x 2x x 4x x 1x x 8x x x 12 = 0 G&R vwo B deel 1 1 Vergelijkingen en ongelijkheden C. von Schwartzenberg 1/25

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1 C. von Schwartzenberg 1/ 1 I, II, IV en V zijn tweedegraadsvergelijkingen. (de hoogste macht van is steeds ; te zien na wegwerken haakjes?) (III is een eerstegraadsvergelijking en VI is een derdegraadsvergelijking) V1a V1b V1c Va Va Vb Vc Vd Ve Vf Neem GR - practicum 1 door. (uitwerkingen aan het eind) Voorkennis: Ontbinden in factoren blz. 16 (op bladzijde 9 in het boek wordt hiernaar verwezen) ( ). V1d 0 ( ). ( 1). V1e 7 ( 7). V1f ( ). 8 ( 8). ❶ ❷ ( ) ( ) = 1 = 8 1. Vb 1 = en 8 =. ❸❹ = ( 1) ( ). Vg = ( ) ( 1). Vh = ( 6) ( ). 1 6 = ( 8) ( 7). 1 0 = ( 6) ( ). Vi 1 = ( ) ( 1) = ( ) ( ). Vj 10 9 = ( 9) ( 1). Vk = ( 19) ( 1). Vl = ( ) ( 1). 1 = ( 6) ( ). 0 = ( 10) ( ). Va Vb Vc ( 1). Vd = ( ) ( ). Ve ( 1). (zie V1b) Vf 1 6 = ( 8) ( 7) ( 1). ( 1). Va Vb Vc 1 ( 1). Vd 8 ( ). 1 6 = ( 6) ( 6) = ( 6). Ve 8 ( 8). 1 8 = ( 1) ( ). Vf 8 0 = ( 10) ( ). a d 6 = 6 ( ) ( ). (1 ) 1 1. e b ( 1) = ( ) ( ). 11 ± f c = ( ) = = 1. = 1 (kan niet) (een kwadraat kan nooit negatief zijn) geen oplossingen. a d 6 = 6 8 ( ) ( ). 0, 6 1 ( 6) ( ) 6 6. b 6 ( 6) c ( ) ( ). e ( ) ( ) ( ). = a =, b = en c = D = b ac = ( ) = 9 16 = b ± D ± = = ± a = 8 = = = 1. f 8 ( ).

2 C. von Schwartzenberg / a d ± ( ) = 1 ( ) = 8 ( ) = 16 ( ) = ± 16 = ± = = 7 1. b e = 9 9 c ( ) = 80 ( ) = 7 a =, b = 9 en c = ( ) = D = b ac = ( 9) = 81 0 = 11 ( ) = ± = ± b ± D 9 ± 11 = = 9 ± 11 = = a 9 11 = 9 11 = = ( 1) = 1 f 8 ( ) = ( 1) = ( ) = ( 1) = ( ) = 1 ( 1) = ± = ± ( ) = ± 1 = ± 1 1 = 1 = = 1 = 1 = = 1 = 6 = a d f 6a 6d b = 1 c = 1 ( ) 1 19 ( 7) ( ) ± = 1 e ( 1) ( 6) a = 1, b = 1 en c = = 1 = 6 D = b ac = ( 1) 1 = = b ± D 1 ± 16 1 ± 16 = = a ( 1) ( 6) = 9 g ( 1) = 6 h ( 1) = = a =, b = 1 en c = = 6 D = ( 1) = 1 16 = 17 a = 1, b = 1 en c = 1 b ± D 1 ± 17 1 ± 17 1 = = D = = 1 = a b ± D 1 ± 1 ± ± 1 = ± = = a ( ) = 16 6b ( ) = 16 kan niet 6c ( ) = ( ) ( ) = 16 (een kwadraat kan niet negatief zijn) ( ) ( ) = 9 = 16 geen oplossingen. 9 = ( 9) ( 1) a = 1, b = 8 en c = D = = 6 6 = 8 b ± D 8 ± 8 8 ± 8 = = a ( ) ( ) = 9 6e ( ) = = 9 = ± 6 = ± 6 = 6 = 6 a =, b = en c = = = 9 D = = = 9 = 9 = 1. b ± D ± = = ± = a 7 = 1 = 7 = = 1.

3 C. von Schwartzenberg / 6f ( ) = ( ) = 6 9 = 7 9 a = 1, b = 7 en c = 9 D = = 9 6 = 1 b ± D 7 ± 1 7 ± 1 = = a g ( 1) = ( ) ( 1) = = a = 1, b = 8 en c = 10 D = = 6 0 = b ± D 8 ± 8 ± = = a h ( ) ( ) = 6 9 = ( 6) ( 1) a 1 6 ( ) ( ). 7b 6 a = 1, b = en c = 6 D = 1 6 = > 0 dus oplossingen. 7c p 6 a = 1, b = p en c = 6 D = p 1 6 = p > 0 dus oplossingen. 8a a = 1, b = 7 en c = p D = ( 7) 1 p = 9 p oplossingen D = 9 p > 0 p > 9 p < 9 = c a =, b = en c = p D = p = 16 1p twee oplossingen D = 16 1p > 0 1p > 16 p < 16 = b a =, b = en c = p D = ( ) p = 8p oplossingen D = 8p > 0 8p > p > = d a =, b = en c = p D = ( ) 1 p = 9 p twee oplossingen D = 9 p > 0 p > 9 p < 9 = a a = 1, b = p en c = D = p 1 = p 100 twee opl. D = p 100 > 0 p > 100 p < 10 p > 10. 9b a = 1, b = p en c = D = p 1 = p 16 geen opl. D = p 16 < 0 p < 16 < p <. 9c a =, b = p en c = D = p = p > 0. (dus voor elke p twee oplossingen) 10a 1 1 p p p =. De vergelijking is: ( ) ( 1) 1 (was bekend). 10b p p 1 p = 1 p =. Dus ( a =, b = 11 en c = 10) D = ( 11) 10 = = 1 b ± D 11 ± 1 = = 11 ± 1 a = 1 = (bekend) 11 1 = 10 = = a 11b 1a 1b 1a De vergelijking is: 0 1 ; deze heeft één oplossing (het is een eerstegraadsvergelijking). p 1 ( a = p 0, b = en c = 1) D = p 1 = 9 p; twee oplossingen D = 9 p > 0 p > 9 p < 9. Dus p < 9 én p 0 p < 0 0 < p < 1. p ( a = p 0, b = en c = ) D = p = 8 p; twee oplossingen D = 8p > 0 8p > p <. Dus p < én p 0 p < 0 0 < p < p ( a = p 0, b = en c = ) D = ( ) p = 9 16 p : twee oplossingen D = 9 16p > 0 16p > 9 p > 9. Dus p > 9 én p 0 9 < p < 0 p > p ( a =, b = 1 en c = p) ; geen oplossing D = 1 p = 1 8p < 0 8p < 1 p > 1 p >

