Representación gráfica de las variaciones de segundo grado
- Ejercicio 281
-
x
2
– 3 x + 2
a = 1 , b = – 3 , c = 2
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
– 3
2 (
1
)
x
=
3
2
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
(
2
)
–
( – 3
)
2
4 (
1
)
y
=
8 – 9
4
y
= –
1
4
⇒ (
3
2
, –
1
4
)
v a l o r m í n i m o
b
2
– 4 a c
=
( – 3
)
2
– 4 (
1
)
(
2
)
= 1 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
– 3 x + 2
= 0
( x – 2
)
( x – 1
)
= 0 {
x
1
= 2
x
2
= 1
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 2 , c u n a d o x = 0
xy0210
3
2
–
1
4
2032 -
x
2
+ 3 x + 2
a = 1 , b = 3 , c = 2
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
3
2 (
1
)
x
= –
3
2
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
(
2
)
–
(
3
)
2
4 (
1
)
y
=
8 – 9
4
y
= –
1
4
⇒ ( –
3
2
, –
1
4
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
3
)
2
– 4 (
1
)
(
2
)
= 1 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
+ 3 x + 2
= 0
( x + 2
)
( x + 1
)
= 0 {
x
1
= – 2
x
2
= – 1
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 2 , c u n a d o x = 0
xy-32-20
–
3
2
–
1
4
-1002 -
x
2
+ 3 x – 10
a = 1 , b = 3 , c = – 10
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
3
2 (
1
)
x
= –
3
2
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
( – 10
)
–
(
3
)
2
4 (
1
)
y
=
– 40 – 9
4
y
= –
49
4
⇒ ( –
3
2
, –
49
4
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
3
)
2
– 4 (
1
)
( – 10
)
= 1 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
+ 3 x – 10
= 0
( x + 5
)
( x – 2
)
= 0 {
x
1
= – 5
x
2
= 2
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = – 10 , c u n a d o x = 0
xy-50-4-6
–
3
2
–
49
4
1-620 -
x
2
+ x – 12
a = 1 , b = 1 , c = – 12
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
1
2 (
1
)
x
= –
1
2
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
( – 12
)
–
(
1
)
2
4 (
1
)
y
=
– 48 – 1
4
y
= –
49
4
⇒ ( –
1
2
, –
49
4
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
1
)
2
– 4 (
1
)
( – 12
)
= 49 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
+ x – 12
= 0
( x + 4
)
( x – 3
)
= 0 {
x
1
= – 4
x
2
= 3
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = – 12 , c u n a d o x = 0
xy-58-40
–
1
2
–
49
4
3048 -
x
2
– 2 x + 1
a = 1 , b = – 2 , c = 1
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
– 2
2 (
1
)
x
= 1
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
(
1
)
–
( – 2
)
2
4 (
1
)
y
=
4 – 4
4
y
= 0 ⇒ ( 1 , 0
)
v a l o r m í n i m o
b
2
– 4 a c
=
( – 2
)
2
– 4 (
1
)
(
1
)
= 0 ⇒
E s c e r o l a s r a í c e s s o n r e a l e s e i g u a l e s
x
2
– 2 x + 1
= 0
( x – 1
)
2
= 0
x – 1
= 0
x
= 1
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t e v a l o e
L a p a r á b o l a c o r t a a l e j e y e n c = 1 , c u n a d o x = 0
xy-1401102134 -
x
2
+ 4 x + 2
a = 1 , b = 4 , c = 2
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
4
2 (
1
)
x
= – 2
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
(
2
)
–
(
4
)
2
4 (
1
)
y
=
8 – 16
4
y
= – 2 ⇒ ( – 2 , – 2
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
4
)
2
– 4 (
1
)
(
2
)
= 8 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
+ 4 x + 2
= 0
x
=
– b ±
b
2
– 4 a c
2 a
x
=
– 4 ±
4
2
– 4 (
1
)
(
2
)
2 (
1
)
x
=
– 4 ±
8
2
x
=
– 4 ± 2
2
2
x
= – 2 ±
2
↔ {
x
1
= – 2 +
2
x
2
= – 2 –
2
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 2 , c u n a d o x = 0
xy-42
– 2 –
2
0
– 2
– 2
– 2 +
2
002 -
–
x
2
– 4 x + 5
a = – 1 , b = – 4 , c = 5
a < 0
⇒
E s n e g a t i v a t i e n e u n v a l o r m á x i m o , e s d e c i r l a p a r á b o l a e s t a i n v e r t i d a
x
= –
b
2 a
x
= –
– 4
2 ( – 1
)
x
= – 2
v a l o r m á x i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 ( – 1
)
(
5
)
–
( – 4
)
2
4 ( – 1
)
y
=
– 20 – 16
– 4
y
