Finding the Least Value of a Quadratic Function: A Practical Guide

Often, in mathematics, we need to determine the least value of a given function. This article focuses on finding the least value of the quadratic function f(x) x2 - 6x. We will explore the process of rewriting the function in standard form, utilizing the vertex formula, and ultimately determining the minimum value of the function.

Revising the Given Function

The initial function provided is:

f(x) x(x - 6)

Firstly, let's rewrite this function in its standard form, which is a quadratic equation in the form of:

ax2 bx c

Step 1: Expansion

We can expand the given function to:

f(x) x2 - 6x

However, we see that it can be further simplified as:

f(x) x2 - 4x - 12

Identifying the Coefficients

For a quadratic function in the standard form ax2 bx c, the coefficients are:

a 1 b -4 c -12

Vertex Formula and Plotting the Parabola

For the parabola defined by the quadratic function, we need to locate the vertex. The x-coordinate of the vertex is calculated using the vertex formula:

Vertex Formula

x -frac{b}{2a}

Substitute the coefficients of our equation:

x -frac{-4}{2 cdot 1} frac{4}{2} 2

Naturally, the y-coordinate (or the value of the function) at this x-value gives us the minimum value of the function. So, we substitute x 2 back into the original equation:

f(2) 2^2 - 4 cdot 2 - 12 4 - 8 - 12 -16

Therefore, the least value of the function f(x) is -16.

Graphical Interpretation

A quadratic function of the form ax2 bx c can also be plotted. For a parabola with a positive coefficient of x2, the graph opens upwards, and its vertex represents the minimum point. Since this function has roots at x -2 and x 6, the vertex occurs at the midway point, which is:

x frac{-2 6}{2} 2

Substituting x 2 into the function to find the value at the vertex:

f(2) 22 - 4 cdot 2 - 12 4 - 8 - 12 -16

Hence, the vertex at (2, -16) is the graph's minimum point.

Conclusion

Thus, by employing the vertex formula and some basic algebraic manipulation, we find that the least value of the quadratic function f(x) x2 - 6x is:

boxed{-16}

Related Quadratic Function Concepts

Understanding this process is essential for any student of algebra and calculus. Here are a few related quadratic function concepts to explore further:

Finding the roots and x-intercepts of a quadratic function. Using the discriminant to determine the nature of the roots. Graphing quadratic functions and identifying their vertex and axis of symmetry.

By delving deeper into these concepts, you will enhance your problem-solving skills and gain a deeper understanding of quadratic functions.