Properties of the Quadratic Functions
A quadratic function has the form
f (x) = a x^{2} + b x + c
Where a, b and c are real numbers and a is not equal to 0.
The domain of this function is the set of all real numbers.
The y intercept of the graph of f is given by f(0) = c.
The x intercepts are found by solving the equation
a x^{2} + b x + c = 0
There are several methods to solve this equation. One of these methods is the use of the quadratic formulas. The two solutions are given by
x_{1} = (  b + √D ) / (2 a)
x_{2} = (  b  √D ) / (2 a)
where D is the discriminant given
D = b^{2}  4 a c
To find the range of the quadratic
function, we first rewrite it in the form
f (x) = a(x  h)^{2} + k

If we expand the square above in f(x) above, we obtain
f (x) = a x^{2}  2a h x + a h^{2} + k

It is the same function as the original one, hence
a x^{2}  2a h x + a h^{2} + k = a x^{2} + b x + c

For the two quadratic expression on both sides of the above equation to be equivalent, they need to have equal coefficients, hence
 2 a h = b (first equation)
and
a h^{2} + k = c (second equation)

Solve the first equation for h to obtain
h =  b / (2a)

Substitute h by b/2a in the second equation and solve it for k to obtain
k = c  b^{2} / (4 a)

Hence any quadratic function can be written in the form f (x) = a(x  h)^{2} + k, with h and k given in terms of a, b and c as shown above. The form above leads to very interesting results. A x changes, The term (x  h)^{2} is either positive or zero. Hence
(x  h)^{ 2} ≥ 0
then two possible cases

1) If a > 0, multiply both sides of the inequality above by a
a (x  h)^{ 2} ≥ 0

Add k to both sides of the inequality
a (x  h)^{ 2} + k ≥ k

The left side is the formula of the
function, f (x) = a(x  h)^{2} + k , hence
f (x) ≥ k

The above result tells us that f (x) has a minimum value equal to k. It also tells us that the range of f (x) is given by
[ k , + ∞)

2) If a < 0, multiply both sides of the inequality (x  h)^{2} ≥ 0 a and change the inequality symbol.
a(x  h)^{2} ≤ 0

The above result tells us that f (x) has a maximum value equal to k. It also tells us that the range of f (x) is given by
( ∞, k]
It is also important to note that k = f(h). The graph of a quadratic function is called a parabola and the point with coordinates (h , k) is called the vertex of the parabola which can be a maximum or a minimum point as we have
shown above. Example 1
f is a quadratic function given by
f (x) = 2x^{2} + 2 x  4
 Find the x and y intercepts of the graph of f.
 Find the vertex of the graph of f.
 Find the domain and range of f.
 Sketch the graph of f.
Solution to Example 1

a  The y intercept is given by
(0 , f(0)) = (0 , 4)

The x coordinates of the x intercepts are the solutions to
2x^{2} + 2 x  4 = 0

The discriminant D of the above quadratic equation is given by
D = (2)^{2}  4(2)(4)
= 36

The solutions are
x_{1} = (2 + 6) / 4
= 1
x_{2} = (2  6) / 4
=  2

The x intercepts are at the points (1 , 0) and (2 , 0).

b  The x coordinate h and the y coordinate k of the
vertex are given by
h = b / 2a
= 2/4
= 1/2
k = f (h)
= f(1/2)
= 2(1/2)^{2} + 2 (1/2)  4
= 9/2
The vertex is at the point (1/2 , 9/2)

c  The domain of f (x) is the set of all real numbers.

Since coefficient a is positive, f has a minimum value equal to k. The range is given by the set of real values in the interval [9/2 , +infinity).

d  Find extra points if necessary;
(2 , f(2)) = (2 , 8) and (3 , f(3)) = ( 3 , 8 ) as an example.
Plot the vertex (lowest
point), the x and y intercepts and the extra points as shown
below. From the vertex, which is a minimum point, the left and right
sides of the graph of f should rise upward.
Matched Problem
f is a quadratic function given by
f (x) = x^{2}  2 x  3
 Find the x and y intercepts of the graph of f.
 Find the vertex of the graph of f.
 Find the domain and range of f.
 Sketch the graph of f.
More references and links to graphing, graphs of functions and quadratic functions
Graphing Functions
Find Vertex and Intercepts of Quadratic Functions  Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.
Quadratic Functions (general form).
Solver to Analyze and Graph a Quadratic Function Graphs of Basic Functions.
