Quadratic Functions and Their Graphs
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Key Questions

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 parÃ¡bola.
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 parÃ¡bola 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 parÃ¡bola 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 parÃ¡bola does not cross the x axis.

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.
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Quadratic Equations and Functions

1Quadratic Functions and Their Graphs

2Vertical Shifts of Quadratic Functions

3Use Graphs to Solve Quadratic Equations

4Use Square Roots to Solve Quadratic Equations

5Completing the Square

6Vertex Form of a Quadratic Equation

7Quadratic Formula

8Comparing Methods for Solving Quadratics

9Solutions Using the Discriminant

10Linear, Exponential, and Quadratic Models

11Applications of Function Models