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Concept

Factored Form of a Quadratic Function

A quadratic function is said to be written in factored form, or intercept form, if it follows a specific format.

$y = a (x - p) (x - q)$

Here, $a,$ $p,$ and $q$ are real numbers with $a \neq = 0 .$ The value of $a$ gives the direction of the parabola. When $a > 0,$ the parabola faces upward, and when $a < 0,$ it faces downward. The zeros of the parabola are $p$ and $q,$ and the axis of symmetry is the vertical line $x = \frac{p + q}{2} .$

Example

Consider the graph of $y = \frac{1}{2} (x - 7) (x - 13) .$

$The graph of parabola y=\dfrac{1}{2}(x-7)(x-13) with the axis of symmetry (x=10) and x-intercepts (x=7 and x=13) shown$

Comparing the generic factored form with the example function, the values of

a,

p,

and

q

can be identified.

Factored Form : Example Function : y = a (x - p) (x - q) y = \frac{1}{2} (x - 7) (x - 13)

These values determine the characteristics of the parabola shown in the graph.

Direction	Zeros	Axis of Symmetry
$a = \frac{1}{2}$	$p = 7$ and $q = 13$	$\frac{p + q}{2}$ $⇓$ $\frac{7 + 1 3}{2} = 10$
Since $\frac{1}{2}$ is greater than $0,$ the parabola opens upward.	The zeros are $7$ and $13 .$ Therefore, the parabola intersects the $x -$ axis at $(7, 0)$ and $(13, 0) .$	The axis of symmetry is the vertical line $x = 10 .$

Extra

Consider other example quadratic functions.

Function 1 : Function 2 : Function 3 : y = 2 (x + 1) (x - 3) y = (x - 5) (x - 9) y = 5 x (x - 2)

Although these functions do not strictly follow the format of the factored form, they are said to be written in factored form because they can easily be rewritten in the desired format.

Function $1$	Function $2$	Function $3$
$y = 2 (x + 1) (x - 3)$ $⇕$ $y = 2 (x - (- 1)) (x - 3)$	$y = (x - 5) (x - 9)$ $⇕$ $y = 1 (x - 5) (x - 9)$	$y = 5 x (x - 2)$ $⇕$ $y = 5 (x - 0) (x - 2)$

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