How can I determine the minimum and maximum values of a quadratic function?

by

Finding the Minimum and Maximum of a Quadratic Function

A quadratic function is typically expressed in the standard form: f(x) = ax² + bx + c, where a, b, and c are constants. The shape of the graph of a quadratic function is a parabola.

To find the minimum or maximum value of this function, follow these steps:

Step 1: Identify the Coefficient ‘a’

The first step is to determine the value of the coefficient a in the quadratic equation. This value dictates whether the parabola opens upwards or downwards:

  • If a > 0, the parabola opens upwards, and the function has a minimum value.
  • If a < 0, the parabola opens downwards, and the function has a maximum value.

Step 2: Find the Vertex

The next step is to calculate the x-coordinate of the vertex using the formula:

x = -b / (2a)

Once you have the x-coordinate, substitute this value back into the original function to find the corresponding y-coordinate:

y = f(-b / (2a))

This gives you the minimum or maximum value of the function.

Step 3: Interpret the Results

After determining the vertex:

  • If a > 0, the vertex represents the minimum value of the function.
  • If a < 0, the vertex represents the maximum value of the function.

Example

Let’s say we have a quadratic function: f(x) = 2x² – 4x + 1. Here, the coefficient a is 2 (which is > 0), so we know it has a minimum value.

1. Calculate the x-coordinate of the vertex:

x = -(-4) / (2 * 2) = 4 / 4 = 1

2. Substitute x back into the function:

y = f(1) = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1

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

Conclusion

Finding the minimum or maximum of a quadratic function involves understanding the direction in which the parabola opens, calculating the vertex using the given formulas, and interpreting the results based on the sign of a. With practice, these steps become straightforward!

Leave a Comment