) Min ( ) 6 Down = (, 6) M ( ) 5 Up = , Min ( ) 7 Up = (, -7) Min ( 1) 7 Up = 1 (1, -7) Min -7-6 ( ) 5 Down = (,5) M 5 - RHHS Mthemtics Deprtment

27 Mthemtics 10 Pge 1 of Qudrtic Formul RHHS Mthemtics Deprtment Recll: Convert the Qudrtic reltions from Stndrd form to Verte form. Emple 1: Solving Qudrtic eqution using the qudrtic formul Solve ech of the following using the qudrtic formul nd leve the nswer in squre root form. ) 0 7 ) 0 c) 0 5 Emple : Find the roots of the following using the qudrtic formul nd ccurte the nswer in two deciml plces. ) 1 ) 7 c) 1 5 (Verte Form) Form) (Stndrd c c c c c c c or c Verte,, : (Qudrtic Formul!!) 0 0 the Qudrtic Formul derive will this 0 nd the roots,let Tofind c c c c c c c

28 Mthemtics 10 Pge of Qudrtic Formul Emple : The height of n oject, in metres, is given the reltion oject in the ir? h 15 8t.9t where t is in seconds. How long is the Emple : For the qudrtic reltion 5 1 (Emple c). Convert the defining eqution of the reltion from stndrd form into verte form.. Use the qudrtic formul to pproimte the roots. c. Determine the verte, is of smmetr nd intercept. d. Sketch the grph using the ke the results from & c e. Stte the mpping rule. RHHS Mthemtics Deprtment

29 Mthemtics 10 Pge of Qudrtic Formul Eercise: 1) Solve ech eqution with the Qudrtic Formul. ) 1 0 ) 0 c) 15 0 d) 1 0 e) 0 0 f) 1 0 g) h) i) j) 8 k) 0 l) m) ) Solve ech of the following using the qudrtic formul nd leve the nswer in squre root form. ) ) 8 11 c) ) For ech of the following. Convert the defining eqution of the reltion from stndrd form into verte form.. Use the qudrtic formul to pproimte the roots. c. Determine the verte, is of smmetr nd intercept. d. Sketch the grph using the ke the results from & c i) 5 1 ii) 6 iii) 16 iv) 1 6 Answers: 1) & -.5 ) 1 & - c).5 & - d) 7 & - e) & -.5 f) 0.5 & -1 g) 1 & - h) -1 i) 17 & -5 j) & -1 k) 8 & -5 l) 1 & - m) 5 & -.5 ) ) 1 c) 1 7 i) Roots: -0. & 5. Verte: (.5, -7.5) Ais of smmetr: =.5 -int: -1 ii) 1 5 Roots: None Verte: (-1, 5) Ais of smmetr: = -1 -int: 6 Roots: - Verte: (-, 0) Ais of smmetr: = - -int: iii) Roots: 6 Verte: (, 18) Ais of smmetr: = -int: 6 iv) 18 i ii iii iv RHHS Mthemtics Deprtment

30 Mthemtics 10 Pge 1 of 5 Optimiztion Prolems Optimiztion: Qudrtic Mimum & Minimum Prolems A qudrtic reltion in stndrd form c cn e rewritten in verte form p q creting perfect squre in the epression, then fctoring the squre. This technique is clled completing the squre. Completing the squre llows ou to find the mimum or minimum vlue of qudrtic reltion lgericll, without using grph which will e used in this lesson to determine the optimum vlue in qudrtic relted sitution. Emple 1: Find the minimum product of two numers whose difference is 8. Emple : A glssworks compn mkes led-crstl owls, tht cretes dil production cost C given C , where is the numer of owls mde. ) How mn owls should e mde to minimize the cost? ) Wht is the cost if this mn owls re mde? RHHS Mthemtics Deprtment

31 Mthemtics 10 Pge of 5 Optimiztion Prolems Emple : A rectngulr fence is to e uilt round plground, one side of the plground is ginst the school. If there is 00 m of fencing ville, wht dimensions would crete the lrgest plground re? School Emple : An enclosure is constructed with the shpe shown nd with perimeter of 600 m. Wht re the vlues of nd so tht the re of the enclosure is mimum? RHHS Mthemtics Deprtment

32 Mthemtics 10 Pge of 5 Optimiztion Prolems Emple 5: Vehicles Incorported currentl sells n verge of 0 compct crs ech week t price of $600 ech. The sles deprtment wnts to increse the price, ut the mrketing deprtment predicts tht for ever $00 increse, sles will fll one cr. If the deler cost (cost to the deler) for ech cr is $000, wht price will mimize profits for Vehicles Incorported? RHHS Mthemtics Deprtment

