Quadratics
Using Factored Form to Find the Zeros, Axis of Symmetry and Optimal Value of Parabolas
Zeros or x-intercepts
Using factored form to determine the zeros of a parabola is much easier than using the vertex form to find the solutions. In order to find the solutions of a quadratic equation, follow these steps:
To find the zeros of the parabola represented by this equation, all that needs to be done is that the 'r' and 's' values in factored form need to be taken out of the brackets, and their signs must be switched. This means that if the 'r' value was positive inside the bracket, it will be negative outside the brackets.
EXAMPLE:
y = 5(x-6)(x+5)
Therefore, the x-intercepts are (6,0) and (-5,0).
Axis of Symmetry
The axis of symmetry is the x-value of the vertex. In order to find the axis of symmetry from factored form, first find the zeros of the parabola and then substitute the zeros into the equation for the axis of symmetry to find the answer.
Equation for axis of symmetry:
x = (r+s)
2
EXAMPLE:
Find the axis of symmetry of the following quadratic equation:
y = -7(x-6)(x+40)
First, find the zeros of the parabola represented in factored form.
Therefore, the zeros are 6 and -40.
Now substitute the zeros into the equation to solve for the axis of symmetry. Note that the order of the zeros doesn't matter, this means that any of the two zeros can be substituted for 'r' and 's'.
x = (r + s)
2
x= [6 + (-40)]
2
x = (-34)
2
x = -17
x = (r + s)
2
x= [(-40) + 6]
2
x = (-34)
2
x = -17
OR
Therefore, the axis of symmetry is x = -17.
Optimal Value
The optimal value is the y-value of the vertex. When finding the optimal value using factored form, sub in the axis of symmetry of the parabola into the factored form equation to solve for y.
EXAMPLE:
Find the optimal value of the quadratic relation:
y = -7(x-6)(x+8)
To find the optimal value, we need to find the axis of symmetry, however, in order to find the axis of symmetry, we need to find the zeros first.
Now sub in the zeros into the equation for the axis of symmetry to find the axis of symmetry.
Therefore, the zeros are (6,0) and (-8,0).
Axis of Symmetry:
x = (r + s)
2
x= [6 + (-8)]
2
x = (-2)
2
x = -1
Therefore, the axis of symmetry is x = -1.
Optimal Value:
All that needs to be done now is that the axis of symmetry will be substituted into the equation to find the optimal value.
y = -7 (x - 6) (x + 8)
y = -7 [(-1) - 6) [(-1) + 8)
y = -7 (-7) (7)
y = 343
Therefore, the optimal value is y = 343.