# Sign of Quadratic Function

Let

*y*=*f*(*x*) =*ax*^{2}+*bx*+*c*, where*a*,*b*,*c*, âˆˆ*R*and*a*â‰ 0.âˆ´

where

*D*is discriminant.The graph of

*y*=*f*(*x*) is parabola whose axis is parallel to the*y*-axis and vertex*A*.For some values of

*x*,*f*(*x*) may be positive, negative, or zero and for*a*> 0, the parabola opens upwards and for*a*< 0, the parabola opens downwards, this gives the following cases:
a > 0 and D < 0, then f(x) > 0 âˆ€ x âˆˆ R |
a > 0 and D > 0 |
a > 0 and D = 0, then f(x) â‰¥ 0 âˆ€ x âˆˆ R |

a < 0 and D < 0 then f(x) < 0 âˆ€ x âˆˆ R |
a < 0 and D > 0 |
a < 0 and D = 0 then f(x) â‰¤ 0 âˆ€ x âˆˆ R |

**If**

*Note:**f*(

*x*) â‰¥ 0, âˆ€

*x*âˆˆ

*R*â‡’

*a*> 0 and

*D*â‰¤ 0 and if

*f*(

*x*) â‰¤ 0 âˆ€

*x*âˆˆ

*R*â‡’

*a*< 0 and

*D*â‰¤ 0.