Quadratic expressions can be written in a number of different forms. One of these

is the

expression's graph if it is written in the intercept form, and hence the name. The

following quadratic expressions are in the intercept forms:

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:

is the

**intercept form**. It is very easy to determine the x-intercepts of a quadraticexpression'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)

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

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

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.

Similarly, for the graph of the quadratic expression

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

of the expression a(x - p)(x - q) at x = (p + q)/2.

For example, for the quadratic expression (x - 3)(x - 4),
x = (p + q)/2

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

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:

- Two x-intercepts
- Vertex of parabola

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)

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