4 C. von Schwartzenberg / 1b p p ( a = p 0, b = 1 en c = p) ( p geeft 1 oplossing, namelijk ) ; twee oplossingen D = 1 p p = 1 p > 0 p > 1 p < 1 p < 1 1 < p < 1 én p 0. 1c p 1 ( a =, b = p en c = 1) ; twee oplossingen D = p 1 = p 8 > 0 p > 8 (grafiek is dalparabool) p < 8 p > 8. 1a 1b p = = p 0 p 6 9 ( a = p 0, b = 6 en c = 9) ; b ± D 6 ± 0 één oplossing D = 6 p 9 = 6 6p 6p = 6 p = 1 = = 6 =. a 1 p 1 ( a = 1, b = p en c = 1) ; één oplossing D = p 1 1 = p p = p = ±. b ± D ± 0 b ± D ± 0 p = b = = = = 1 en p = b = = = = 1. a 1 a 1 1a 1b 16a 16b * 10 heeft één oplossing, omdat de grafiek van f en de horizontale lijn y = 10 één snijpunt hebben. (zie fig. 1.1a) 10 heeft ook één oplossing, omdat de grafiek van f en de horizontale lijn y = 10 één snijpunt hebben. 10 heeft twee oplossingen, omdat de grafiek van f en de lijn y = 10 twee snijpunten hebben. (zie figuur 1.1b) 10 heeft geen oplossingen, omdat de grafiek van f en de lijn y = 10 geen snijpunten hebben a 17d 6 ± = ± 81 = ±. 17b 17e = =. 1 = 97 = 96 =. 17c 17f 0, = ± a 18d 1 = = 1 ± 1 = ± 1. 8 = 1 8 = 1 1 = = b 18e = = geen oplossing. 6 7 = 97 6 = ± c 18f 1 = 9 = ,1 1 = ,1 = a ( ) 7 = 7 ( ) ( ) = 10 = ± 10 ± b 6 ( 1) = 1 ( 1) = ( 1) = 1 = =

5 C. von Schwartzenberg / 19c 1 ( 1) = 8 ( 1) = 16 1 = ± 16 = ± = 1 ± 1 ± 1 = 1 1 = 1. 19d ( ) = 19 1 ( ) = 81 ( ) = = = = 6 6 = = a 0c = 17 ±. ( ) = 1 ( ) = = = 1 = b 0d = 167 = 17 = (1 ) = (1 ) = 1 (1 ) = 6 1 = ± 6 = 1 ± = 1 ± ab ( ) = ( ) ( 1) 1. a c 6 ( 6) ( ) ( ). 1 1 ( 1) ( 6) ( ) 6. b d 6 6 ( 6) ( 6) ( 1) (noem tijdelijk t ) t 1t 6 ( t ) ( t 9) t = t = 9 ± ±. a c 10 9 (noem tijdelijk t ) t 10t 9 ( t 9) ( t 1) t = 9 t = 1 ± ± = (noem tijdelijk t ) t 10t 16 ( t 8) ( t ) t = 8 t = ± 8 ±. b d 8 9 (noem tijdelijk t ) t 8t 9 ( t 9) ( t 1) t = 9 t = 1 (kan niet) ± ( 10 ) ( ) ( ) (dubbel). ab 11 1 (noem tijdelijk p) p 11p 1 ( a =, b = 11 en c = 1) D = ( 11) 1 = = b ± D 11 ± p = = = 11 ± a p = = = p = 11 = 6 = = 1 1 ± ± 1 1.

6 C. von Schwartzenberg 6/ a c 6a 6c 6 = (noem tijdelijk t ) 6t 7t ( a = 6, b = 7 en c = ) b = (noem tijdelijk t ) t t ( a =, b = 1 en c = ) D = ( 7) 6 = 9 8 = 1 D = ( 1) = 1 = b ± D 7 ± 1 t = = = 7 ± 1 b ± D 1 ± t = = = 1 ± a 6 1 a t = = = t = 7 1 = 6 = t = = = 1 t = 1 = 1 (k.n.) ± ± 1. ± = d 7 (noem tijdelijk t ) t 7t ( a =, b = 7 en c = ) D = 7 = 9 = 81 b ± D 7 ± 81 t = = = 7 ± 9 a t = = = t = 7 9 = 16 = (k.n.) ± 1 = ± = (noem tijdelijk t ) 16t 16t ( a = 16, b = 16 en c = ) D = ( 16) 16 = 096 b ± D 16 ± 096 t = = = 16 ± 6 a t = = = t = 16 6 = 7 = 9 ± = ± = ± ± = ± = ± 1. 1 = 1 (noem tijdelijk t ) t t 1 ( a =, b = en c = 1) 6b 1 = (noem tijdelijk t ) t 1t 18 ( a =, b = 1 en c = 18) D = ( ) 1 = 61 D = 1 18 = 809 b ± D ± 61 t = = = ± 19 b ± D 1 ± 809 t = = = 1 ± a 8 a = = 9 19 = = 17 = 1 1 = = 1 =... (k.n.) ± ± 1 ±.. 6 = 6 (noem tijdelijk t ) t t ( a =, b = en c = ) 6d = (noem tijdelijk t ) 6t t ( a = 6, b = en c = 7) D = ( ) = 16 D = ( ) 6 7 = 6 b ± D ± 16 t = = = ± b ± D ± 6 t = = = ± 08 a 8 a = = = 08 7 = = 08 = 16 = = = 8 = = 8. 7a De getallen 7 en 7. 7b 1 = 7 1 = 7 = 8 = 6. 8a 1 = 8 8b = 1 1 = 8 1 = 8 = 1 = 1 = 9 = 7 9 = 1 7 = 1. ± ±. 8c = 11 8d = 11 = 11 = 16 = 6 8 (k.n.) ± (k.n.) 16 ±.

7 C. von Schwartzenberg 7/ 9a 9c = 1 9b = 1 = 1 = (k.n.) ± 10. = 6 9d 6 = (noem tijdelijk t ) t t 6 t t 6 ( t 6) ( t 1) ( t ) ( t ) t = 6 1 (k.n.) ± 6 ± ±. = 1 = 1 = 1 = = = (stel t ) t 10t t 10t ( t 1) ( t ) ( t 6) ( t ) t = a = (kwadrateren) = b = heeft geen oplossing, omdat een wortel niet negatief kan zijn. 1a 1c 1 (kwadrateren) 1 1 ( 7) ( ) 7 (voldoet) (voldoet niet). (kwadrateren) = 0 = 0 = ( ) (voldoet) (voldoet). 1b 1d = 8 0 (kwadrateren) ( ) = ( a = 9, b = 8 en c = 0) D = ( 8) 9 0 = 6 70 = 78 b ± D 8 ± 78 = = 8 ± 8 a = 6 = (voldoet) 8 8 = 0 =... (voldoet niet) = 18 7 (kwadrateren) ( ) = ( ) ( ) (voldoet) (voldoet niet). a c (kwadrateren) (voldoet). 9 = (kwadrateren) = 7 9 ( a =, b = 7 en c = 9) D = ( 7) 9 = 1 b b ± D 7 ± 1 = = 7 ± a 8 7 = 7 = 9 (voldoet) 7 = = 1 (voldoet niet) (kwadrateren) = 0 = ( ) (voldoet) 6 1 (voldoet). d = 1 = (kwadrateren) 1 = (voldoet). a 10 b 10 (kwadrateren) = ( a =, b = 1 en c = 100) D = ( 1) 100 = 81 b ± D 1 ± 81 = = 1 ± 9 a = 6 1 (voldoet niet) 1 9 = = (voldoet) = (kwadrateren) 1 = 0 = 1 0 = ( ) ( ) (voldoet) (voldoet niet).

8 C. von Schwartzenberg 8/ c 6 6 (kwadrateren) = 6 ( a =, b = en c = 6) D = ( ) 6 = 9 b ± D ± 9 = = ± 7 a 8 7 = = (voldoet niet) 7 = 18 = 1 (voldoet) d 10 8 = (kwadrateren) 6 = 6 = (voldoet). a ( ) 6 (stel p) p p 6 ( p ) ( p ) p = p = b ( voldoet niet, > 0 of < 0 kan ook niet) (kwadrateren). a 9 8 (stel t ) t 9t 8 ( t 8) ( t 1) b 7 = (stel t ) t 8t 7 t = 8 t = 1 (kwadrateren) ( t 7) ( t 1) 8 = 6 1 = 1 t = 7 1 (kwadrateren) 6 = 1 = 1. 7 = 79 1 = 1 79 = 9 1 = 1. c 8 8 = 6 d (stel t ) (stel t ) t t 8t 6t 8 ( a = 8, b = 6 en c = 8) ( t ) ( t 1) D = ( 6) 8 8 = 969 t = 1 (kwadrateren) b ± D 6 ± 969 t = = = 6 ± 6 = 10 1 = 1 a = 1 = = 18 = = = 1 (kwadr.) = 6 ( 1 ) = = 1 = a 6c 0 = ( t ) t 11t 0 ( t 6) ( t ) t = 6 (kwadrateren) 6 = 6 = = (stel t ) t 7t 10 ( t ) ( t ) t = (kwadr.) = =. 6d 6b 1 = ( t ) t 16t 1 ( t 1) ( t 1) t = 1 1 (kwadrateren) 1 = = = 1 = 1. = (stel t ) t 1 0t ( a =, b = 10 en c = ) D = ( 1 0) = 1069 b ± D 10 ± 1069 t = = = 10 ± 10 a 6 = = = 10 ( 1 ) = = 1 = (kwadr.) = (kwadrateren) = 1 ( 9) ( 16) 9 (voldoet niet) 16 (voldoet). 1 1 (stel t ) t t 1 ( t ) ( t ) t = t = (k.n.) = 16.

