Finding the Minimum Value of y x^2 - x

This article explores the process of finding the minimum value of the quadratic function y x^2 - x using different mathematical methods. Whether you are a beginner in algebra or a seasoned calculus student, this guide will help you understand the underlying principles and techniques used in determining the minimum value of such functions.

Introduction to the Problem

Consider the function y x^2 - x. The problem of finding its minimum value is a classic application in both calculus and algebra. Letrsquo;s explore the different ways to approach this problem and understand the steps involved.

Using Calculus

One of the most intuitive ways to solve this problem is by using calculus. The concept of derivatives plays a crucial role in finding the minimum (or maximum) value of a function.

To find the minimum value using calculus, we start by taking the derivative of the function:

Take the derivative of y x^2 - x

y' 2x - 1

Set the derivative equal to zero to find critical points:

2x - 1 0

2x 1

x 0.5

Substitute x 0.5 into the original function to find the minimum value:

y (0.5)2 - 0.5

y 0.25 - 0.5

y -0.25

Therefore, the minimum value of y x^2 - x is -0.25, which occurs at x 0.5.

Using Algebra: Completing the Square

Algebra provides another elegant method to find the minimum value of the quadratic function y x^2 - x. By completing the square, we can transform the quadratic into a form that clearly shows its minimum value and corresponding point.

Start with the function:

y x^2 - x

Complete the square:

y x^2 - x (1/2)^2 - (1/2)^2

y (x - 1/2)^2 - 1/4

The expression (x - 1/2)^2 is always non-negative and reaches its minimum value of 0 when x 1/2.

Substitute x 1/2 into the equation:

y 0 - 1/4

y -1/4

Therefore, the minimum value of y x^2 - x is -1/4, which occurs at x 1/2.

Using Algebra: Vertex Form

Another approach using algebra involves directly using the vertex form of the quadratic equation. The vertex of a parabola occurs at x -b/2a for the function y ax^2 bx c.

Identify the coefficients: a 1, b -1, and c 0.

Find the vertex using the vertex formula:

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

x 1/2

Substitute x 1/2 into the function to find the minimum value:

y (1/2)^2 - 1/2

y 1/4 - 1/2

y -1/4

Thus, the minimum value of y x^2 - x is -1/4, which occurs at x 1/2.

Conclusion

Both calculus and algebra provide effective methods to find the minimum value of the quadratic function y x^2 - x. By completing the square, finding the vertex, or using the derivative, we can determine that the minimum value is -1/4, which occurs at x 1/2. Understanding these techniques not only helps in solving specific problems but also enhances your overall mathematical skills.