Quadratic Functions and Their Graphs
Key Questions

There are two forms a quadratic function could be written in: standard or vertex form. Here are the following ways you can determine the vertex and direction dependent on the form:
Standard Form (
#f(x)=ax^2 + bx + c# )
1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
2. Vertex can be found by#x= b/(2a)# and then plugging in that value to find y.
Here is an example:
#y = 2x^2 + 4x  3# , Faces downward since a = 2.
to find the vertex:#x= (4)/(2(2))=(4)/ (4) = 1#
then plug that value into the equation#y = 2x^2 + 4x  3#
#y = 2(1)^2 + 4(1)  3 #
#y = 2(1) +4(1)  3 #
#y = 2 + 4 3 #
#y = 1 #
Vertex is (1,1)Vertex Form (
#y=a(xh)^2 +k# )
1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
2. Vertex is (h,k).
Here is an example:
#y = 3(x2)+6# Faces down since a = 3 and the vertex is (2, 6). 
A quadratic function is one of general form:
#y=ax^2+bx+c# where a, b and c are real numbers.
This function can be plotted giving a PARABOLA (a curve in the shape of an upward or downward U)
To find the x intercepts you must put y=0; in this way you fix at zero the coordinate y of the points you are seeking.
You are left with finding the coordinate x of the points.
If y=0 you are left with:#0=ax^2+bx+c# which is a second degree equation.
By solving this equation you'll find two values of x (x1 and x2) that together with y=0 will give you the intercepts:
intercept 1: (x1 , 0)
intercept 2: (x2 , 0)Remember that a second degree equation can also have solutions:
 coincident (the intercept is the VERTEX of the parabola)
 imaginary (The parabola does not cross the x axis)
Depending upon the discriminant of the equation. 
The graphs of quadratic functions are called parabolas, and they look like the letter "U" right side up or up side down.
I hope that this was helpful.

A quadratic function has the general form:
#y=ax^2+bx+c# (where
#a,b 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=b/(2a)# and#y=Delta/(4a)#
Where#Delta=b^24ac# 3) point of intercept with the y axis. This is the point where the parábola crosses the y axis and has coordinates:
#(0,c)# 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:#ax^2+bx+c=0# , which will give the x coordinates of these points (2 solutions)
Depending on the discriminant#Delta=b^24ac# if it is <0 the parábola does not cross the x axis.