The vertex of a quadratic in intercept form can be found by first determining the x-coordinate of the axis of symmetry, then substituting that value back into the original quadratic equation to find the corresponding y-coordinate.
Understanding Quadratic Intercept Form
The intercept form of a quadratic equation is given by:
f(x) = a(x - p)(x - q)
where p and q are the x-intercepts of the parabola.
Finding the Axis of Symmetry
According to the reference, “Because of symmetry, the axis of symmetry is halfway between the x-intercepts." Therefore, the x-coordinate of the axis of symmetry (and thus the x-coordinate of the vertex) can be found by averaging the x-intercepts:
x-coordinate of vertex = (p + q) / 2
Calculating the Vertex
- Identify the x-intercepts: From the equation f(x) = a(x - p)(x - q), note the values of p and q.
- Calculate the axis of symmetry: Find the average of p and q using the formula (p+q)/2. This will be the x-coordinate of your vertex.
- Substitute to find the y-coordinate: Substitute the x-coordinate of the vertex back into the original intercept form equation (f(x) = a(x - p)(x - q)) to solve for the corresponding y-coordinate.
- State the vertex: The vertex is written as the coordinate (x-coordinate, y-coordinate) you have found.
Example:
Let's say we have the quadratic equation f(x) = 2(x - 1)(x + 3).
- The x-intercepts are p = 1 and q = -3.
- The x-coordinate of the vertex is (1 + (-3)) / 2 = -1.
- To find the y-coordinate, substitute x = -1 into the original equation: f(-1) = 2(-1 - 1)(-1 + 3) = 2(-2)(2) = -8.
- Therefore, the vertex is (-1, -8).
Summary
The process for finding the vertex when the quadratic is in intercept form involves using the symmetry of the parabola. The x-coordinate of the vertex is the average of the x-intercepts and substituting that value back into the original equation will yield the y-coordinate.
