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0 D W H P i W L F D A u t o r i a : C a r l o s H e n r i q u e D i a s Te m a 0 4 F u n ç ã o Q u a d r á t i c a e A p l i c a ç õ e s 7 H P D � � � F u n ç ã o Q u a d r á t i c a e A p l i c a ç õ e s A u t o r i a : C a r l o s H e n r i q u e D i a s C o m o c i t a r e s s e d o c u m e n t o : D I A S , C a r l o s H e n r i q u e . M a t e m á t i c a : F u n ç ã o Q u a d r á t i c a e A p l i c a ç õ e s . C a d e r n o d e A t i v i d a d e s . A n h n a g u e r a P u b l i c a ç õ e s : V a l i n h o s , 2 0 1 4 . Í n d i c e � � � � � � $ Q K D Q J X H U D � ( G X F D F L R Q D O � � 3 U R L E L G D � D � U H S U R G X o m R � ¿ Q D O � R X � S D U F L D O � S R U � T X D O T X H U � P H L R � G H � L P S U H V V m R � � H P � I R U P D � L G r Q W L F D � � U H V X P L G D � R X � P R G L ¿ F D G D � H P � O t Q J X D � S R U W X J X H V D � R X � T X D O T X H U � R X W U R � L G L R P D � P á g . 1 6 P á g . 1 7 P á g . 1 8 P á g . 1 7 P á g . 1 2P á g . 1 1 A C O M PA N H E N A W E B P á g . 3 C O N V I T E À L E I T U R A P á g . 4 P O R D E N T R O D O T E M A � C o n t e ú d o N e s t a a u l a , v o c ê e s t u d a r á : • A p l i c a ç õ e s d a s f u n ç õ e s q u a d r á t i c a s e m m o d e l o s q u e e n v o l v e m c u s t o , r e c e i t a e l u c r o . • O p r o c e s s o p a r a e n c o n t r a r o b r e a k - e v e n p o i n t n o m o d e l o q u e e n v o l v e f u n ç ã o q u a d r á t i c a . • O p o n t o d e m á x i m o d a f u n ç ã o q u a d r á t i c a r e c e i t a . • O p o n t o d e m á x i m o d a f u n ç ã o q u a d r á t i c a l u c r o . • R e s o l u ç ã o d e p r o b l e m a s a p l i c a d o s . H a b i l i d a d e s $ R � ¿ Q D O � � Y R F r � G H Y H U i � V H U � F D S D ] � G H � U H V S R Q G H U � D V � V H J X L Q W H V � T X H V W } H V � • e � S R V V t Y H O � F R Q V W U X L U � P R G H O R V � P D W H P i W L F R V � D S O L F D G R V � j � J H V W m R � H P S U H V D U L D O � R X � j � F R Q W D E L O L G D G H � X W L O L ] D Q G R � D � I X Q o m R � q u a d r á t i c a ? • C o m o e n c o n t r a r o b r e a k - e v e n p o i n t e m u m m o d e l o q u e e n v o l v e f u n ç ã o q u a d r á t i c a ? • C o m o a f u n ç ã o r e c e i t a t o r n a - s e u m a f u n ç ã o q u a d r á t i c a ? • 5 H F H L W D � P i [ L P D � V L J Q L ¿ F D � O X F U R � P i [ L P R " C ú C O N V I T E À L E I T U R A � F u n ç ã o Q u a d r á t i c a e A p l i c a ç õ e s I n t r o d u ç ã o 1 R � 7 H P D � � � � Y R F r � H V W X G R X � D � F D U D F W H U L ] D o m R � G D V � I X Q o } H V � S R O L Q R P L D L V � G R � � o � J U D X � � 1 H V W H � W H P D � � Y R F r � H V W X G D U i � D � D S O L F D o m R � G H V V D V � I X Q o } H V � H P � S U R E O H P D V � T X H � H Q Y R O Y H P � D � i U H D � G H � J H V W m R � H P S U H V D U L D O � H � F R Q W D E L O L G D G H � � S R U � H [ H P S O R � � R � H V W X G R � G D V � I X Q o } H V � U H F H L W D � � F X V W R � H � O X F U R � � 3 D U D � I D F L O L W D U � � D � I X Q o m R � S R O L Q R P L D O � G R � � o � J U D X � S R G H � V H U � F K D P D G D � G H � I X Q o m R � T X D G U i W L F D � � W H U P R � T X H � V H U i � X W L O L ] D G R � P X L W D V � Y H ] H V � E x e m p l o P r á t i c o E m u m a l o j a , o p r e ç o d e u m c a l ç a d o p o d e v a r i a r d e a c o r d o c o m a d e m a n d a � � ( P � J H U D O � � D � T X D Q W L G D G H � G H P D Q G D G D � d e u m b e m a u m e n t a à m e d i d a q u e o p r e ç o p o r u n i d a d e d i m i n u i . A s s i m , o p r e ç o d o c a l ç a d o p o d e s e r r e l a c i o n a d o p o r u m a