= 9 ⇒ ( – 2 , 9
)
v a l o r m á x i m o
b
2
– 4 a c
=
( – 4
)
2
– 4 ( – 1
)
(
5
)
= 36 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
–
x
2
– 4 x + 5
= 0
x
2
+ 4 x – 5
= 0
( x + 5
)
( x – 1
)
= 0 {
x
1
= – 5
x
2
= 1
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 5 , c u n a d o x = 0
xy-67-50-291027 -
x
2
– 6 x + 3
a = 1 , b = – 6 , c = 3
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
– 6
2 (
1
)
x
= 3
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
1
)
(
3
)
–
( – 6
)
2
4 (
1
)
y
=
12 – 36
4
y
= – 6 ⇒ ( 3 , – 6
)
v a l o r m í n i m o
b
2
– 4 a c
=
( – 6
)
2
– 4 (
1
)
(
3
)
= 24 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
x
2
– 6 x + 3
= 0
x
=
– ( – 6
)
±
( – 6
)
2
– 4 (
1
)
(
3
)
2 (
1
)
x
=
6 ±
36 – 12
2
x
=
6 ±
24
2
x
=
6 ± 2
6
2
x
= 3 ±
6
{
x
1
= 3 +
6
x
2
= 3 –
6
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 3 , c u n a d o x = 0
xy03
3 –
6
03-6
3 +
6
063 -
2
x
2
+ x – 6
a = 2 , b = 1 , c = – 6
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
1
2 (
2
)
x
= –
1
4
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
2
)
( – 6
)
–
(
1
)
2
4 (
2
)
y
=
– 48 – 1
8
y
= –
49
8
⇒ ( –
1
4
, –
49
8
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
1
)
2
– 4 (
2
)
( – 6
)
= 49 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
2
x
2
+ x – 6
= 0
2
x
2
+ 4 x – 3 x – 6
= 0
2 x ( x + 2
)
– 3 ( x + 2
)
= 0
( x + 2
)
( 2 x – 3
)
= 0 {
x
1
= – 2
x
2
=
3
2
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = – 6 , c u n a d o x = 0
xy
–
5
2
4-20
–
1
4
–
49
8
3
2
024 -
–
x
2
+ 2 x + 15
a = – 1 , b = 2 , c = 15
a < 0
⇒
E s n e g a t i v a t i e n e u n v a l o r m á x i m o , l a p a r á b o l a s e a b r e h a c i a a b a j o
x
= –
b
2 a
x
= –
2
2 ( – 1
)
x
= 1
v a l o r m á x i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 ( – 1
)
(
15
)
–
(
2
)
2
4 ( – 1
)
y
=
– 60 – 4
– 4
y
= 16 ⇒ ( 1 , 16
)
v a l o r m á x i m o
b
2
– 4 a c
=
(
2
)
2
– 4 ( – 1
)
(
15
)
= 64 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
–
x
2
+ 2 x + 15
= 0
x
2
– 2 x – 15
= 0
x
2
+ 3 x – 5 x – 15
= 0
x ( x + 3
)
– 5 ( x + 3
)
= 0
( x + 3
)
( x – 5
)
= 0 {
x
1
= – 3
x
2
= 5
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 15 , c u n a d o x = 0
xy-49-301165069 -
2
x
2
– x – 15
a = 2 , b = – 1 , c = – 15
a > 0
⇒
E s p o s i t i v a t i e n e u n v a l o r m í n i m o
x
= –
b
2 a
x
= –
– 1
2 (
2
)
x
=
1
4
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 (
2
)
( – 15
)
–
( – 1
)
2
4 (
2
)
y
=
– 120 – 1
8
y
= –
1
4
⇒ ( –
1
4
, –
121
8
)
v a l o r m í n i m o
b
2
– 4 a c
=
( – 1
)
2
– 4 (
2
)
( – 15
)
= 121 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
2
x
2
– x – 15
= 0
2
x
2
– 6 x + 5 x – 15
= 0
2 x ( x – 3
)
+ 5 ( x – 3
)
= 0
( x – 3
)
( 2 x + 5
)
= 0 {
x
1
= 3
x
2
= –
5
2
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 15 , c u n a d o x = 0
xy-36
–
5
2
0
–
1
4
–
121
8
30
7
2
6 -
– 3
x
2
+ 7 x + 20
a = – 3 , b = 7 , c = 22
a < 0
⇒
E s n e g a t i v a t i e n e u n v a l o r m á x i m o , e s d e c i r l a p a r á b o l a s e a b r e h a c i a a b a j o
x
= –
b
2 a
x
= –
7
2 ( – 3
)
x
=
7
6
v a l o r m í n i m o p a r a x
y
=
4 a c –
b
2
4 a
y
=
4 ( – 3
)
(
22
)
–
(
7
)
2
4 ( – 3
)
y
=
– 264 – 49
– 12
y
=
313
12
⇒ (
7
6
,
313
12
)
v a l o r m í n i m o
b
2
– 4 a c
=
(
7
)
2
– 4 ( – 3
)
(
22
)
= 313 ⇒
E s p o s i t i v a l a s r a í c e s s o n r e a l e s y d e s i g u a l e s
– 3
x
2
+ 7 x + 20
= 0
3
x
2
– 7 x – 20
= 0
3
x
2
– 12 x + 5 x – 20
= 0
3 x ( x – 4
)
+ 5 ( x – 4
)
= 0
( x – 4
)
( 3 x + 5
)
= 0 {
x
1
= 4
x
2
= –
5
3
E l t r i n o m i o e s 0 c u a n d o x t o m a e s t o s v a l o r e s
L a p a r á b o l a c o r t a a l e j e y e n c = 20 , c u n a d o x = 0
xy-2-6
–
5
3
0
7
6
313
12
40
39
9
-6