How to Find the Vertex of a Quadratic Equation in Intercept Form?

The vertex of a quadratic equation in intercept form can be found by first determining the x-coordinate of the vertex (which lies on the axis of symmetry) and then substituting that x-value back into the equation to find the corresponding y-coordinate.

Here's a breakdown of the process:

Understanding Intercept Form

A quadratic equation in intercept form is written as:

f(x) = a(x - p)(x - q)

Where:

• a is a constant coefficient
• p and q are the x-intercepts (roots) of the parabola.

Steps to Find the Vertex

1. Find the x-coordinate of the vertex (axis of symmetry): Since the parabola is symmetrical, the x-coordinate of the vertex is exactly halfway between the two x-intercepts. Calculate it using the formula:

   x_vertex = (p + q) / 2

2. Find the y-coordinate of the vertex: Substitute the x-coordinate you just found back into the original quadratic equation in intercept form:

   y_vertex = a(x_vertex - p)(x_vertex - q)

   This will give you the y-coordinate of the vertex.

3. The vertex: The vertex is then represented as the point (x_vertex, y_vertex).

Example

Let's say you have the quadratic equation: f(x) = 2(x - 1)(x + 3)

1. Find x_vertex: p = 1 and q = -3. So, x_vertex = (1 + (-3)) / 2 = -1

2. Find y_vertex: y_vertex = 2((-1) - 1)((-1) + 3) = 2(-2)(2) = -8

3. The vertex: The vertex of the parabola is (-1, -8).

Key Takeaway

The x-coordinate of the vertex is the average of the x-intercepts p and q, and the y-coordinate is found by plugging this x-value back into the original equation.