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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≠ 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= p+q2.
Consider the graph of y= 12(x-7)(x-13).
Comparing the generic factored form with the example function, the values of a, p, and q can be identified. Factored Form:& y= a(x- p)(x- q) Example Function:& y= 1/2(x- 7)(x- 13) These values determine the characteristics of the parabola shown in the graph.
| Direction | Zeros | Axis of Symmetry |
|---|---|---|
| a= 1/2 | p= 7 and q= 13 | p+ q/2 ⇓ 7+ 13/2= 10 |
| Since 12 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. |
| 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=5x(x-2) ⇕ y= 5(x- 0)(x- 2) |