9 C. von Schwartzenberg 9/ 8a 8b De kruisproducten bij de tabel geeft ( ) ( ) ( ). 9a = 10 = = 1 1 = 1 1 9b 9c d 1 = ( ) = 10 ( 1) ( ) ( 1) = ( ) ( 1) ( ) = ( 1) 1 = = = = = 1 ( 1) 7 10 = 9 = 1 ( ) ( ) = 1 9 (vold.). (vold.). 1 (vold.). (vold.) (vold.). VOLDOET NIET ALS EEN NOEMER NUL WORDT!!! 9e = f = ( ) = ( 18) ( 1) ( ) ( ) = ( ) ( ) = ( a =, b = 1 en c = 18) 7 1 ( a = 7, b = en c = 1) D = ( 1) 18 = D = ( ) 7 1 = 89 b ± D 1 ± = = 1 ± b ± D ± a = = ± a = 18 =, (vold.) 1 = 8 = (vold.). 17 = = (vold.) 17 = 8 = (vold.) a 1 1 = 1. (vold.) (vold.) 0b = = 1 ( ) ( 1) 1. (vold.) (vold.) 0c = = 1 1. (vold.) (vold.) (vold.) 0d 1 = = ( ) ( 1) 1. (vold.) (vold. niet) 1a 1c a 10 = = ( 10) = ( 1) 10 = (vold.) (vold.) 1b 8 = = = ( ) ( ). (vold.) (vold.) (vold.) 10 = 6 1 1d = 1 1 = ( 1) ( 1) ( 10) = ( 1) (6 1) = ( 1) 7 0 = ( 1) 18 6 = ( 1) 7 0 = 18 6 = 8 0 = 71 (stel t ) 0 = 6 0 t 71t ( a =, b = 71 en c = ) 0 = 1 0 (stel t ) D = ( 71) t 1t 0 ( a =, b = 1 en c ) b ± D 71 ± 0 t = = = 71 ± D = ( 1) 0 = = 9 a t = 71 = 1, 71 b ± D 1 ± 9 = t = = = 1 ± a 1, 1,. t = 1 = 1 =, (vold.) (vold.) (vold.) (vold.),,. (vold.) (vold.) (vold.) (vold.) en y = invullen in y = 1 geeft 0 = 1 (klopt). y -as b invullen in l geeft y = 1 = 1 = (, ) ligt op l. c y = 1 y = 1 y = 1. -as y = 6 y = 1 y y = y -6 0 y 1 0 y 0 1 y 0

10 C. von Schwartzenberg 10/ a y 6; dus snijpunt met de -as: (6, 0). y = y = 8; dus snijpunt met de y -as: (0, 8). b 8 (8, ) ligt niet op l = (18, 16) ligt op l. 0 8 = ( 0, 8) ligt op l. c 16 p = p = 0 p = 1 1. d q 8 = q = 168 q =. a l : y = gaat door (0, ) en (1, 1); m: y = gaat door (0, ) en (, 0). (de grafieken van de lijnen in de figuur hiernaast) b Het snijpunt is (, 1). c y = 1 is zowel oplossing van y = als van y =. y -as l m -as 6a y = 8 y = 1 = 0 y = 1 y = 1 y = 17 y = 17 = 1. 6b y = 7 y = 1 y = 8 y = = 7 y = 7 = c y = 8 y = 1 = 9 6 y = 1 y = 1 y = y = = 1. 7a y = 7 y = 16 y = Nee, er is geen variabele geëlimineerd. 7b y = 7 y = 16 7y = 9 Nee, er is geen variabele geëlimineerd. 8a y = 7 1 y y = 7 10 y 7 = 7 1 y y y =. 8b y = 6 1 y = 19 y = 6 1 y = = y = 19 y = 19 = y. 8c y = 1 y = y = 6 y = 1 9 = 7 1 y = 1 y = 1 y = 1. 9a y = 69 1 y = 7 y = 69 1y = 1y = 10 y = 8 = 7 y = b y = 19 y = 8 0y = 76 0y = 17 = 99 1 y = y = y y =. 9c 0, 8 0,y = 1 0 0, 0,y = 1, 0 6y 6 6y 0 = y 6 6y 6y = 18 y =. 0 y = 1 y = y = y = 8 7 = 1 y = 8 y = 8 y = 6 y = 9. 1ab 1 y = invullen: = 1 b 1 c = 1 b c = b c. y = invullen: = b c = b c 1 = b c. b c = 1c b c = 1 b = b = c = b c = c =.

11 C. von Schwartzenberg 11/ (1, 8) op parabool 8 = a 1 c; (, 17) op parabool 17 = a c. a c = 8 a c = 17 a = 9 a = c = 8 a c = 8 c =. (, 8) op k a b = 8; (, 8) op l b a = 8. a b = 8 a b = 8 1 a b = 16 a b = 8 a = 8 a = 8 16 b 8 b 16 8 = = = =. a b = 8 a (, 1) op parabool 1 = p q; (, 1) op lijn 1 = p q. p q = p q = 1 6p = 6 p = 1 q = p q q. b De parabool y = snijden met de lijn y =. = 6 ( ) ( ) = = (was gegeven) y 9 = = y = = 1. (, 10) op parabool 10 = a b c ❶; (0, ) op parabool = c ❷; (, ) op parabool = a 9 b c ❸. ❷ invullen in ❶ en ❸ geeft: 10 = a b a b = 1 a b = 7 en = 9a b 9a b = 1. a b = 7 9a b = 1 1 6a b = 1 9a b = 1 1a = 0 a = 8 b = 7 a b = 7 b = 1 8 = 1 b = 1. 6 y = 1 y = 10 y = 1 1 y = 10 y = 1 1 y 1 = y = 10 =. y = 10 7a y = 9 ❶ y = ❷ ❷ in ❶ geeft: ( ) = = 9 10 = 1 = 1, in ❷ y = 1, =. 1 y = 1 ❶ 7b 6y = 8 ❷ ❶ in ❷ geeft: 6 ( 1 1) = 8 6 = 8 6 = 1 = in ❶ y = = c y ❶ y = 9 ❷ ❶ in ❷ geeft: (y ) y = 9 1y 9 y = 9 19y = 8 y = in ❶ = = 7. 8a ❶ ❷ y = y = ❶ in ❷ geeft: ( ) = = 0 = 6 0 = ( ) ( ) = in ❶ = in ❶ y = 9 = 6 y = = 1. 8b ❶ ❷ y = y = y = y = ❷ in ❶ geeft: ( ) = = ( ) in ❷ = in ❷ y = y = 9 =.