e q u a ç ã o , d e f o r m a a p e r m i t i r q u e o v e n d e d o r d e t e r m i n e u m p r e ç o p a r a u m a d e m a n d a . P o r e x e m p l o , o v e n d e d o r p e r c e b e T X H � R � S U H o R � G R � F D O o D G R � S � S R G H � V H U � U H O D F L R Q D G R � S H O D � T X D Q W L G D G H � G H P D Q G D G D � [ � G R � V H J X L Q W H � P R G R � p = - 3 x + 3 0 0 ( Q W m R � � S D U D � Y H Q G H U � � S R U � H [ H P S O R � � � � � F D O o D G R V � � [ � � � � � � � R � S U H o R � S R U � F D O o D G R � V H U i � p = - 3 � � � � � � � � � � � � � � � � � � � � � U H D L V � ( Q W U H W D Q W R � � V H � H O H � G H V H M D � D X P H Q W D U � V X D V � Y H Q G D V � H � F R P H U F L D O L ] D � � � � F D O o D G R V � � [ � � � � � � � R � S U H o R � V H U i � � p = - 3 � � � � � � � � � � � � � � � � � � � � � � U H D L V 2 E V H U Y H � T X H � � S D U D � Y H Q G H U � � � � F D O o D G R V � � R � S U H o R � G H Y H � V H U � G H � 5 � � � � � � � � � � S D U D � Y H Q G H U � � � � � R � S U H o R � G H Y H � V H U � G H � 5 � � � � � � � � � � O b v i a m e n t e , m e n o r o p r e ç o , m a i o r o n ú m e r o d e c a l ç a d o s v e n d i d o s . P a r a c a l c u l a r a r e c e i t a r e l a t i v a à v e n d a d o s c a l ç a d o s , o v e n d e d o r m u l t i p l i c a a q u a n t i d a d e v e n d i d a p e l o p r e ç o d e c a d a F D O o D G R � � ' H V W H � P R G R � � D � I y U P X O D � T X H � I R U Q H F H � D � U H F H L W D � U H O D W L Y D � j � Y H Q G D � G H � F D O o D G R V � p � R � S U H o R � S � Y H ] H V � D � T X D Q W L G D G H � [ � G H � P O R D E N T R O D O T E M A � c a l ç a d o s v e n d i d o s , o u s e j a , R = p x . P o r é m , c o m o o p r e ç o j á é c a l c u l a d o p e l a r e l a ç ã o p = - 3 x + 3 0 0 , s u b s t i t u i n d o p p o r � � � [ � � � � � � � W H P � V H � R = p [ � � � � � [ � � � � � x R = - 3 x � + 3 0 0 x 3 H U F H E D � T X H � � V H � R � Y H Q G H G R U � G H V H M D � Y H Q G H U � � � � F D O o D G R V � � R � S U H o R � � F R P R � Y H U L ¿ F D G R � D Q W H U L R U P H Q W H � � V H U i � G H � 5 � � � � � � � � � � H � a r e c e i t a r e l a t i v a d e s t a v e n d a s e r á : R = p [ � � � � � � � � � � � � � � U H D L V � � A r e c e i t a t a m b é m p o d e s e r c a l c u l a d a : R = p x = - 3 � � � + 3 0 0 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � U H D L V � A f u n ç ã o R = - 3 x � � � � � [ � T X H � G H W H U P L Q D � D � U H F H L W D � S D U D � [ � V D S D W R V � Y H Q G L G R V � p � X P D � I X Q o m R � T X D G U i W L F D � � 2 � J U i ¿ F R � G D � S D U i E R O D � D V V R F L D G D � D � H V V D � I X Q o m R � p � U H S U H V H Q W D G R � V H J X L Q G R � D V � H W D S D V � D � V H J X L U � � o � � ' H W H U P L Q D U � F R H ¿ F L H Q W H V � � a f u n ç ã o é R = - 3 x � + 3 0 0 x , e n t ã o , a = - 3 , b = 3 0 0 e c = 0 . � o � � C o n c a v i d a d e � G D � S D U i E R O D � � Q H V W H � F D V R � � F R P R � D � � � � � � D � � � � � D c o n c a v i d a d e d a p a r á b o l a é v o l t a d a p a r a b a i x o . 3 o � � , Q W H U F H S W R � F R P � R � H L [ R � \ � a p a r á b o l a c o r t a o e i x o y e m 0 , p o i s c = 0 . � o � � , Q W H U F H S W R � F R P � R � H L [ R � [ � d e v e - s e r e s o l v e r a e q u a ç ã o - 3 x � � � � � [ � � � 3 H O D � I y U P X O D � G H � % K D V N D U D � P O R D E N T R O D O T E M A � P o r t a n t o , a p a r á b o l a c o r t a o e i x o d a s a b s c i s s a s e m x � = 0 e x � � � � � � � 5 o � � 9 p U W L F H � G D � S D U i E R O D � a c o o r d e n a d a d o v é r t i c e d a p a r á b o l a é d e t e r m in a d a a p a r t i r d e d o i s v a l o r e s , x v e y v : $ � F R R U G H Q D G D � G R � Y p U W L F H � G D � S D U i E R O D � p � � � � � � � � � � � � o � � & R Q V W U X L U � R � J U i ¿ F R � � c o l o c a m - s e o s p o n t o s n o p l a n o c a r t e s i a n o a p a r t i r d a s i n f o r m a ç õ e s o b t i d a s n o s p a s s o s a n t e r i o r e s . ( P � V H J X L G D � � W U D o D � V H � D � F X U Y D � T X H � S D V V D � S H O R V � S R Q W R V � F i g u r a 4 . 