Quadratic function

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✪ Quadratic Function :
       f(x) = ax2 + bx + c 
or y =  ax2 + bx + c 
Graphical view of quadratic equation :
y = ax2 + bx + c = 0
, ( a ≠ 0 )
 y= a[(x + b/2a)2 — D/4a2 ]
( y + D/4a ) = a(x + b/2a)2  . . .(1)
let y + D/4a = Y & x + b/2a = X
∴ Y = aX2 
this equation is look like a parabola.
so , shape of the curve is parabolic.
Axis of parabola :
Y = aX2 
X=0 , or x + b/2a = 0
x = – b/2a i.e parallel to y—axis.
Opening of parabola :
Depend on ‘a’ —
1. if a > 0 —> opening upwards.
2. if a < 0 —> opening downwards.
3. if a=0 —> it becomes straight line –
bx + c =0 , but here a ≠ 0
Note: focus on some points :
Nature of the graph depends on ‘a’ & ‘D’
Case 1: focus on ‘a’
1. if a > 0 —> opening upwards.
2. if a < 0 —> opening downwards.
3. if a=0 —> it becomes straight line –
bx + c =0 , but here a ≠ 0
Case 2: focus on ‘D’
1. if D > 0 —> cut at two points ( two real & unequal roots).
2. if D < 0 —> neither cut point nor touchpoint ( imaginary roots ).
3. if D=0 —> one touch point or at this point tangent at x-axis (both roots are equal & real ).
All Possibility of graph :
We know that graph is depends on “a” & “ D ”
a= { > 0 or < 0 }.     i.e   2 case
D = { < 0 , > 0 or =0 }.  i.e  3 case
So , total number of case 
                               = {a} x { D }
                                       =  2 x 3
                                       =   6
—————————————————–

a/D

D > 0

D=0

D<0

a>0

a>0 , D>0

a>0,D = 0

a>0,D<0

a<0

a<0 , D>0

a<0,D=0

a<0,D<0

Here you can see six possibilities of graph :
Case 1 :  a > 0  &  D > 0
  a> 0 : { parabola opening  upwards }
  D > 0 : {  parabola cut at x-axis in two distinct points }
Case 1 :  a > 0  &  D > 0
f(x) > 0 for all x ∈ (- ∞ , 𝞪 ) ⋃ ( β , ∞).
f(x) < 0 for all x ∈ (𝞪 , β ).
Case 2 :  a > 0  &  D = 0
  a> 0  { parabola opening  upwards }
  D = 0 {  parabola touch at x-axis  }
Case 2 :  a > 0  &  D = 0  
f(x) > 0 for all x except vertex of parabola.
Case 3 :  a > 0  &  D < 0
  a> 0  { parabola opening  upwards }
  D < 0 {  parabola neither touch nor cut x-axis  }
Case 3 :  a > 0  &  D < 0
f(x) > 0 for all x .
Case 4 :  a < 0  &  D > 0
  a< 0  { parabola opening  downwards }
  D > 0 {  parabola cut at x-axis in two points  }
Case 4 :  a < 0  &  D > 0
f(x) < 0 for all x ∈ (- ∞ , 𝞪 ) ⋃ ( β , ∞).
f(x) > 0 for all x ∈ (𝞪 , β ).
Case 5 :  a < 0  &  D = 0
  a < 0  { parabola opening  downwards }
  D = 0 {  parabola touch at x-axis  }
Case 5 :  a < 0  &  D = 0  
f(x) < 0 for all x except vertex of parabola.
Case 6 :  a < 0  &  D < 0
  a < 0  { parabola opening  downwards }
  D < 0 {  parabola neither touch nor cut at x-axis  }
Case 6 :  a < 0  &  D < 0
f(x) < 0 for all x .

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