12 C. von Schwartzenberg 1/ 8c 9 y y y 8 y 8 = = ❶ ❷ in geeft: ( 8) 6 0 (vermenigvuldigen met ) 0 6 (stel t ) t 0t 6 ( t 16) ( t ) t = 16 t = ❷ ❶ ❷ ❷ ❷ ❷ in in in in y 8 y 8 y 8 y 8 = = = = = = = =. kun je algebraïsch oplossen door t te stellen. Je krijgt ( t ) ( t 1) kun je niet algebraïsch oplossen. ( ) kun je niet algebraïsch oplossen. ( ) ( ) ( 1) kun je algebraïsch oplossen. 60a 1, 1, en. 60b 1 geeft y = ( 1) ( 1) ( 1) 1 6 (klopt), 1 geeft y = (klopt), geeft y = 6 (klopt) en geeft y = 6 (klopt). 61a, en. 61b. 6a Neem GR - practicum door. (uitwerkingen aan het eind) (intersect met ZStandard) 0, 79 1, 79. 6b 1 (intersect met ZStandard) 0, 8,. 6c 0, = (intersect), 1 1, 76 1,6. 6d 0,, (intersect) 1, 1 1,. 6a 0, 0 (intersect) 1, 8,8 9, 6. 6b 0,1 0,1 1 0 = (intersect) 10, 1,6 11,7.

13 C. von Schwartzenberg 1/ 6a 6b 9 (intersect),,67 0, 8 0, 8, 67,. 9 (intersect),10,87 0, 1 0, 66,,9. 6a = (intersect) 1, 8 1,6 1. 6b (intersect), 1 0, 76,. 6c (intersect) 0,8, 1,8. 6d 10 (intersect lukt niet) 10 (intersect) 1, 6 1, 8,1. 66a 6 (intersect) 1. 66b Uit de plot volgt nu: 1 < <. (voor tussen 1 en ) 67a 67b 67c 1 (intersect), 1, 1. 1 (zie plot), 1, (intersect), 6, 6. > 11 (zie plot) <, 6 >, (intersect), 1, (zie plot), 1, 7. 67d, (intersect) 6,., < (zie plot) < 6, < <.

14 C. von Schwartzenberg 1/ 68a = 1 (niet met intersect!!!) 1 ( 7) ( ) 7 < 1 (zie plot) < < 7. 68b = (niet intersect!!!) ( a =, b = en c = ) D = ( ) = 9 16 = b ± D ± = = ± a = 8 = = = 1. (zie plot) 1. 68c 6 (niet intersect) 6 ( a =, b = 1 en c = 6) D = 1 6 = 1 8 = 9 b ± D 1 ± 9 = = 1 ± 7 a 1 7 = 6 = = 8 =. 6 (zie plot) d (niet intersect) ( ) ( ) ( 1) 1. > (zie plot) < < 0 > 1. 69a 69b 69c 0, = 1 (intersect) 0, 6, 66 1, 69. 0, (zie plot) 0, 6, 66 1, 69. 0, 8 = 7 (intersect) 0, 0,,9,78. 0, 8 7 (zie plot) 0, 0,,9, (intersect),, 06 0, 69 1,, (zie plot),, 06 0, 69 1,, 76., 69d 10 = 8 (intersect) 1, 1,10 1, 69, (zie plot) 1, 1,10 1, 69,1. 70a p p ( a = 1, b = p en c = p) D = p 1 p = p p. 70b Twee oplossingen D = p p > 0. 71a 71b 71c p p ( a = 1, b = p en c = p) D = p 1 p = p 1 p. D p 1 p = p ( p 1) (of intersect) p p = 1. Twee oplossingen D > 0 (zie plot) p < 0 p > 1. p ( p ) 0, ( a = p 0, b = p en c,) D = ( p ) p 0,. D p 10p 16 = ( p 8) ( p ) (of intersect) p = 8 p =. Twee oplossingen D > 0 (zie plot) p < 0 0 < p < p > 8. ( p geeft 1 oplossing) p ( p ) ( a = p 0, b = p en c = ) D = ( p ) p. D p 10p 9 = ( p 9) ( p 1) (of intersect) p = 9 p = 1. Geen oplossingen D < 0 (zie plot) 9 < p < 1. ( p geeft 1 oplossing)

15 C. von Schwartzenberg 1/ 7a 7b 7c 1 ( p ) 1 1 ( a = 1, b = p en c = 1 ). 1 D = ( p ) 1 1 = ( p ) 9. D ( p ) = 9 p = ± 7 p = ± 7 p = 9 p = (kan niet) (of met intersect) p = ±. Twee oplossingen D > 0 (zie plot) p < p >. p p 16 ( p p 16) p p 16. Drie oplossingen als p p 16 ( a = p 0, b = p en c = 16) twee oplossingen heeft. D = ( p ) p 16 = p 6 p. D p 6 p = p ( p 6) p p = 6 (of met intersect) p p = 6 =. Drie oplossingen D > 0 (zie plot) p < p > 0. ( p geeft 16 1 oplossing) p p 1 p (p ) 1 ( p (p ) 1 ) p (p ) Eén oplossing als p (p ) ( a = p 0, b = p en c = ) geen oplossing heeft. 1 D = (p ) p = (p ) p. D (p ) p (eact of intersect) p = 1 p = 1. Eén oplossing D < 0 (zie plot) 1 < p < ( p geeft ( ) oplossingen)

