In the factored form of the quadratic expression, the expression is in the form of two (or more) brackets (eq. 1).
Eq. 1: \(y = a(x+b)(x+c)\)
In the factored form, the b and c coefficients describe where the x-intercepts of the parabola are (Figure 1, Figure 2).
Figure 1: Changing the b value of the factored form of the quadratic expression. The b value of the factored form relates to where one of the x intercepts of the parabola is. The x intercept is located at the negative of the b value, with positive b values indicating positive x intercepts, and vice-versa. As the b value changes, the location of the vertex also changes, but not the stretch or compression of the parabola. Rather the location of the parabola changes to ensure a consistent shape of parabola with different changes in x intercepts.
Figure 2: Changing the c value of the factored form of the quadratic expression. The c value of the factored form describes where one of the x intercepts of the parabola is. The x intercept is located at the negative of the c value, with positive c values indicating positive x intercepts, and vice-versa. As the c value changes, the location of the vertex also changes, but not the stretch or compression of the parabola. Rather the location of the parabola changes to ensure a consistent shape of parabola with different changes in x intercepts.
The a coefficient on the other hand describes how steep the parabola is (Figure 3). However unlike the vertex form of the quadratic equation, it isn't as clear to see the relationship between this value and how steep the curve is.
Figure 3: Changing the a value of the factored form of the quadratic expression. As the a value of the factored form changes, the stretch of the parabola increases, like the a value of other forms of quadratic expressions. Similarly, as the a value diverges from 0, the greater the stretch of the parabola is. Negative a values create parabolas that open downwards, while positive a values create parabolas that open upwards.
The factored form is useful for analyzing where a parabola crosses the x-intercepts. This can be useful if the x-intercept is significant, this could be for example, if the x-intercept intercept is the ground, and the expression describes a flight path of an object for example (Figure 3).
Not every expression has a factored form, some expressions do not have x-intercepts, and so they don’t have a factored form (Figure 4). In these cases another form of the expression is more appropriate to describe the parabola.