1 � * U i ¿ F R � G D � I X Q o m R � 5 � � � [ � + 3 0 0 x . e � S R V V t Y H O � S H U F H E H U � S H O R � J U i ¿ F R � D S U H V H Q W D G R � Q D � ) L J X U D � � � � � T X H � D � U H F H L W D � P i [ L P D � R F R U U H � T X D Q G R � D � Y H Q G D � G H � F D O o D G R V � p � L J X D O � D � � � � � [ � � � � H � R � Y D O R U � G D � U H F H L W D � P i [ L P D � F R U U H V S R Q G H Q W H � p � G H � 5 � � � � � � � � � � � � \ � � � � � � � � � 2 � Y p U W L F H � G D � S D U i E R O D � I R U Q H F H � P O R D E N T R O D O T E M A � D � O R F D O L ] D o m R � G D � P i [ L P D � U H F H L W D � � [ v � � H � D � U H F H L W D � P i [ L P D � � \ v � � � ( Y L G H Q W H P H Q W H � � V H � D � S D U i E R O D � H V W L Y H U � F R P � F R Q F D Y L G D G H � Y R O W D G D � S D U D � F L P D � � H V V D � F R R U G H Q D G D � U H S U H V H Q W D U i � R � P t Q L P R � G D � I X Q o m R � � ( P � S U R E O H P D V � S U i W L F R V � S D U D � D � O R F D O L ] D o m R � G R � m á x i m o o u m í n i m o d e u m a f u n ç ã o q u a d r á t i c a , b a s t a d e t e r m i n a r o v é r t i c e d a p a r á b o l a . $ Q D O L V D Q G R � D J R U D � S R U � R X W U R � S R Q W R � G H � Y L V W D � H � D L Q G D � F R P � U H O D o m R � D R � P H V P R � F D O o D G R � � R � Y H Q G H G R U � S H U F H E H � T X H � R � F X V W R � G H � I D E U L F D o m R � p � G D G R � S R U � & � � � [ � � � � � � � $ V V L P � � S R U � H [ H P S O R � � S D U D � D � I D E U L F D o m R � G H � � � � F D O o D G R V � � R � F X V W R � V H U i � & � � � � � � � � � � � � � � � � � � � � � � � � � � � � U H D L V � $ � I X Q o m R � F X V W R � & � � � [ � � � � � � p � S R O L Q R P L D O � G R � S U L P H L U R � J U D X � � H � D � P R Q W D J H P � G R � J U i ¿ F R � G H V V H � W L S R � G H � I X Q o m R � I R L � H V W X G D G D � Q R � 7 H P D � � � � 2 � J U i ¿ F R � G D � I X Q o m R � F X V W R � H V W i � U H S U H V H Q W D G R � Q D � ) L J X U D � � � � � F i g u r a 4 . 2 � * U i ¿ F R � G D � I X Q o m R � & � � � [ � � � � � � 1 D � ) L J X U D � � � � � � p � S R V V t Y H O � F R Q ¿ U P D U � R � H [ H P S O R � D Q W H U L R U � � S D U D � � � � F D O o D G R V � S U R G X ] L G R V � � R � F X V W R � V H U i � G H � 5 � � � � � � � � � � � 2 � Y H Q G H G R U � S R G H � G H W H U P L Q D U � R � O X F U R � D R � S U R G X ] L U � H � F R P H U F L D O L ] D U � F D O o D G R V � � 3 R U � H [ H P S O R � � F R P R � M i � V H � Y H U L ¿ F R X � D Q W H U L R U P H Q W H � � S D U D � � � � F D O o D G R V � I D E U L F D G R V � H � Y H Q G L G R V � � R � F X V W R � H � D � U H F H L W D � V m R � � U H V S H F W L Y D P H Q W H � � 5 � � � � � � � � � � � H � 5 � � � � � � � � � � � � $ V V L P � � R � l u c r o a s s o c i a d o s e r á : / � � 5 � � � � � � � � � � ± � 5 � � � � � � � � � � � 5 � � � � � � � � � P O R D E N T R O D O T E M A � ' H � I R U P D � J H Q p U L F D � � D � I X Q o m R � O X F U R � p � H V F U L W D � X W L O L ] D Q G R � D � U H O D o m R � / � � 5 � ± � & � � $ V V L P � L = - 3 x � � � � � [ � ± � � � � � [ � � � � � � � � � L = - 3 x � � � � � [ � ± � � � � [ � � � � � � L = - 3 x � � � � � [ � � � � � A s s i m c o m o a f u n ç ã o r e c e i t a , a f u n ç ã o l u c r o L = - 3 x � � � � � [ � � � � � � W D P E p P � p � X P D � I X Q o m R � T X D G U i W L F D � � H � R � J U i ¿ F R � G D � S D U i E R O D � D V V R F L D G D � D � H V V D � I X Q o m R � p � U H S U H V H Q W D G R � V H J X L Q G R � D V � H W D S D V � D � V H J X L U � � o � � ' H W H U P L Q D U � F R H ¿ F L H Q W H V � � a f u n ç ã o é L = - 3 x � � � � � [ � � � � � � � H Q W m R � � D � � � � � E � � � � H � F � � � � � � � o � � & R Q F D Y L G D G H � G D � S D U i E R O D � � D � F R Q F D Y L G D G H � G D � S D U i E R O D � p � Y R O W D G D � S D U D � E D L [ R � � S R L V � � Q H V W H � F D V R � � D � � � � � � D � � � � 3 o � � , Q W H U F H S W R � F R P � R � H L [ R � \ � � D � S D U i E R O D � F R U W D � R � H L [ R � \ � H P � � � � � � � � S R L V � F = � � � � � . � o � � , Q W H U F H S W R � F R P � R � H L [ R � [ � d e v e - s e r e s o l v e r a e q u a ç ã o - 3 x � � � � � [ � � � � � � � � 3 H O D � I y U P X O D � G H � % K D V N D U D � P o r t a n t o , a p a r á b o l a c o r t a o e i x o d a s a b s c i s s a s e m x � � � � � � H � [ � � � � � � 5 o � � 9 p U W L F H � G D � S D U i E R O D � a c o o r d e n a d a d o v é r t i c e d a p a r á b o l a é d e t e r m i n a d a a p a r t i r d e d o i s v a l o r e s , x v e y v : P O R D E N T R O D O T E M A � $ � F R R U G H Q D G D � G R � Y p U W L F H � G D � S D U i E R O D � p � � � � � � � � � � � � o � � & R Q V W U X L U � R � J U i ¿ F R � � c o l o c a m - s e o s p o n t o s n o p l a n o c a r t e s i a n o a p a r t i r d a s i n f o r m a ç õ e s o b t i d a s n o s p a s s o s a n t e r i o r e s . ( P � V H J X L G D � � G H Y H � V H � W U D o D U � D � F X U Y D � T X H � S D V V D � S R U � H V V H V � S R Q W R V � F i g u r a 4 . 3 � * U i ¿ F R � G D � I X Q o m R � / � � � � [ � � � � � [ � � � � � � 1 D � ) L J X U D � � � � � � R � O X F U R � P i [ L P R � R F R U U H � T X D Q G R � [ � � � � R X � V H M D � � T X D Q G R � � � � F D O o D G R V � V m R � S U R G X ] L G R V � H � � � � V m R � F R P H U F L D O L ] D G R V � � 2 � Y D O R U � F R U U H V S R Q G H Q W H � D R � O X F U R � P i [ L P R � p � G H � 5 � � � � � � � � � � 2 � Y p U W L F H � G D � S D U i E R O D � I R U Q H F H X � D � L Q I R U P D o m R � V R E U H � R � P i [ L P R � d a f u n ç ã o . P O R D E N T R O D O T E M A � � $ L Q G D � F R P � U H O D o m R � D R � J U i ¿ F R � G D � ) L J X U D � � � � � � R V � Y D O R U H V � G H � [ � S D U D � R V � T X D L V � R � O X F U R � p � Q X O R � R F R U U H P � T X D Q G R � [ � � � � � R X � [ � � � � � � & R P R � M i � H [ S O L F D G R � Q R � 7 H P D � � � � R � O X F U R � ] H U R � U H S U H V H Q W D � D � V L W X D o m R � H P � T X H � D � U H F H L W D � p � L J X D O � D R � F X V W R � � R X � V H M D � � V L W X D o m R � d o b r e a k - e v e n p o i n t � � & R Q V W U X L Q G R � R � J U i ¿ F R � G D � I X Q o m R � U H F H L W D � H � G D � I X Q o m R � F X V W R � H P � X P � P H V P R � V L V W H P D � G H � H L [ R V � � p � p o s s í v e l o b s e r v a r e s t a s i t u a ç ã o : F i g u r a 4 . 