16 C. von Schwartzenberg 16/ D1a D1d D1g D1i Da Da Db Diagnostische toets ( 1) 1 = = 16 ±. ( ) D1b 9 = ( ) ( 1) 1. D1e ( a = 1, b = 7 en c = 1) D = ( 7) 1 1 = 9 = 101 b ± D 7 ± ± 101 = = a ( ) ( ) = 7 = 7 ( a = 1, b = 1 en c = ) D = 1 1 = 1 1 = 1 b ± D 1 ± 1 1 ± 1 = = a p Db ( a =, b = en c = p) D = p = 16 8p < 0 8p < 16 p > 16 =. 8 ( ) = 81 = ± 81 = ± 9 = ± 9 1, ±, 6. D1j D1h D1c p 7 Dc ( a =, b = p en c = 7) D = p 7 = p > 0 p > p < 18 p > 18. p 8 p p = 1; 1 ( 6) ( ) 6 (was bekend). p ( a = p 0, b = en c = ); twee oplossingen D = p = 0p > 0 0p > p < = 1. 0 ( p geeft een eerstegraadsvergelijking met één oplossing) ( a =, b = 1 en c = ) D = ( 1) = 1 = b ± D 1 ± = = 1 ± a 6 1 = 6 = 1 1 = = D1f ( ) ( 1) 1 = 1 1. ( ) ( 1) = ( ) = 6 1 ( a = 1, b = en c = 1) D = ( ) 1 1 = 9 = 1 b ± D ± 1 ± 1 = = a ( ) ( ) ( 1) ( 1) = ( ) ( ) 9 ( 1) = = = 8 8 = (kan niet) geen oplossingen. p 6 1 ( a = p 0, b = 6 en c = 1) D = ( 6) p 1 = 6 8p 6 = 8p 6 = = p; 8 b ± D 6 ± 0 = = 6 = 1 = ; a 1, voor p 6 1. Da Dd = 86 = =. 1 ( ) = 1 ( ) = 1 16 = ± 1 = ± 1 16 ± Db De 6 = 9 = 1 ±. 100 ( 1) = 68 = ( 1) 1 = = = 1 1. Dc Df 19 = = ( ) = 10 = 10 =

17 C. von Schwartzenberg 17/ Da Dc De 6 (stel t ) t 6t Db 6 1 (stel t ) t 6t 1 ( a =, b = 6 en c = 1) ( t ) ( t 1) D = ( 6) 1 = 6 0 = 16 t = t = 1 b ± D 6 ± t = = = ± = 1 6 = = 1 a ± ± 1 = ± 1. ± 1 = ± 1 ± 1. 6 Dd 6 ( 6 ) ( 6 ) ( ) ( 1) 6 ( a = 1, b = 6 en c = ) (dubbel) 1. b ± D 6 ± 8 6 ± 8 D = 6 1 = 6 8 = 8 = = a = 10 8 Df 6 10 (stel t ) 8 (stel t ) t 10t ( a =, b = 10 en c = ) t t D = ( 10) = = 6 ( t 7) ( t 6) b ± D 10 ± t = = = ± = 10 8 = = 1 7 (k.n.) 6 a ± D6a = 1 D6b = 17 = 1 = 1 17 (kan niet) ± = ±. = 17 = 17 = = 1,,. D7a D7c 1 = D7b = (kwadrateren) = 16 = = (voldoet). 6 6 = (kwadrateren) = 1 6 ( 9) ( ) 9 (voldoet) (voldoet niet). = (kwadrateren) 9 = ( ) ( a = 9, b = en c = 100) D = ( ) = b ± D ± = = ± 6 a = (voldoet) 6 = 0 = (voldoet niet) D7d (kwadrateren) 9 0 = 17 ( a =, b = 17 en c = ) D = ( 17) = b ± D 17 ± = = 17 ± 1 a = = (voldoet niet) 17 1 = = 1 (voldoet) D8a ( t ) t 0t 189 ( t 7) ( t 7) t = 7 t = 7 (k.n.) (kwadrateren) 7 = = 9 (voldoet). D8b 1 = (stel t ) t 8t 1 ( t 6) ( t ) t = 6 (kwadrateren) 6 = 6 = 6 (voldoet) (voldoet). D9a 6 18 (teller en noemer 0) (voldoet, want de noemer 0). 1

18 C. von Schwartzenberg 18/ D9b 6 D9c = = ( ) ( ) = 7 (vold.) (vold.)., (vold.). D9e 1 = D9f 1 ( 1) ( ) = ( ) ( 1) 8 = 1 1 ( a = 1, b = 1 en c = 1) D = ( 1) 1 1 = 169 = 16 b ± D 1 ± 16 1 ± 16 = = a (vold.) (vold.). D9d = 1 1 = (vold.) (vold.) (vold.). = 1 = 7 ( ) = 7 ( ) 8 16 = ( a = 8, b = 7 en c = 1) D = ( 7) 8 1 = 1681 b ± D 7 ± 1681 = = 7 ± 1 a = 8 = (vold.) 7 1 = = 1 (vold.) y = 7 1 y = 7 D10a D10b D11 (, 18) op parabool 18 = a b ; y = y = 8 (, 0) op parabool 0 = a 16 b. y = 7 6y = 1 a b = 18 6y 1 6y = 8a b 11y = = 8 1a = 18 y = 7 1 = 7 y = 7 a = 1 y = 7 y 7 = 8 = 6 b 18 y = = a b = 18 b = 1. y = 1. b = 6. D1a y = ❶ y = ❷ ❷ in ❶ geeft: ( ) = 1 = = 9 in ❷ y = = 6. D1b y = 10 ❶ y = 6 ❷ ❷ in ❶ geeft: ( 6) = = ( a =, b = 10 en c = 8) D = ( 10) 8 = b ± D 10 ± = = 10 ± a 6 10 = 1 = 10 = 8 = in = ❷ 1 in ❷ y 8 6 y = = = =. 9 9 D1a, (intersect) 1,7 0,86 0,69 1,9. D1b 1 (intersect),11 0, 6 1, 6 1, 89. D1a 6 1 (intersect), 89 0, 09, (zie plot), 89 0, 09, 80.

