Question

12. Explain how the factored form of a quadratic equation, Y=a ( X−S )( X− T), can be used to calculate the value of the vertex. Be sure to describe the mathematical steps involved in this process

Answer 1

The factored form of a quadratic equation is helpful in determining the vertex of the parabola because it allows us to easily identify the x-coordinates of the two points where the parabola intersects the x-axis. These two x-intercepts, S and T, are equidistant from the vertex.

To calculate the x-coordinate of the vertex, we simply need to find the midpoint of S and T. This can be done using the formula x = (S + T)/2.

Once we have the x-coordinate of the vertex, we can substitute it into the original equation to find the y-coordinate. This will give us the coordinates of the vertex (x, y).

For example, if we have the factored form Y = 2 (X−3)(X−7), we know that the x-intercepts are at X=3 and X=7. Therefore, the x-coordinate of the vertex is x = (3 + 7)/2 = 5.

To find the y-coordinate of the vertex, we can substitute x = 5 into the original equation:

Y = 2 (5−3)(5−7)

Y = 2 (2)(-2)

Y = -8

Therefore, the vertex of the parabola is located at (5, -8).