How to Find the Min and Max of a Quadratic Function

To determine the minimum or maximum of a quadratic function, you first need to understand its general form, which is:

f(x) = ax² + bx + c

In this equation, a, b, and c are constants, and the value of a helps define the shape of the parabola.

1. Identify the Coefficient a:

If a > 0, the parabola opens upwards, indicating that it has a minimum point. If a < 0, the parabola opens downwards, which means it has a maximum point.

2. Use the Vertex Formula:

The vertex of the parabola, which represents the minimum or maximum value of the function, can be found using the formula:

x = -b / (2a)

Calculate x using the values of b and a from your quadratic function.

3. Calculate the Corresponding f(x) Value:

Once you have the value of x, substitute it back into the original function to find the minimum or maximum value of the function:

f(x) = a(-b / (2a))² + b(-b / (2a)) + c

This will give you the y-coordinate of the vertex, which is the minimum (if a > 0) or maximum (if a < 0) value of the function.

Example:

For the quadratic function f(x) = 2x² – 4x + 1, here’s how you would find the minimum:

1. Identify a = 2 (which is greater than 0, so there’s a minimum).
2. Calculate x = -(-4) / (2 * 2) = 1.
3. Substitute x back into the function: f(1) = 2(1)² – 4(1) + 1 = -1.

So, the minimum value of the function is -1 at x = 1.