[Back] [VVHS Math - Mr. Davis]
Quadratic functions and applications are a large part of the mathematics of the world around us. Any general problem involving areas is a quadratic problem. The force of gravity, which basically holds the universe as we know it together, can be modeled with a quadratic function.
The general or standard form of a quadratic function is y = ax2 + bx + c, or in function form, f(x) = ax2 + bx + c, where x is the independent variable, y is the dependent variable, and a, b, and c are constants.
Another important form of equation used is the
vertex form, y = a(x - h)2 + k, where (h, k) is the vertex of the parabola.(fig. 2,3) In both
forms, a determines the size and direction of the parabola.
The larger the absolute
value of a, the steeper (or thinner) the parabola is, since the value of y is increased
more quickly. All parabolas are actually similar in a geometric sense, just as all circles
and squares are similar figures, with apparent size determined by the constant a.
If a is positive, the parabola opens upward, if negative, the parabola opens downward.
Since the graph of any function in the form 
is a parabola, the parts of the graph
of the parabola are determined by the values of a, b, and c.
The most meaningful points of the graph of a parabola are:
x-intercepts: The x-intercepts, if any, are also called the roots of
the function. They are meaningful specifically as the zeroes
of the function, but also represent the two roots for any
value of 
.
y-intercept: The importance of the y-intercept is usually as an
initial value or initial condition for some state of an experiment,
especially one where the independent variable represents time.
Vertex: The vertex represents the maximum (or minimum) value
of the function, and is very important in calculus and many natural
phenomena.
Given 
, the y-intercept is immediately seen to be c, too easy!
y-intercept = ( 0, c )
Given 
, the x-intercepts are the solutions to the equation 
, the infamous
quadratic equation. The most common methods to solve this equation are by factoring, completing the square,
or by the quadratic formula. Completing the square is no longer as commonly used.
So, given 
, the x-intercepts can be found in basically two ways, factoring or the quadratic formula.
Factoring: If 
can be factored into the form 
,
then the x-intercepts are 
and 
.
Example: 
Quadratic Formula: For any function in the form
, x-intercepts are given by
Example: 
Or, about 
More examples (first set y = 0 to find the x-intercepts, then solve):
Factoring Completing the Square Quadratic Formula
Since these methods aren't easy, if a parabola has NO real x-intercepts to be found, or in other words imaginary
roots, it's nice to know that fact ahead of time. The discriminant, or 
, is often used to determine if
there are x-intercepts (also called the roots of the function) or not. Notice in the 3rd example above, once it's
found that the discriminant is 4-12=-8, the solutions must therefore be imaginary numbers (complex conjugates,
that is), so there are no x-intercepts.
Given 
, the vertex can be found by completing the square, or by using the expression derived
from completing the square on the general form. Go here for the full story on completing the square.
Quick Formula: For any function in the form
,
the vertex is 
(Note: typically just -b/2a is calculated and plugged in for x to find y.)
Example: 
| General Quadratic function: |  |
| Standard form: |  |
| Vertex form, vertex ( h, k ) : |  |
| Vertex, in terms of a, b, and c : |  |
| X-intercepts, in terms of a, b, and c : |  |
X-intercepts, if is factorable: |
 |
| Y-intercept: |  |