Quadratic Functions

Quadratic functions are the playground for many deeper algebraic concepts. In competitions, we look beyond the vertex formula to inequalities, root distribution, and functional properties.

Standard Forms

General Form: $f (x) = a x^{2} + b x + c$
Vertex Form: $f (x) = a (x - h)^{2} + k$ , where $(h, k)$ is the vertex.
Factored Form: $f (x) = a (x - r_{1}) (x - r_{2})$ , where $r_{1}, r_{2}$ are roots.

Analysis Techniques

1. Extreme Values

For

f (x) = a x^{2} + b x + c

If $a > 0$ , the minimum is at $x = - b /2 a$ .
If $a < 0$ , the maximum is at $x = - b /2 a$ .
The value is $f (- b /2 a) = c - \frac{b ^{2}}{4 a} = \frac{4 a c - b ^{2}}{4 a}$ .

2. The Discriminant $Δ$

Δ = b^{2} - 4 a c

determines the nature of the roots:

$Δ > 0$ : Two distinct real roots.
$Δ = 0$ : One real double root (tangent to x-axis).
$Δ < 0$ : Two complex conjugate roots (no x-intercepts).

Tip (Range of Quadratics)

To find the range of a rational function

y = \frac{Q _{1} ( x )}{Q _{2} ( x )}

, rearrange it into a quadratic equation in

x

with coefficients depending on

y

, then require

Δ_{x} \geq 0

Inequality Conditions

For

a x^{2} + b x + c > 0

to hold for all real $x$ :

$a > 0$ (parabola opens up)
$Δ < 0$ (no roots, never crosses zero)

Practice Problems

Exercise (Problem 1)

Find the minimum value of

f (x) = 2 x^{2} - 8 x + 5

Exercise (Problem 2)

Find all

k

such that the inequality

k x^{2} - 2 x + (k - 1) < 0

holds for all real

x

Exercise (Problem 3)

If the roots of

x^{2} - p x + q = 0

differ by 1, prove that

p^{2} - 4 q = 1