33 Mthemtics 10 Pge of 5 Optimiztion Prolems Eercise 1. The generl eqution of thrown oject is given h h0 v0t 5t, where the vlues of h 0 nd v 0 represent the initil height nd initil speed of the oject. i) Determine the eqution representing the height of rock tht is thrown upwrd from cliff tht is 15 m high, t n initil speed of 10 m/s. ii) Determine the mimum height of the rock.. A ll is thrown from n prtment uilding. Its height, in metres, fter t seconds, is given h 5t 10t 5. i) Determine the initil height of the ll. ii) Determine the mimum height of the ll. ii) Determine the length of time it tkes for the ll to rech tht height.. Two numers differ 8. Their product is to hve the lest vlue possile. Determine the numers.. The sum of the se nd the height of tringle is 15 cm. Wht is the gretest possile re for tringle hving this propert. 5. A rectngulr lot is ordered on one side strem nd on the other three sides fencing. If there is 600 metres of fence ville, determine the dimensions of the lot with the gretest re. 6. A rectngulr field is enclosed fence nd divided into two lots nother section of fence prllel to two of its sides. If the 600 metres of fence tht is used must enclose mimum re, wht re the dimensions of the field? 7. A fence is to e uilt round the re shown in the digrm. Determine the vlues of nd tht would produce minimum re if the perimeter is 00 metres. 8. If the totl costs re C R 150, nd totl revenues re, where represents the totl numer of merchndize sold. ) Find the rek-even point(s). (Revenue = Cost) or (Profit = 0) ) Write the profit function, nd find wht level production mimizes the profit? c) Wht is the mimum profit? 9. Assume tht compn knows tht the cost to produce items is given the cost function C dollrs. It lso knows tht the revenue from items is given the revenue function R Find the mimum profit the cn epect nd how mn of these items the hve to produce nd sell to mke this mimum profit. RHHS Mthemtics Deprtment

34 Mthemtics 10 Pge 5 of 5 Optimiztion Prolems 10. From producing certin product, if totl costs cn e represented C totl revenues cn e represented possile profit., nd the R 1600, find the rek-even point(s) nd the mimum 11. An uditorium hs sets for 100 people. For the pst severl ds, the uditorium hs een filled to cpcit for ech show. Tickets currentl cost $5.00 nd the owner wnts to increse the ticket prices. He estimtes tht for ech $0.50 increse in price, 100 fewer people will ttend. Wht ticket price will mimize the profit? 1. A grocer sells 50 loves of red d. The cost is $0.65 lof. The grocer estimtes tht for ech $0.05 price increse, fewer loves of red will e sold. Grph, nd then determine wht cost will mimize the profit? 1. A us compn trnsports 500 people d etween Morse Rd. nd high St. The one-w fre is $0.50. The owner estimtes tht for ech $0.10 price increse, 50 pssengers will e lost. Grph nd then determine wht price will mimize their profit? 1. A cit trnsit sstem crries 800 us riders per d for fre of $1.85. The cit hopes to reduce cr pollution getting more people to ride the us, while mimizing the trnsit sstem s revenue t the sme time. A surve indictes tht the numer of riders will increse 800 for ever $0.05 decrese in the fre. Wht fre will produce the gretest revenue? 15. A senior s dnce clu gs $5 cover chrge nd verges 00 customers on Frid nights. Over the pst severl months, the clu hs chnged nd cover price severl times to see how it ffects the numer of customers. The hve discovered tht for ever increse of $0.50 in the cover chrge, the numer of customers decreses 0, find the cover chrge tht mimize the revenue. 16. The cost C, in dollrs, of operting concrete-cutting mchine is modelled C.n 66n 655, where n is the numer of minutes the mchine is run. How long must the mchine run for the operting cost to e minimum? Wht is the minimum cost? 17. A us compn hs 000 pssengers dil, ech ping fre of $. For ech $0.15 increse, the compn estimtes tht it will lose 0 pssengers. If the compn needs to tke in $10 50 per d to st in usiness, wht fre should e chrged? Answers 1) h 5t 10t 15; m height is 0m ) 5m; 0m; 1s ), - 5 ) or ) m; 5000 m 6) m ), 9 9 8) (50, 5000), (10, 100) ) 0 units c) M Profit is $00 9) $00 10) (0, 1600), (80, 11600) 11) $5,50; M income = $6050 1) $0.95; M income = $6.10 1) $0.75; M income = $81.5 1) $ ) Keep the sme price, no chnge 16) 15 min; Min cost is $160 17) $.75 (800 tickets) or $1.5 (7. tickets) RHHS Mthemtics Deprtment

35 Mthemtics 10 Pge 1 of Optimiztion Prolems (More) RHHS Mthemtics Deprtment Recll: Convert the Qudrtic reltions from Stndrd form to Verte form. Emple 1: Determine the Verte from Stndrd Form i) Convert the following qudrtic reltions to verte form to determine the verte. ii) Use the short cut (verte formul) to determine the verte from stndrd form. ) 15 1 ) 8 Emple : Optimiztion Prolems Find two rel numers nd tht sum to 50 nd tht hve product tht is mimum. (Verte Form) Form) (Stndrd c c c c c c c or c Verte,, : (Qudrtic Formul!!) 0 0 the Qudrtic Formul derive will this 0 nd the roots,let Tofind c c c c c c c

