Finding The Vertex Of Quadratic Function F(x) = (x-6)(x+2) A Comprehensive Guide

In the realm of mathematics, quadratic functions hold a prominent position, particularly within algebra and calculus. Their characteristic U-shaped curves, known as parabolas, exhibit intriguing properties, one of which is the vertex. The vertex represents the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards. Understanding how to determine the vertex is crucial for analyzing and interpreting quadratic functions. This article delves into the process of finding the vertex of the quadratic function f(x) = (x - 6)(x + 2), providing a comprehensive explanation and step-by-step guidance.

Understanding Quadratic Functions and the Vertex

Before we embark on finding the vertex of the given quadratic function, let's establish a solid understanding of quadratic functions and the concept of the vertex. A quadratic function is a polynomial function of degree two, generally expressed in the form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the curve changes direction. It is the lowest point on the graph if the parabola opens upwards (a minimum) and the highest point if the parabola opens downwards (a maximum).

The vertex plays a pivotal role in understanding the behavior of a quadratic function. It provides information about the function's range, symmetry, and extreme values. For instance, if the vertex is at the point (h, k), then the range of the function is y ≥ k if the parabola opens upwards, and y ≤ k if the parabola opens downwards. The vertical line x = h is the axis of symmetry for the parabola, dividing it into two symmetrical halves.

Methods for Finding the Vertex

There are several methods to determine the vertex of a quadratic function. We will explore three common approaches:

1. Using the Vertex Formula: This method involves using a specific formula to directly calculate the coordinates of the vertex. The vertex formula is derived from completing the square and is a quick and efficient way to find the vertex.
2. Completing the Square: This method involves rewriting the quadratic function in vertex form, which directly reveals the coordinates of the vertex. Completing the square is a powerful technique with applications beyond finding the vertex.
3. Finding the Axis of Symmetry: This method involves finding the axis of symmetry first and then using it to determine the vertex. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

Finding the Vertex of f(x) = (x-6)(x+2)

Now, let's apply these methods to find the vertex of the quadratic function f(x) = (x - 6)(x + 2).

Method 1: Using the Vertex Formula

The vertex formula provides a direct way to calculate the coordinates of the vertex. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex, denoted as h, is given by:

h = -b / 2a

The y-coordinate of the vertex, denoted as k, is found by substituting the value of h back into the function:

k = f(h)

To use the vertex formula, we first need to expand the given function f(x) = (x - 6)(x + 2) into the standard quadratic form:

f(x) = x² + 2x - 6x - 12

f(x) = x² - 4x - 12

Now we can identify the coefficients: a = 1, b = -4, and c = -12. Applying the vertex formula, we find the x-coordinate of the vertex:

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

Next, we substitute h = 2 back into the function to find the y-coordinate of the vertex:

k = f(2) = (2)² - 4(2) - 12 = 4 - 8 - 12 = -16

Therefore, the vertex of the quadratic function f(x) = (x - 6)(x + 2) is (2, -16).

Method 2: Completing the Square

Completing the square is another powerful technique for finding the vertex. This method involves rewriting the quadratic function in vertex form:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola. To complete the square for the function f(x) = x² - 4x - 12, we follow these steps:

1. Group the x² and x terms:

   f(x) = (x² - 4x) - 12

2. Take half of the coefficient of the x term, square it, and add and subtract it inside the parentheses:

   The coefficient of the x term is -4. Half of -4 is -2, and squaring -2 gives 4. So we add and subtract 4 inside the parentheses:

   f(x) = (x² - 4x + 4 - 4) - 12

3. Rewrite the expression inside the parentheses as a squared term:

   f(x) = (x - 2)² - 4 - 12

4. Simplify:

   f(x) = (x - 2)² - 16

Now the function is in vertex form, f(x) = (x - 2)² - 16. Comparing this to the vertex form f(x) = a(x - h)² + k, we can see that h = 2 and k = -16. Therefore, the vertex is (2, -16), which confirms our result from the vertex formula method.

Method 3: Finding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. The equation of the axis of symmetry is given by x = h, where h is the x-coordinate of the vertex. For a quadratic function in the standard form f(x) = ax² + bx + c, the axis of symmetry can be found using the formula:

x = -b / 2a

This is the same formula we used to find the x-coordinate of the vertex in Method 1. Once we find the axis of symmetry, we can substitute the x-value back into the function to find the y-coordinate of the vertex.

For the function f(x) = x² - 4x - 12, we already found that a = 1 and b = -4. Therefore, the axis of symmetry is:

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

So the axis of symmetry is the line x = 2. This means the x-coordinate of the vertex is 2. To find the y-coordinate, we substitute x = 2 back into the function:

f(2) = (2)² - 4(2) - 12 = 4 - 8 - 12 = -16

Thus, the vertex is (2, -16), consistent with our previous findings.

Conclusion

In this article, we explored the concept of the vertex of a quadratic function and demonstrated three methods for finding it: the vertex formula, completing the square, and finding the axis of symmetry. Applying these methods to the quadratic function f(x) = (x - 6)(x + 2), we consistently found the vertex to be (2, -16). Understanding how to determine the vertex is essential for analyzing and interpreting quadratic functions, as it provides valuable information about the function's extreme values, range, and symmetry. By mastering these techniques, you can gain a deeper understanding of quadratic functions and their applications in various mathematical and real-world contexts. This knowledge will prove invaluable in further mathematical studies and problem-solving endeavors involving quadratic relationships. Remember, the vertex is a key feature of a parabola, unlocking insights into its behavior and properties. Whether you choose the vertex formula, completing the square, or finding the axis of symmetry, the ability to accurately determine the vertex will enhance your understanding of quadratic functions and their significance in mathematics and beyond.