4 � * U i ¿ F R � G D � I X Q o m R � U H F H L W D � H � G D � I X Q o m R � F X V W R � H P � X P � P H V P R � V L V W H P D � G H � H L [ R V � 2 E V H U Y H � T X H � � Q D � ) L J X U D � � � � � � D � U H J L m R � V R P E U H D G D � H V W i � O L P L W D G D � S H O R V � Y D O R U H V � G H � [ � P D L R U H V � T X H � � � � H � P H Q R U H V � T X H � � � � � e � Q H V W D � U H J L m R � T X H � D � U H F H L W D � p � P D L R U � T X H � R � F X V W R � � R X � V H M D � � H P � T X H � K i � O X F U R � � $ L Q G D � Q D � ) L J X U D � � � � � � S R G H � V H � R E V H U Y D U � T X H � D �U H F H L W D � P i [ L P D � p � U H S U H V H Q W D G D � S H O R � S R Q W R � � � � � � � � � � � � ( Q W U H W D Q W R � � H V V H � S R Q W R � H V W i � I R U D � G D � U H J L m R � G H � O X F U R � � R X � V H M D � � F R Q V H J X L U � D � U H F H L W D � P i [ L P D � � Q H V W H � F D V R � � V L J Q L ¿ F R X � R E W H U � S U H M X t ] R � � ( P � W H U P R V � S U i W L F R V � � D � U H F H L W D � P i [ L P D � Q H P � V H P S U H � U H S U H V H Q W D U i � R � O X F U R � P i [ L P R � � j V � Y H ] H V � � S R G H � D W p � U H S U H V H Q W D U � X P D � V L W X D o m R � G H � S U H M X t ] R � � F R P R � P R V W U D � D � ) L J X U D � � � � � � , V V R � D F R Q W H F H � S R U T X H � R � O X F U R � G H S H Q G H � Q m R � V y � G D � I X Q o m R � U H F H L W D � � P D V � t a m b é m d a f u n ç ã o c u s t o . P O R D E N T R O D O T E M A � � A c e s s e o s i t e M u n d o E d u c a ç ã o . • C o n t é m u m a b r e v e e x p l i c a ç ã o s o b r e a s f u n ç õ e s c u s t o , r e c e i t a e l u c r o , j u n t a m e n t e a e x e m p l o s J U i ¿ F R V � / L Q N � � � K W W S � � � Z Z Z � P X Q G R H G X F D F D R � F R P � P D W H P D W L F D � I X Q F R H V � F X V W R � U H F H L W D � O X F U R � K W P ! � � $ F H V V R � H P � � � � P D L � � � � � � � A c e s s e o s i t e B r a s i l E s c o l a . • C o n t é m u m a b r e v e e x p l i c a ç ã o t e ó r i c a e e x e m p l o s s o b r e a s f u n ç õ e s c u s t o , r e c e i t a e l u c r o . / L Q N � � � K W W S � � � Z Z Z � E U D V L O H V F R O D � F R P � P D W H P D W L F D � P D W H P D W L F D � Q D � H F R Q R P L D � I X Q F D R � F X V W R � I X Q F D R � U H F H L W D � � K W P ! . $ F H V V R � H P � � � � P D L � � � � � � � A c e s s e o s i t e d a B i b l i o t e c a V i r t u a l d a A n h a n g u e r a . • 1 R � F D P S R � G H � S H V T X L V D � � G L J L W H � f u n ç õ e s . A p a r e c e r ã o v á r i a s p r o d u ç õ e s a c a d ê m i c a s c o m a p l i c a ç õ e s d a s f u n ç õ e s p o l i n o m i a i s . / L Q N � � � K W W S � � � Z Z Z � D Q K D Q J X H U D � F R P � E L E O L R W H F D V � E L E O L R W H F D � Y L U W X D O � F X U V R � H D G � D G P L Q L V W U D F D R ! � � $ F H V V R � H P � � � � P D L � � � � � � � $ V V L V W D � D R � Y t G H R � � F u n ç ã o L u c r o . • E s t e v í d e o m o s t r a a r e s o l u ç ã o d e u m e x e r c í c i o q u e e n v o l v e a a p l i c a ç ã o d e f u n ç õ e s q u a d r á t i c a s n a f o r m u l a ç ã o d a r e c e i t a , c u s t o e l u c r o . / L Q N � � � K W W S � � � Z Z Z � \ R X W X E H � F R P � Z D W F K " Y \ 2 V G & 6 � 8 = � N ! � � $ F H V V R � H P � � � � P D L � � � � � � � � 7 H P S R � � � � � � � A C O M PA N H E N A W E B � � , Q V W U X o } H V � $ J R U D � � F K H J R X � D � V X D � Y H ] � G H � H [ H U F L W D U � V H X � D S U H Q G L ] D G R � � $ � V H J X L U � � Y R F r � H Q F R Q W U D U i � D O J X P D V � T X H V W } H V � G H � P ~ O W L S O D � H V F R O K D � H � G L V V H U W D W L Y D V � � / H L D � F X L G D G R V D P H Q W H � R V � H Q X Q F L D G R V � H � D W H Q W H � V H � S D U D � R � T X H � H V W i � V H Q G R � S H G L G R � A G O R A É A S U A V E Z Q u e s t ã o 1 6 L J D � R V � S D V V R V � � D � V H J X L U � � S D U D � G H G X ] L U � X P D � L P S R U W D Q W H � F D U D F W H U t V W L F D � V R E U H � D V � S D U i E R O D V � • ( Q F R Q W U H � R V � L Q W H U F H S W R V � G D � I X Q o m R � I � [ � [ � � � � [ � � � � F R P � R � H L [ R � [ � � R X � V H M D � � U H V R O Y D � D � H T X D o m R � [ � � � � [ � � � � � � � • D e t e r m i n e a m é d i a a r i t m é t i c a d a s d u a s s o l u ç õ e s e n c o n t r a d a s , o u s e j a , 2 xx x 21 � . • D e t e r m i n e a c o o r d e n a d a x v d a p a r á b o l a , o u s e j a , r e s o l v a : a � b x v � . L e m b r e - s e d e q u e x v � I R U Q H F H � D � O R F D O L ] D o m R � G R � S R Q W R � G H � m á x i m o o u m í n i m o d a p a r á b o l a . C o m p a r e o r e s u l t a d o e n c o n t r a d o p e l a m é d i a a r i t m é t i c a e o v é r t i c e d a p a r á b o l a . O q u e s e p o d e c o n c l u i r ? $ W H Q o m R � � $ V � T X H V W } H V � G H � � � D � � � G H Y H P � V H U � U H V S R Q G L G D V � F R P � E D V H � Q R � H Q X Q F L D G R � D � V H J X L U � & H U W R � S U R G X W R � p � I D E U L F D G R � F R P � F X V W R � H � U H F H L W D � V H J X Q G R � D V � I X Q o } H V � & � � � � � [ � � � � � � � H � 5 � � � � [ � � � � � [ � � U H V S H F W L Y D P H Q W H � � & R Q V L G H U H � T X H � � S D U D � H V W H � S U R G X W R � � D � T X D Q W L G D G H � F R P H U F L D O L ] D G D � p � L J X D O � j � T X D Q W L G D G H � I D E U L F D G D � � 2 � J U i ¿ F R � � D � V H J X L U � � P R V W U D � R � F R P S R U W D P H Q W R � d a f u n ç ã o r e c e i t a e d a f u n ç ã o c u s t o . � � A G O R A É A S U A V E Z Q u e s t ã o 2 $ � S D U W L U � G R � J U i ¿ F R � � D � U H F H L W D � P i [ L P D � R E W L G D � Q D � F R P H U F L D O L ] D o m R � G H V W H � S U R G X W R � p � a ) � 5 � � � � � � � � � � � � b ) � 5 � � � � � � � � � � � � c ) � 5 � � � � � � � � � � � � d ) � 5 � � � � � � � � � � � � e ) � 5 � � � � � � � � � � � � � � A G O R A É A S U A V E Z Q u e s t ã o 3 $ � T X D Q W L G D G H � F R P H U F L D O L ] D G D � T X H � I R U Q H F H � D � U H F H L W D � P i [ L P D � p � a ) � � � � b ) � � � � � c ) � � � � � d ) 3 0 0 . e ) � � � � � Q u e s t ã o 4 $ V � T X D Q W L G D G H V � F R P H U F L D O L ] D G D V � T X H � U H S U H V H Q W D P � R � b r e a k - e v e n p o i n t s ã o : a ) � � � � H � � � � � b ) � � � � H � � � � � c ) � � � � H � � � � � d ) � � � � � H � � � � � e ) � � � � � H � � � � � � � 4 X H V W m R � � O s v a l o r e s d e r e c e i t a q u e r e p r e s e n t a m o s d o i s p o n t o s b r e a k - e v e n p o i n t s ã o : a ) � 5 � � � � � � � � � � � � H � 5 � � � � � � � � � � � � b ) � 5 � � � � � � � � � � � � H � 5 � � � � � � � � � � � � c ) � 5 � � � � � � � � � � � � H � 5 � � � � � � � � � � � � d ) � 5 � � � � � � � � � � � � H � 5 � � � � � � � � � � � � e ) � 5 � � � � � � � � � � � � H � 5 � � � � � � � � � � � � 2 E V H U Y D o m R � � Q H V W H � F D V R � � R V � G R L V � Y D O R U H V � G H � U H F H L W D � V m R � L J X D L V � D R V � G R L V � Y D O R U H V � G H � F X V W R � � S R L V � V H � W U D W D � G R � b r e a k - e v e n p o i n t . Q u e s t ã o 6 ( Q F R Q W U H � D � I X Q o m R � O X F U R � G H � X P � S U R G X W R � T X H � p � F R P H U F L D O L ] D G R � V H J X Q G R � D � I X Q o m R � U H F H L W D � 5 � � � � [ � � � � � [ � H � I D E U L F D G R � V H J X Q G R � D � I X Q - o m R � F X V W R � & � � � � � [ � � � � � � � $ W H Q o m R � � $ V � T X H V W } H V � G H � � � D � � � � G H Y H P � V H U � U H V S R Q G L G D V � F R P � E D V H � Q R � H Q X Q F L D G R � D � V H J X L U � 8 P � I H L U D Q W H � Q R W D � T X H � R � S U H o R � G R � T X L O R � G R � W R P D W H � Y D U L D � G H � D F R U G R � F R P � D � U H O D o m R � S � � � � [ � � � � � 2 � J D V W R � W R W D O � G R � I H L U D Q W H � F R P � R V � W R - P D W H V � � J D V W R � G D � F R P S