36 Mthemtics 10 Pge of Optimiztion Prolems (More) Emple : Optimiztion Prolems Lst er the erook t Centrl High cost $75 nd onl 500 were sold. A student surve found tht for ever $5 reduction in price, 100 more students will u erooks. Wht price should e chrged to mimize the revenue from erook sles? Emple : Optimiztion Prolems Mr wnts to fence rectngulr grden to keep the deer from eting her fruit nd vegetles. One side of her grden ttches her shed wll so she will not need to fence tht side. However, she lso wnts to use mteril to seprte the rectngulr grden in two sections. She cn fford to u 80 totl feet of fencing to use for the perimeter nd the section dividing the rectngulr grden. Wht dimensions will mimize the totl re of the rectngulr grden? Emple 5: Optimiztion Prolems A piece of wire 0 feet long is cut into two pieces nd ech piece is ent to form squre. Determine the length of the two pieces so tht the sum of the res of the two squres is minimum. RHHS Mthemtics Deprtment

37 Mthemtics 10 Pge 1 of Discriminnts of Qudrtic Functions & Equtions Recll: Qudrtic formul is used to find the roots of the qudrtic reltion To find the roots, let 0 nd c 0, then c. c Note tht in the qudrtic formul, the vlue of c determines the numer nd tpe of roots qudrtic eqution hs. This epression is clled the discriminnt. When using the qudrtic formul, If c 0, then the qudrtic eqution hs rel roots. If c 0, then the qudrtic eqution hs 1 rel root. If c 0, then the qudrtic eqution hs no rel roots. Emple 1: Determine how mn roots ech eqution hs. ) ) c 0 c 0 c 0 no rel roots 1 rel root rel roots Emple : For, determine how mn -intercepts ech reltion hs. Determine the ke fetures (intercepts, is of smmetr, verte) of the prol. Sketch the prol. RHHS Mthemtics Deprtment

38 Mthemtics 10 Pge of Discriminnts of Qudrtic Functions & Equtions Eercise For ech of the following: ) Convert the defining eqution of the function from stndrd form into verte form. ) Sketch grph of the prol. Use the 5 point technique. c) Use the qudrtic formul to pproimte the roots. d) Compre the numer of roots with the numer of -intercepts. ) 5 1 ) 6 c ) 8 16 RHHS Mthemtics Deprtment

39 Mthemtics 10 Pge of Discriminnts of Qudrtic Functions & Equtions RHHS Mthemtics Deprtment

40 Mthemtics 10 Pge of Discriminnts of Qudrtic Functions & Equtions Answers ) ) c) d) :.5, 7.5 V : 1,5 V :,0 V :,6 V int : 0. & 5. int : 1 rel roots int : NA int : 5 No rel roots int : int :16 1rel root int : int : rel roots 6 6 RHHS Mthemtics Deprtment

41 Mthemtics 10 Pge 1 of Chpter Review 1. Fill in the lnks for ech of the following qudrtic functions. ) ( ) verte direction of opening ) ( 1) is of smmetr -intercept c) ( 5) 6 min/m vlue shpe compred to (circle pproprite tpe) grph of d) 1 verte is of smmetr e) ( ) 5 direction min/m point of opening (circle pproprite tpe) f) ( ) -intercept verte g) direction of opening min/m point (circle pproprite tpe) h) 1 -intercepts (if n) verte. Drw ccurte grphs of the following functions. Clerl indicte the 5 ke points used. ) ( 1) 5 ) RHHS Mthemtics Deprtment

42 Mthemtics 10 Pge of Chpter Review. Determine the equtions of the following prols for the given informtion. ) verte ( 0,0) pssing through the point (,1) 1 ) verte, ) nd -intercept ( c) is of smmetr, with mimum vlue of, with the sme shpe s d) minimum point of ( 1, 9) with -intercepts of,. A rectngulr prking lot is to e fenced on three sides leving the fourth side open to the street. If there is 800 metres of fencing ville, determine the dimensions tht would produce the mimum re. 5. Solve ech of the following: ) 8 0 ) ( 5)( ) 6 c ) 5 0 d) 5 6 e) ( 7) 5 f ) ( 1) 6. The sum of numer nd its squre is 90. Wht is the numer? Write full solution. 7. Proper solutions would e required for ech of the following: ) Two numers differ 6. If their product is minimum, determine these numers. ) A footll is kicked so tht its height fter t seconds is given, h 8t 5t. Determine the mimum height of the ll, nd the length of time it is in the ir. 8. A rectngulr lot is 15 m longer thn it is wide. If the re is 1000 m, determine the length. 9. Find the nd intercepts of the grph of 8 Answers 1 ) ) m 1 1 ) c) d) ) ) 8 7 e) f) 11 7 c) or 1 d) 6 or ) -10 or ) ) 5 h t ; m height is 9. m 5 5 8) 5 0 m 9) or ; RHHS Mthemtics Deprtment