Content-
✪ 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 }
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 }
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 }
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 }
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 }
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 }
Nice sir…. your explanation is quite good..
It is also helpful for me