# What are the important features of the graphs of quadratic functions?

Dec 16, 2014

A quadratic function has the general form:
$y = a {x}^{2} + b x + c$

(where $a , b \mathmr{and} c$ are real numbers) and is represented graphically by a curve called PARABOLA that has a shape of a downwards or upwards U.
The main features of this curve are:
1) Concavity: up or down. This depends upon the sign of the real number $a$:

2) Vertex. The vertex is the highes or lowest point of the parábola.
the coordinates of this point are:
$x = - \frac{b}{2 a}$ and $y = - \frac{\Delta}{4 a}$
Where $\Delta = {b}^{2} - 4 a c$

3) point of intercept with the y axis. This is the point where the parábola crosses the y axis and has coordinates: $\left(0 , c\right)$

4) Possible points of intercept with the x axis (there also can be none). These are the points where the parábola crosses the x axis.
They are obtained by putting y=0 and solving for x the 2nd degree equation: $a {x}^{2} + b x + c = 0$, which will give the x coordinates of these points (2 solutions)
Depending on the discriminant $\Delta = {b}^{2} - 4 a c$ if it is <0 the parábola does not cross the x axis.