U D � G R � W R P D W H � F R P � R � S U R G X W R U � H � R � W U D Q V S R U W H � � p � G D G R � S H O D � U H O D o m R � & � � � [ � � � � � 1 R V � G R L V � F D V R V � � [ � U H S U H V H Q W D � D � T X D Q W L G D G H � H P � T X L O R J U D P D V� G H � W R P D W H V � 4 X H V W m R � � 6 D E H Q G R � T X H � D � I X Q o m R � U H F H L W D � 5 � p � G D G D � S H O D � U H O D o m R � 5 � � S � � � [ � � S U H o R � Y H ] H V � D � T X D Q W L G D G H � G H � W R P D W H � F R P H U F L D O L ] D G D � � � R E W H Q K D � D � I X Q o m R � U H F H L W D � H � H V E R F H � R � J U i ¿ F R � G H V V D � I X Q o m R � F R P � R � J U i ¿ F R � G D � I X Q o m R � F X V W R � � , Q G L T X H � R � b r e a k - e v e n p o i n t . A G O R A É A S U A V E Z � � Q u e s t ã o 8 $ � S D U W L U � G D � I X Q o m R � U H F H L W D � H � G R � J U i ¿ F R � R E W L G R � Q D � T X H V W m R � D Q W H U L R U � � U H V S R Q G D � D � � 4 X D O � D � T X D Q W L G D G H � H P � T X L O R J U D P D V � G H � W R P D W H � D � V H U � F R P H U F L D O L ] D G D � S D U D � T X H � D � U H F H L W D � V H M D � P i [ L P D " � E � � 4 X D O � D � U H F H L W D � P i [ L P D " Q u e s t ã o 9 2 E W H Q K D � D � I X Q o m R � O X F U R � H � G H W H U P L Q H � D � � 4 X D O � D � T X D Q W L G D G H � H P � T X L O R J U D P D V � G H � W R P D W H � D � V H U � F R P H U F L D O L ] D G D � S D U D � T X H � R � O X F U R � V H M D � P i [ L P R " � E � � 4 X D O � R � O X F U R � P i [ L P R " 4 X H V W m R � � � ( V E R F H � R � J U i ¿ F R � G D � I X Q o m R � O X F U R � L Q G L F D Q G R � R V � S U L Q F L S D L V � S R Q W R V � � L Q F O X V L Y H � R � b r e a k - e v e n p o i n t . A G O R A É A S U A V E Z N e s t e t e m a , v o c ê a p r e n d e u s o b r e a s a p l i c a ç õ e s d a s f u n ç õ e s q u a d r á t i c a s e m m o d e l o s q u e e n v o l v e m r e c e i t a e O X F U R � � $ O p P � L V V R � � Y R F r � D S U H Q G H X � D � P R Q W D U � R � J U i ¿ F R � G H V V D V � I X Q o } H V � � S R V V L E L O L W D Q G R � D � O R F D O L ] D o m R � G R V � S R Q W R V � G H � P i [ L P R � G D � U H F H L W D � H � G R � O X F U R � � Y H U L ¿ F D Q G R � D V � G L I H U H Q o D V � W H y U L F D V � H Q W U H � D P E R V � � 9 R F r � W D P E p P � D S U H Q G H X � D � H Q F R Q W U D U � R � S R Q W R � G H � e q u i l í b r i o e n t r e a s f u n ç õ e s c u s t o e r e c e i t a , o b r e a k - e v e n p o i n t � � S H U P L W L Q G R � D � O R F D O L ] D o m R � G R � L Q W H U Y D O R � G H � O X F U R � ( V E R F H � R � J U i ¿ F R � G D � I X Q o m R � O X F U R � L Q G L F D Q G R � R V � S U L Q F L S D L V � S R Q W R V � � L Q F O X V L Y H � R b r e a k e v e n p o i n t N ê d b l i d f d á i d l l i F I N A L I Z A N D O � � 0 8 5 2 / 2 � � $ I U k Q L R � & D U O R V � � % 2 1 ( 7 7 2 � � * L i F R P R � � M a t e m á t i c a A p l i c a d a a A d m i n i s t r a ç ã o , E c o n o m i a e C o n t a b i l i d a d e � � � � � H G � � 6 m R � 3 D X O R � � & H Q J D J H � / H D U Q L Q J � � � � � � � 7 $ 1 � � 6 R R � 7 D Q J � � M a t e m á t i c a A p l i c a d a à A d m i n i s t r a ç ã o e E c o n o m i a � � � � � H G � � 6 m R � 3 D X O R � � 3 L R Q H L U D � � � � � � � R E F E R Ê N C I A S $ E V F L V V D V � � Q R � V L V W H P D � F D U W H V L D Q R � � R � H L [ R � G D V � D E V F L V V D V � p � R � H L [ R � [ � � D T X H O H � F R P X P H Q W H � U H S U H V H Q W D G R � Q D � K R U L ] R Q W D O � & R Q F D Y L G D G H � � Q D � S D U i E R O D � � D � F R Q F D Y L G D G H � p � R � O D G R � H P � T X H � K i � F D Y L G D G H � � G H S U H V V m R � R X � Y D O H � B r e a k - E v e n P o i n t � � H [ S U H V V m R � L Q J O H V D � T X H � V L J Q L ¿ F D � S R Q W R � G H � H T X L O t E U L R � ' H P D Q G D � � p � D � T X D Q W L G D G H � G H � X P � E H P � R X � V H U Y L o R � T X H � R V � F R Q V X P L G R U H V � G H V H M D P � D G T X L U L U � S R U � X P � S U H o R � G H ¿ Q L G R � H P � G D G R � m e r c a d o , d u r a n t e u m a u n i d a d e d e t e m p o . 