19 C. von Schwartzenberg 19/ D1b 1 (intersect) 0, 1 0,7,78,91. > 1 (zie plot) < 0, 1 0,7 < <,78 >,91. D1a = (niet met intersect!!!) D1b ( a =, b = en c = ) D = 0 b ± D ± 00 = = ± 0 a 6 0 = 18 = 0 = = 11 = (zie plot). 6 (niet met intersect!!!) ( 6) ( ) ( ). 6 < 0 (zie plot) < 0 < <. D16a p p p = p ( p ) p ( a = p 0, b = p en c = p). D = ( p ) p p = ( p ) 16 p. D ( p ) 16p (intersect) p = p,. Geen oplossing D < 0 (zie plot) p < p > 0,. ( p geeft 1 oplossing) D16b p p ( p p ) p p. Eén oplossing als p p ( a = p 0, b = p en c = ) geen oplossing heeft. D = ( p) p = p 8 p. D p 8p p ( p ) (of intersect) p p =. D < 0 (zie plot) < p < 0. ( p geeft ook 1 oplossing) Dus één oplossing als < p 0.

20 C. von Schwartzenberg 0/ G1a G1d Ga Gb Gc Gemengde opgaven 1. Vergelijkingen en ongelijkheden 7 = G1b G1c ( ) ( 6) = 9 7 ( a =, b = 1 en c = ) 6 1 = 9 (7 ) D = 1 = 1 = 1 7 = b ± D 1 ± 1 = = ± ( 7) ( ). a = = 1 1 = 6 = 1 1. ( ) ( 1) = 1 G1e ( ) = 6 G1f ( ) = 7 9 ( 1) = 1 = ± 6 ( ) = = = ± 6 1 = = = ( a = 1, b = en c = ) 1 1 ( 9) ( 1) 1. D = 1 = 9 1 = b ± D ± 1 ± 1 = = a p 6 p. ( a = p 0, b = 6 en c = p) D = 6 p p = 6 1 p. D 6 1p 6 = 1p p = p = ±. D > 0 (zie plot) < p <. ( p 6 1 oplossing) 6 p 6 6p 6p 6p 6 p p 6 ( p ) ( p ) p = p =. p = geeft: ( 6) ( 9) 6 (bekend) 9. p p ( a = p 0, b = p en c = ) D = ( p) p = p 16 p. D p 16p p ( p ) p p =. ( p (kan niet) geen oplossing) p = 8 1 ( 1) ( 1) 1 (dubbel). p = geeft: ( 6) ( ) 6 (bekend). Ga Gc Ge 6 6 (stel t ) Gb 7 = 18 t 6t = 18 (stel t ) ( t ) ( t 1) t 7t 18 t 7t 18 ( a = 1, b = 7 en c = 18) t = 1 ( t 9) ( t ) D = ( 7) 1 18 < 0 (geen oplossingen) 1 = 1. 9 (k.n.). 10 = Gd 10 ( 1) = (stel t ) = ( 1) 10t 17t 67 ( a = 10, b = 17 en c = 67) 1 = ± D = ( 17) = 669 = 1 ± b ± D 17 ± 669 t = = = 17 ± a = = = 17 =... (k.n.) Gf 108 (stel t ) ( 16 8) (stel t ) t t 108 t 16t 8 ( t 1) ( t 9) ( t 1) ( t ) 1 (kwadrateren) 9 (k.n.) 1 1 ± 1 ±. 1 (voldoet).

21 C. von Schwartzenberg 1/ Gg (stel t ) 6t 10t 6 t t ( a =, b = en c = ) D = = 809 b ± D ± 809 t = = = ± a 6 = 8 (kwadr.) =... (k.n.) (voldoet). Gh ( 1) ( 1) (stel ( 1) = t ) t t ( t ) ( t 1) ( 1) = ( 1) = 1 1 = ± 1 = ± 1 = 1 ± = 1 ± Ga = 1 6 = = 6 ( 1) = 6 6 = 6 6 = 1 1 (voldoet). Gb Gd 1 = ( 1) ( ) = ( ) ( ) = ( a = 1, b = 11 en c = 6) D = ( 11) 1 6 = 1 Gg b ± D 11 ± 1 11 ± 1 = = a (voldoet) (voldoet). = (kwadrateren) ( ) 0 = 1 1 ( a =, b = 1 en c = 1) D = 1 1 = 1 8 = 9 = 1 = 7 (kwadrateren) = 9 = 7 7 = 1 (voldoet). Ge b ± D 1 ± 9 = = 1 ± a 1 = = 1 (vold. niet) 1 = = 1 (vold.). 1 8 = 1 1 = 7 1 = 7 (kwadrateren) 1 = 9 = (voldoet). Gc = 1 ( ) ( ) = ( 1) 10 = 8 = = 1 1 (voldoet). 8 Gf 1 1 ( ) ( 1) = ( ) ( 1) (vold.) 1 (vold.). Gh = 8 1 (kwadrateren) ( ) = ( a = 9, b = 8 en c = 1) D = ( 8) 9 1 = 6 6 = 100 b ± D 8 ± 100 = = 8 ± 10 a = 1 (vold.) 8 10 = = 1 (vold. niet) y = 1 Ga y = y = 1y = 96 1y = 91 y = 7 = y =. Gb y = 1 y = 1 6y = 10 6y = 90 = 1 6 y = 1 y = 1 y = 10 y =. G6 (, ) op grafiek = a 16 b 6; (, 1) op parabool 1 = a b 6. 16a b = 68 1 a b = 16a b = 68 8a b = a = 7 a = 1 b = a b = b = 10 b =.

