Finding The Vertex Of Quadratic Function F(x) = (x - 4)(x + 2)

In mathematics, quadratic functions play a crucial role, appearing in various applications from physics to engineering. One of the key features of a quadratic function is its vertex, which represents the minimum or maximum point of the parabola. Understanding how to find the vertex is essential for analyzing and interpreting quadratic functions. This article will guide you through the process of finding the vertex of a quadratic function, using the example of f(x) = (x - 4)(x + 2).

Understanding Quadratic Functions and Their 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 U-shaped curve. The vertex of the parabola is the point where the parabola changes direction. If a > 0, the parabola opens upwards, and the vertex represents the minimum point. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point.

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) are the coordinates of the vertex. This form is particularly useful because it directly reveals the vertex. However, quadratic functions are often given in standard form (f(x) = ax² + bx + c) or factored form (f(x) = a(x - r₁)(x - r₂)). Therefore, we need methods to convert these forms to vertex form or to find the vertex directly.

Methods to Find the Vertex

There are several methods to find the vertex of a quadratic function, including:

1. Converting to Vertex Form: This involves completing the square to rewrite the quadratic function in the form f(x) = a(x - h)² + k.
2. Using the Vertex Formula: The x-coordinate of the vertex can be found using the formula h = -b / 2a, and the y-coordinate can be found by substituting h back into the function, k = f(h).
3. Finding the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, and its equation is x = -b / 2a. The vertex lies on this line.
4. Using the Factored Form: If the quadratic function is given in factored form, f(x) = a(x - r₁)(x - r₂), the x-coordinate of the vertex is the average of the roots, h = (r₁ + r₂) / 2.

Finding the Vertex of f(x) = (x - 4)(x + 2)

Let's apply these methods to find the vertex of the given quadratic function, f(x) = (x - 4)(x + 2). This function is given in factored form, which makes it convenient to use the method involving the roots.

1. Expanding to Standard Form

First, we expand the factored form to obtain the standard form of the quadratic function. This will help us identify the coefficients a, b, and c.

f(x) = (x - 4)(x + 2)

f(x) = x² + 2x - 4x - 8

f(x) = x² - 2x - 8

Now, we have the quadratic function in the standard form f(x) = ax² + bx + c, where a = 1, b = -2, and c = -8.

2. Using the Vertex Formula

The vertex formula provides a direct way to find the coordinates of the vertex. The x-coordinate h is given by:

h = -b / 2a

Substituting the values of a and b, we get:

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

h = 2 / 2

h = 1

Now, we find the y-coordinate k by substituting h = 1 into the function f(x):

k = f(1)

k = (1)² - 2(1) - 8

k = 1 - 2 - 8

k = -9

Thus, the vertex of the quadratic function is (1, -9).

3. Using the Factored Form

Alternatively, we can use the factored form f(x) = (x - 4)(x + 2) to find the vertex. The roots of the quadratic function are the values of x that make f(x) = 0. Setting each factor to zero, we find the roots:

x - 4 = 0 => x = 4

x + 2 = 0 => x = -2

The roots are r₁ = 4 and r₂ = -2. The x-coordinate of the vertex is the average of the roots:

h = (r₁ + r₂) / 2

h = (4 + (-2)) / 2

h = 2 / 2

h = 1

This matches the x-coordinate we found using the vertex formula. Now, we find the y-coordinate by substituting h = 1 into f(x):

k = f(1)

k = (1 - 4)(1 + 2)

k = (-3)(3)

k = -9

Again, we find that the vertex is (1, -9).

4. Completing the Square (Converting to Vertex Form)

Another method is to complete the square to convert the quadratic function to vertex form. Starting with the standard form f(x) = x² - 2x - 8, we complete the square as follows:

f(x) = (x² - 2x) - 8

To complete the square, we take half of the coefficient of the x term, square it, and add and subtract it within the parentheses. The coefficient of x is -2, so half of it is -1, and squaring it gives 1.

f(x) = (x² - 2x + 1 - 1) - 8

f(x) = (x² - 2x + 1) - 1 - 8

f(x) = (x - 1)² - 9

Now, the quadratic function is in vertex form f(x) = a(x - h)² + k, where a = 1, h = 1, and k = -9. Therefore, the vertex is (1, -9), which confirms our previous results.

Conclusion

In conclusion, we have demonstrated multiple methods to find the vertex of the quadratic function f(x) = (x - 4)(x + 2). Whether using the vertex formula, the factored form, or completing the square, the vertex is consistently found to be (1, -9). This point represents the minimum value of the function, as the parabola opens upwards (a > 0). Understanding these methods is crucial for analyzing quadratic functions and their applications in various fields. By mastering these techniques, you can confidently tackle a wide range of problems involving quadratic functions and their vertices.

Therefore, the correct answer is C. (1, -9).

Choosing the Correct Answer

Based on our calculations using all methods, we have consistently found the vertex to be (1, -9). Therefore, the correct answer is:

C. (1, -9)

This comprehensive guide should help you understand how to find the vertex of a quadratic function, particularly when it is given in factored form. Remember to practice these methods to build your proficiency and confidence in solving related problems. Whether you're a student learning algebra or someone applying these concepts in a professional setting, a strong understanding of quadratic functions and their vertices is invaluable.