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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. |
Consider other example quadratic functions. Function1:& y=2(x+1)(x-3) Function2:& y=(x-5)(x-9) Function3:& y=5x(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=5x(x-2) ⇕ y= 5(x- 0)(x- 2) |