Quadratic function 1 Comment / Math / By Science++ 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 ≠ 0Note: 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 ≠ 0Case 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—————————————————–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 }f(x) < 0 for all x . ✪ 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 ≠ 0Note: 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 ≠ 0Case 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/DD > 0D=0D<0a>0a>0 , D>0a>0,D = 0a>0,D<0a<0a<0 , D>0a<0,D=0a<0,D<0Here 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 } f(x) < 0 for all x .
[email protected] 23rd April 2020 at 11:39 am Nice sir…. your explanation is quite good.. It is also helpful for me Log in to Reply
Nice sir…. your explanation is quite good..
It is also helpful for me