Finding The Vertex Of A Quadratic Function A Step By Step Guide

In mathematics, quadratic functions play a vital role, and understanding their properties is crucial for various applications. One of the key features of a quadratic function is its vertex, which represents the point where the parabola reaches its maximum or minimum value. This article delves into a step-by-step guide on how to find the vertex of a quadratic function, providing a clear understanding of the concepts and techniques involved.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in the form:

$f(x) = ax^2 + bx + c$

where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the curve changes direction, representing either the minimum value (if a > 0) or the maximum value (if a < 0) of the function.

Methods to Find the Vertex

There are several methods to determine the vertex of a quadratic function, each with its own advantages and applications. Let's explore the most commonly used techniques:

1. Using the Vertex Formula

The vertex formula is a direct and efficient method to find the vertex of a quadratic function. For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the vertex (h, k) can be calculated using the following formulas:

h = -b / 2a k = f(h)

where h represents the x-coordinate of the vertex, and k represents the y-coordinate of the vertex. This formula provides a straightforward approach to determine the vertex coordinates without the need for completing the square or other complex methods.

To illustrate this method, let's consider an example. Suppose we have the quadratic function:

f(x) = 2x² + 8x - 3

To find the vertex, we first identify the coefficients a, b, and c:

a = 2, b = 8, c = -3

Now, we can apply the vertex formula to calculate the x-coordinate h:

h = -b / 2a = -8 / (2 * 2) = -2

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

k = f(-2) = 2(-2)² + 8(-2) - 3 = -11

Therefore, the vertex of the quadratic function is (-2, -11).

2. Completing the Square

Completing the square is another powerful technique to rewrite a quadratic function in vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola. To complete the square, we manipulate the standard form of the quadratic function by adding and subtracting a constant term to create a perfect square trinomial.

Let's demonstrate this method with an example. Consider the quadratic function:

f(x) = x² - 6x + 5

To complete the square, we first focus on the quadratic and linear terms (x² - 6x). We take half of the coefficient of the linear term (-6), which is -3, and square it, resulting in 9. We then add and subtract 9 within the expression:

f(x) = (x² - 6x + 9) - 9 + 5

Now, the expression within the parentheses is a perfect square trinomial, which can be factored as:

f(x) = (x - 3)² - 4

This is the vertex form of the quadratic function, where a = 1, h = 3, and k = -4. Therefore, the vertex of the parabola is (3, -4).

3. Using 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 x-coordinate of the vertex lies on the axis of symmetry. For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the equation of the axis of symmetry is:

x = -b / 2a

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

Let's illustrate this method with an example. Consider the quadratic function:

f(x) = -3x² + 12x - 8

To find the vertex, we first determine the axis of symmetry:

x = -b / 2a = -12 / (2 * -3) = 2

This means the axis of symmetry is the vertical line x = 2. Now, we substitute x = 2 into the function to find the y-coordinate of the vertex:

f(2) = -3(2)² + 12(2) - 8 = 4

Therefore, the vertex of the quadratic function is (2, 4).

Applying the Concepts: Solving the Given Problem

Now, let's apply these methods to solve the given problem:

Find the vertex of the quadratic function:

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

First, we need to expand the function to bring it into the standard form:

f(x) = x² - 2x - 8

Now, we can use the vertex formula to find the vertex. Identify the coefficients:

a = 1, b = -2, c = -8

Calculate the x-coordinate h:

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

Calculate the y-coordinate k:

k = f(1) = (1)² - 2(1) - 8 = -9

Therefore, the vertex of the quadratic function is (1, -9), which corresponds to option A.

Conclusion

Finding the vertex of a quadratic function is a fundamental concept in mathematics with diverse applications. This article has explored three primary methods to determine the vertex: the vertex formula, completing the square, and using the axis of symmetry. Each method offers a unique approach to solving the problem, providing flexibility and a deeper understanding of quadratic functions. By mastering these techniques, you can confidently find the vertex of any quadratic function and apply this knowledge to various mathematical and real-world scenarios.

• Find the vertex
• Quadratic function
• Vertex formula
• Completing the square
• Axis of symmetry
• Parabola
• Maximum value
• Minimum value
• Vertex form
• Standard form

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