22 C. von Schwartzenberg / a b = 10 7 G7a 8,6a 7,0b = a 7 b = 100 8,6a 7,0b = 118 1, 6a = 1 a = 8, 7 8,7 b = 10 a b = 10 b = 6, 6. G7b Stel hij neemt ml van 1% en y ml van 0%. y = y = y = y = 100 1y = 00 y = = 600 y = 600. G8a 1 (intersect) 0,17, 8. > 1 (zie plot) < 0,17 >, 8. G8b 0, = (intersect) 1, 0, 6, 67. 0, (zie plot) 1, 0, 6, 67. G8c 8 (intersect), 1, 71. < 8 (zie plot), 1 < <, 71. G8d = 6 (intersect), 78 0, 1 0,9, 1, 9. > 6 (zie plot),78 < < 0, 1 < < 0,9, 1 < <, 9. G9a G9b p p 1 p (p 1) 1 ( p (p 1) 1 ) p (p 1) Drie oplossingen als p (p 1) ( a = p 0, b = p 1 en c = ) twee oplossingen heeft. 1 D = (p 1) p = (p 1) 9 p. D (p 1) 9p (eact of intersect) p = 1 p = 1. Drie oplossingen D > 0 (zie plot) p < 0 0 < p < 1 p > ( p geeft ( ) oplossingen) p p p ( p) ( p ( p) ) p ( p). Eén oplossing als p ( p) ( a = p 0, b = p en c = ) geen oplossing heeft. D = ( p) p = ( p) 16 p. D ( p) 16p (eact of intersect) p = 1 p =. Eén oplossing D < 0 (zie plot) 1 < p <. ( p geeft ( ) oplossingen)

23 C. von Schwartzenberg / 1a 1b TI-8 1. Berekeningen op het basisscherm, 6 1, 7 86,19. 1c 6, 80, 6. 1d 1,8 : 0,01. 11, 8,7 1,0. a 1,1 6,97. c 1,8 :,1 1,. b 1,1,9. d 1,8 :,1, 9. a b, 8 = 11, 7. c 8,1 :1,6 68,. 8, 91,1 1,, 8. d 8,1 1,, 7 : 8 19, 07. a b (, 7) =, 9. c ( 1, 8) = 10, 976. d,7 =, 9. 1, 8 = 10, 976. a , 6. c ,0. 18 b 100 0,. d , 8. 1, ,6 6a 8,,6 6b 1,1 8, 7,0 1,87. 6c 1,80. 1, 7,,9,. 6d 8,1 1,88,6 7, 86,91. 1,6,9 7a 1 = 11. 7c b (1 ) = 11. 7d = a : =. 8c ( 1 1 ) = b ( 1 1 ) : 1 = 9. 8d 1 : = a (1 ) =. 9b 9 ( ) = 81. 9c :1 1 = ab 1c 1d TI-8. Formules, grafieken en tabellen Zie de eerste drie schermen hiernaast. Kies WINDOW: [, 1] [ 10, 1]. (gebruik deze notatie) (schaalstreepjes uit met Xscl en Yscl ) Zie de schermen hieronder. Neem (bijvoorbeeld) WINDOW: [ 1, 1] [ 10, 1]. ab Zie de schermen hiernaast (neem Zoom 6 = ZStandard). c Met ZStandard is er één snijpunt te zien. abcd Zie de schermen hieronder; het linker snijpunt: ( 1,70; 0,76) en de top van y: (,; 6, 99).

24 C. von Schwartzenberg / a b Zie de schermen hiernaast. f ( ), ; f ( 1, ) =,88; f (0, 8) = 0, 78; f (8,) = 101, 97. (de optie value in è ( `$ ) werkt als $ ). (zie de schermen hieronder) cde f (1) = 16,; f ( 17) =,1; f (1) = 70,1; f (10) = 189. a y 1( ) = 1, ; y 1(0, ) =, 0068; y 1(1, 8) = 8,1888; y 1() = 79. b y ( ) = 11, 7; y (0, ) =, 8; y (1, 8) =,18; y () = 1,. cd y 1(1) = 97, ; y 1( 18) 69, ; y (1) = 118, ; y ( ) =, 8. 6ab Zie hieronder: y 1(,1) = 1,887. 6c y 1(1, 7) =, 98 en y (1, 68) = 96, 8. (zie hieronder) 7 Haal de antwoorden uit de tabel hiernaast f ( ),8,8 7, - -,8-1, 8,8, 8 9 y 1, en y = 0,. (zie hiernaast) optie intersect S1( 6, 6;, 9) en S(1, 86;, ). 0, = (intersect in ZStandard) 0, 8, 8. (zie de schermen hieronder) 10a 10b 0, =, 6 (intersect in ZStandard) 1,1 6,1. 0, 1 = (intersect op [ 10, 10] [ 10, 0]) 6.

25 C. von Schwartzenberg / 10c 10d 0, 0 0, = (intersect op [ 10, 0] [ 10, 10]) 8, 18,. 0, 10 = (intersect op [ 8, ] [ 100, 10])),9,1,90. (zie de schermen hieronder) 11ab 0, 6 = 0, 1 (intersect in ZStandard),7 7, 0. (zie de schermen hieronder) 11c 0, 6 = 0, 1 (intersect in ZStandard) 0, 9,1. (zie de schermen hierboven) (het is niet nodig om grafieken uit te zetten; kies met : of ; bij First curve? en/of Second curve? de juiste formules) 1abc 0,9 (intersect in ZStandard), 1,0. (zie de schermen hieronder) 1d 1, (intersect in ZStandard) 0, 68,. (zie de schermen hieronder) 1a 1 (intersect in ZStandard) 0,19,19. (zie de schermen hieronder) 1b 0,8 (intersect in ZStandard),66 1, 1. (zie de schermen hieronder) 1c 0, 1, 1, 6 (intersect in ZStandard) 1, 66 0, 8, 8. (zie de schermen hieronder) 1ab (intersect op [, 1] [ 00, 1000]), 7. (zie de schermen hieronder) 1c 1a (intersect) 1, 88. (zie het scherm hiernaast) WINDOW: [ 1, ] [0, 0] ; vanaf 1, 9 (intersect en plot). 1b 0, 78 1,11 (intersect).