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Quadrilatic Function

1) The document analyzes different forms of quadratic equations, including equations missing the middle (bx) term and equations of the form (x-c)2. 2) For equations of the form y=ax2+c, the direction of opening is up if a>0 and down if a<0. The vertex is (c,0) and the axis of symmetry is x=0. 3) For equations of the form y=(x-c)2, the direction of opening is always up. The vertex is (-c,0) and the axis of symmetry is x=-c. There is one x-intercept at (-c,0).

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Catalin Blesnoc
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190 views6 pages

Quadrilatic Function

1) The document analyzes different forms of quadratic equations, including equations missing the middle (bx) term and equations of the form (x-c)2. 2) For equations of the form y=ax2+c, the direction of opening is up if a>0 and down if a<0. The vertex is (c,0) and the axis of symmetry is x=0. 3) For equations of the form y=(x-c)2, the direction of opening is always up. The vertex is (-c,0) and the axis of symmetry is x=-c. There is one x-intercept at (-c,0).

Uploaded by

Catalin Blesnoc
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Having in view that quadrilatic function has the general form as y=a x2 +b x +c where a , b , c ∈ R and a ≠ 0, I analysed the

followig forms:

A) In the following form of quadrilatic ecuation the midle term (from a x 2+ b x+ c ) is missing (b x term)

y=x 2 +2 y=x 2 +5 y=x 2−12


a=1 ; b=0 ,c =2 a=1 ; b=0 ,c =5 a=1 ; b=0 ,c =−12

1 f ( x )=−x 2 +3 3
f ( x )=x 2− f ( x )=−x 2−
2 a=−1 ; b=0 , c=3 4
−1 −3
a=1 ; b=0 ,c = a=−1 ; b=0 , c=
2 4

Function Direction of opening Vertex Axis of symmetry X-Intercept


2
y=x +2 up ( 2 ; 0) x=0 no points
y=x 2 +5 up (5 ; 0) x=0 no points
y=x 2−12 up (−12 ; 0 ) x=0 two points .....
1 up −1 x=0 two points .....
y=x 2−
2 ( 2
;0)
y−x 2 +3 down (3 ; 0) x=0 two points ....
3 down −3 x=0 no points
y=−x2 −
4 ( 4
;0)
General Conclusion
General form: f ( x )=a x 2 +c

If a> 0 , direction of opening is up


If a< 0 , direction of opening is down

Vertex is ( c ; 0 )
Axis of symmetry x=0

If a> 0 and c >0 , there is no point of intersection of the Graph with Ox axes
If a< 0 and c <0 , there is no point of intersection of the Graph with Ox axes
B) quadrilatic ecuation

2 2 2
y= ( x −2 ) y= ( x −4 ) 1
y= x− ( ) 2

2 2 2
y= ( x +2 ) y= ( x +3 ) 3
( )
y= x +
2
Function Direction of opening Vertex Axis of symmetry X-Intercept Y-Intercept
y= ( x −2 ) 2 up ( 2 ; 0) x=2 one point, in ( 2 ; 0 ) (0 ; 4 )

y= ( x −4 )
2 up ( 4 ; 0) x=4 one point in( 4 ; 0 ) ( 0 ; 16 )
2
1
1
( 12 ; 0) ( 12 ; 0) (0 ; 14 )
up
( )
y= x−
2
x=
2
one point in

y= ( x +2 )
2 up (−2 ; 0 ) x=−2 one point in (−2 ; 0 ) (0 ; 4 )

y= ( x +3 )2 up (−3 ; 0 ) x=−3 one point in (−3 ; 0 ) (0 ; 9)


2
2 −2
( −23 ; 0 ) (0 ; 49 )
up −2
( )
y= x +
3 ( 3
;0 ) x=
3
one point in

General Conclusion
2
General form: y= ( x ± c )

direction of opening is always up

Vertex is ( ∓ c ; 0 )
Axis of symmetry x=∓c
X-Intercept only in one point ( ∓ c ; 0 )
Y-Intercept only in one point ( 0 ; c 2 )

C) quadrilatic ecuation

y=x 2 y=3 x 2 y=0.5 x 2


y=−x2 y=−2 x 2 y=
−9 2
x
2

Function Direction of opening Vertex Axis of symmetry X-Intercept Y-Intercept


y=x 2 up ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
y=3 x 2 up ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
y=0.5 x 2 up ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
y=−x2 down ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
y=−2 x 2 down ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
−9 2 down ( 0 ; 0) x=0 one point, in ( 0 ; 0 ) no points
y= x
2

General Conclusion
General form: y=a x2
If a> 0 , direction of opening is up
If a< 0 , direction of opening is down

Vertex is always ( 0 ; 0 )
Axis of symmetry x=0
X-Intercept only in one point ( 0 ; 0 )
Y-Intercept no intercept

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