Intercept form of quadratic expressions

Quadratic expressions can be written in a number of different forms. One of these
is the intercept form. It is very easy to determine the x-intercepts of a quadratic
expression's graph if it is written in the intercept form, and hence the name. The
following quadratic expressions are in the intercept forms:

• (x + 1)(x + 2)
• (x - 2)(x - 1)
• 2(x + 3)(x + 4)

As you can notice, each of these expressions is written as a
product of two linear expressions. Additionally, there can be a number
present before the two expressions, as in example 3 above. Thus, assuming 'a', 'p'
and 'q' to be any numbers, you can generalize the intercept form as follows:

a(x - p)(x - q)

Unique characteristics, importance of the intercept form ...

Getting x-intercepts:

In any quadratic expression written in the intercept form a(x - p)(x - q), the numbers 'p' and 'q' represent
the x-intercepts of the graph corresponding
to it.

Thus in the quadratic expression
2(x - 1)(x - 2), the x-intercepts
of its corresponding graph are 1 and 2. Similarly, for the quadratic expressions

• (x - 3)(x - 4), the x-intercepts are 3 and 4
• (x + 2)(x + 3), the x-intercepts are -2 and -3
• 3(x - 1/2)(x + 1/3), the x-intercepts are 1/2 and -1/3

Getting axis of symmetry:

The line of symmetry of a parabola (the graph of a quadratic expression) is the
line passing through its vertex, and around which the parabola is symmetrical. The
equation for the line of symmetry of a quadratic expression's parabola is of due
importance in mathematics. It can be obtained easily when the quadratic expression
is written in its intercept form.

For the quadratic expression a(x
- p)(x - q), its line of symmetry is given by:

x = (p + q)/2

For example, for the quadratic expression
(x - 3)(x - 4), the axis of symmetry for its corresponding parabola is

x = (3 + 4)/2, that is x = 7/2

Similarly, for the quadratic expression

• (x + 3)(x + 4), equation for axis of symmetry is x = -7/2
• 3(x - 1/2)(x + 1/3), equation for axis of symmetry is x = -1/12
• 3(x + 1)(x - 1), equation for axis of symmetry is x = 0

Getting the coordinates of the vertex

The coordinates of any point on a graph are in the form of (x, y), where 'x' is the x-coordinate (ie,
abscissa) and 'y' is the y-coordinate (ie ordinate)
of the point.

For the quadratic expression a(x - p)(x - q), the x-coordinate of its vertex is

x = (p + q)/2

The y-coordinate of the vertex is equal to the value
of the expression a(x - p)(x - q) at x = (p + q)/2.

For example, for the quadratic expression (x - 3)(x - 4),

x-coordinate of vertex = (3 + 4)/2 = 7/2, and

y-coordinate of vertex = (7/2 - 3)(7/2 - 4) = -1/4

Hence the coordinates of the vertex are (7/2, -1/4)

Similarly, for the graph of the quadratic expression

• 3(x + 3)(x + 4), vertex is (-7/2, -1/4)
• 2(x + 1/2)(x - 1/3), vertex is (-1/12, -25/72)
• 3(x - 1)(x + 1), vertex is (0, -3)

Graphing a quadratic expression in the intercept form

In order to graph a quadratic expression, you require three points' coordinates:

1. Two x-intercepts
2. Vertex of parabola

By following the above mentioned procedure, the x-intercepts and coordinates of vertex
can be obtained. Then, the parabola can be graphed on a coordinate plane by plotting
these three points and joining them with the help of a free hand curve.

For example, for the quadratic expression
(x - 3)(x - 4),

• x-intercepts are 3 and 4, and
• Coordinates of vertex are (7/2, -1/4)

Hence the graph of the expression (x - 3)(x - 4) is: