12. Explain how the factored form of a quadratic equation, = ( − )( − ), can be used to calculate the value of the vertex. Be sure to describe the mathematical steps involved in this process
The factored form of a quadratic equation, = ( − )( − ), contains information about the x-intercepts of the parabola. We know that the x-coordinate of the vertex of a parabola lies at the midpoint of the two x-intercepts.
Therefore, to calculate the x-coordinate of the vertex, we can use the formula:
x-coordinate of vertex = ( -b ) / 2a
where a and b are the coefficients of the quadratic equation in standard form, ax^2 + bx + c.
Once we have the x-coordinate of the vertex, we can substitute it back into the quadratic equation to find the y-coordinate of the vertex.
For example, consider the quadratic equation = (2x - 3)(x + 4).
The x-intercepts can be found by setting each factor equal to zero and solving for x:
2x - 3 = 0 or x + 4 = 0
Solving for x, we get:
x = 3/2 or x = -4
The midpoint of 3/2 and -4 is (3/2 - 4) / 2 = -5/4. Therefore, the x-coordinate of the vertex is -5/4.
Substituting this back into the quadratic equation, we get:
y = (2(-5/4) - 3)(-5/4 + 4) = 25/8
Therefore, the vertex of the parabola is (-5/4, 25/8).