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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.

Here, and are real numbers with The value of gives the direction of the parabola. When the parabola faces upward, and when it faces downward. The zeros of the parabola are and and the axis of symmetry is the vertical line with equation


Consider the graph of

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 and can be identified.
These values determine the characteristics of the parabola that can be seen in the graph.
Direction Zeros Axis of Symmetry

Since is greater than the parabola opens upward. The zeros are and Therefore, the parabola intersects the axis at and The axis of symmetry is the vertical line


Consider other example quadratic functions.
Although these functions do not strictly follow the format of the factored form, they are said to be written in factored form. This is because they can easily be rewritten in the desired format.
Function Function Function