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Intercept Form of a Quadratic Function
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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(xp)(xq)

Here, a, p, and q are real numbers with a0. 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 with equation

Example

Consider the graph of

The graph of the given parabola with the axis of symmetry and x-intercepts shown
Comparing the generic factored form with the example function, the values of a, p, and q can be identified.
These values determine the characteristics of the parabola that can be seen in the graph.
Direction Zeros Axis of Symmetry
p=7 and q=13

Since 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.
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 1 Function 2 Function 3
y=2(x+1)(x3)

y=2(x(-1))(x3)
y=(x5)(x9)

y=1(x5)(x9)
y=5x(x2)

y=5(x0)(x2)
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