2 U G H Q D G D V � n o s i s t e m a c a r t e s i a n o , o e i x o d a s o r d e n a d a s é o e i x o y , a q u e l e c o m u m e n t e r e p r e s e n t a d o n a v e r t i c a l . 7 $ 1 � � 6 R R � 7 D Q J � M a t e m á t i c a A p l i c a d a à A d m i n i s t r a ç ã o e E c o n o m i a � � � � � H G � � 6 m R � 3 D X O R � � 3 L R Q H L U D � � � � � � � $ E L L W W L L G E L p L O W W G K L W O G L O S S Á R I O � � G A B A R I T O Q u e s t ã o 1 5 H V S R V W D � A s o l u ç ã o d a e q u a ç ã o x � � � � [ � � � � � � � p � [ � � � � � R X � [ � � � � � � � $ � P p G L D � G H V V H V � S R Q W R V � p � � � 0 �� x � . O p o n t o q u e r e p r e s e n t a X v 6 22 1 2 )( �� . P o r t a n t o , a c o o r d e n a d a d o v é r t i c e d a p a r á b o l a p o d e s e r c a l c u l a d a p o r m e i o d a m é d i a a r i t m é t i c a d o s i n t e r c e p t o s d a I X Q o m R � F R P � R � H L [ R � [ � � 2 E V H U Y H � R � J U i ¿ F R � Q u e s t ã o 2 5 H V S R V W D � � A l t e r n a t i v a D . 2 E V H U Y H � R � J U i ¿ F R � � � G A B A R I T O Q u e s t ã o 3 5 H V S R V W D � � A l t e r n a t i v a C . 2 E V H U Y H � R � J U i ¿ F R � Q u e s t ã o 4 5 H V S R V W D � A l t e r n a t i v a C . 2 E V H U Y H � R � J U i ¿ F R � 4 X H V W m R � � 5 H V S R V W D � A l t e r n a t i v a D . � � 2 E V H U Y H � R � J U i ¿ F R � Q u e s t ã o 6 5 H V S R V W D � � / � � 5 � & � � � � [ � � � � � [ � ± � � � � � [ � � � � � � � � � / � � � � [ � � � � � [ � � � � [ � � � � � � � � / � � � � [ � � � � � [ � � � � � � 4 X H V W m R � � 5 H V S R V W D � � D � � 5 � � � � � [ � � � � � � � [ � � 5 � � � � [ � � � � � � [ � E � � $ � S D U W L U � G R V � S R Q W R V � L P S R U W D Q W H V � G D � S D U i E R O D � � R � J U i ¿ F R � ¿ F D U i � G A B A R I T O � � G A B A R I T O Q u e s t ã o 8 5 H V S R V W D � � D � � 1 R � J U i ¿ F R � G D � 4 X H V W m R � � � � D � F R R U G H Q D G D � G R � Y p U W L F H � p � � � � � � � � � 3 R U W D Q W R � � D � T X D Q W L G D G H � H P � T X L O R J U D P D V � G H � W R P D W H � D � V H U � F R P H U F L D O L ] D G D � S D U D � T X H � D � U H F H L W D � V H M D � P i [ L P D � p � [ v � � � N J � � E � � $ � U H F H L W D � P i [ L P D � T X H � p � G D G D � H P � U H D L V � p � \ v � � 5 � � � � � � � � 2 E V H U Y H � R � J U i ¿ F R � Q u e s t ã o 9 5 H V S R V W D � � / � � 5 � ± � & � � � � [ � � � � [ � ± � � � [ � � � � � � / � � � � [ � � � � [ � � � � D � � $ � T X D Q W L G D G H � T X H � I R U Q H F H � O X F U R � P i [ L P R � p � G D G D � S H O R � a � b x v � � � ( Q W m R � � D � � � � � � E � � � � H � F � � � � � � � � �� � � x v � � � � x v � � 3x v $ V V L P � � D � F R P H U F L D O L ] D o m R � G H � � � N J � I R U Q H F H � R � O X F U R � P i [ L P R � E � � 2 � O X F U R � P i [ L P R � S R G H � V H U � H Q F R Q W U D G R � W R P D Q G R � [ � � � � / � � � � � � � � � � � � � � � � / � � � � � � � � � � � � / � � 5 � � � � � � � � 4 X H V W m R � � � 5 H V S R V W D � � $ � S D U W L U � G R V � S R Q W R V � L P S R U W D Q W H V � G D � S D U i E R O D � S D U D � D � I X Q o m R � / � � � � [ � � � � [ � � � � � R � J U i ¿ F R � ¿ F D U i � G A B A R I T O
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