Quadratic Functions

When $\Delta=0,$ the equation $ax^2+bx+c=0$ has two equal real roots $x=-\fracb2a,-\fracb2a,$and $f(x)=a\left(x+\fracb2a\right)^2$, which is a parabola touching x-axis.

Case 1

Thus when $\Delta=0,$ a>0, we have a parabola touching x-axis and face upwards. The minimum value of $f(x)=ax^2+bx+c$ is zero (when $x=-\fracb2a)$ and maximum value is $+\infty. $Note that $ax^2+bx+c\geq0$ (for all real x).

When $\Delta=0,$ a<0, we have a parabola touching x-axis and face downwards. The maximum value of $f(x)=ax^2+bx+c$ is zero (when $x=-\fracb2a)$ and minimum value is $+\infty.$ Note that $ax^2+bx+c\leq0$ (for all real x).

When $\Delta<0,$ the equation $ax^2+bx+c=0$ has no real roots, and $f(x)=a\left[\left(x+\fracb2a\right)^2-\frac\Delta4a^2\right],$ which is a parabola wit vertex at $\left(-\fracb2a,-\frac\Delta4a\right).$

Thus, when $\Delta<0,$ a>0, we have a parabola fully lying above x-axis and facing upwards. Since it doesn't meet x-axise, there is no real x such that f(x)=0 i.e. $ax^2+bx+c=0$ has no real roots. Note that minimum value of $ax^2+bx+c$ is -$\frac\Delta4a$(a positive number). Thus $ax^2+bx+c>0$ in this case, for all real x.

When $\Delta<0,$ a<0, we have a parabola lying fully below x-axis and facing downwards. The maximum value of $f(x)$ is $-\frac\Delta4a$(a negative number). Thus $ax^2+bx+c<0$ in this case, for all real x

When $\Delta>0,$ the equation has two distinct real roots, say $\alpha,\beta(\alpha<\beta).$ Then $f(x)=a\left[\left(x+\fracb2a\right)^2-\frac\Delta4a^2\right],$ is a parabola wit vertex at $\left(-\fracb2a,-\frac\Delta4a\right).$

Thus when $\Delta>0,$ $a>0$, the vertex lies below x-axis but parabola faces upwards. Thus it meets x-axis at two points, which give us two distinct real Quadratic Functions roots of $ax^2+bx+c=0.$ Note that minimum value of f(x) is $-\frac\Delta4a$. Also, $ax^2+bx+c=0$ at $x=\alpha,\beta;ax^2+bx+c<0$ when $\alpha<x<\beta,$ and $ax^2+bx+c>0$ otherwise.

When $\Delta>0,a<0,$ the vertex lies above x-axis (as $-\frac\Delta4a>0)$, but the parabola faces downwards. Thus it meets x-axis at two points, which give us distinct real roots of $ax^2+bx+c=0$. Note that maximum value of f(x) is $-\frac\Delta4a$. Also, $ax^2+bx+c=0$ at $x=\alpha,\beta;$ $ax^2+bx+c>0$ when $\alpha<x<\beta$ and $ax^2+bx+c<0$ otherwise.

From the above discussion of Quadratic Functions , we see that:

(i) When $\Delta=0$ : $ax^{2}+bx+c>0$ if $a>0$ ; $ax^{2}+bx+c<0$ if $a<0,$

and $ax^{2}+bx+c=0$ at $x=-\frac{b}{2a}.$ Thus $ax^{2}+bx+c$ has the same sign as a (except at $x=-\frac{b}{2a}$ where it is zero).

(ii) When $\Delta<0:ax^{2}+bx+c>0$ if $a>0$ ; $ax^{2}+bx+c<0$ if $a<o.$ Thus $ax^{2}+bx+c$ has the same sign as a.

(iii) When $\Delta>0$ : $ax^{2}+bx+c=0$ at $x=\alpha,\beta;ax^{2}+bx+c$ has sign opposite to a when $\alpha<x<\beta$ , otherwise it has same sign as a.

In short, $ax^{2}+bx+c$ has same sign as a except when $ax^{2}+bx+c=0$ has two real distinct roots $\alpha,\beta$ and $\alpha<x<\beta$.

Corollary 1. $ax^{2}+bx+c>0$ for all $x\in R$,

iff $\Delta=b^{2}-4axc<0,$ $a>0$.

Corollary 2. $ax^{2}+bx+c<0$ for all $x\in R$ iff $a<0,$ $\Delta<0$.

Corollary 3. $ax^{2}+bx+c\geq0$ for all real x iff $a>0,$ $\Delta\leq0$.

Corollary 4. $ax^{2}+bx+c\leq0$ for all real x iff $a<0,$ $\Delta\leq0$.

Maximum/Minimum value of $ax^{2}+bx+c$

From previous graphs, we see that maxima/minima occurs at $x=-\frac{b}{2a}$, and its value is $-\frac{\Delta}{4a}$. When $a>0$, we have a minima ; when $a<0$, we have maxima.

graphing quadratic functions , solving quadratic functions , graphing functions