Finding The Vertex Of F(x)=x²+12x A Step-by-Step Guide

Jul 14, 2025 by ADMIN 55 views

Unveiling the Vertex of a Quadratic Function f(x) = x² + 12x: A Comprehensive Guide

Finding the vertex of a quadratic function is a fundamental concept in algebra, with wide-ranging applications in various fields, from physics and engineering to economics and computer science. The vertex represents the maximum or minimum point of the parabola defined by the quadratic function, making it a crucial feature to understand its behavior. In this article, we will delve into the process of determining the vertex of the function f(x) = x² + 12x, providing a step-by-step explanation and exploring different methods to arrive at the solution. We will also discuss the significance of the vertex and its implications in real-world scenarios.

The question at hand is: What is the vertex of the function f(x) = x² + 12x? The provided options are:

A. (-6, -36) B. (-6, 0) C. (6, 0) D. (6, -36)

To solve this, we need to understand the standard form of a quadratic function and how to extract the vertex coordinates from it. The standard form of a quadratic function is given by f(x) = ax² + bx + c, where a, b, and c are constants. The x-coordinate of the vertex, often denoted as h, can be found using the formula h = -b / 2a. Once we have the x-coordinate, we can find the y-coordinate, denoted as k, by substituting h back into the original function, i.e., k = f(h). This comprehensive guide will walk you through each step, ensuring a clear understanding of the process and the underlying mathematical principles.

Understanding Quadratic Functions and Their Properties

Before diving into the specific problem, let's establish a solid understanding of quadratic functions and their key properties. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable x is 2. The general form of a quadratic function is 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 that opens either upwards or downwards depending on the sign of the coefficient a.

If a > 0, the parabola opens upwards, and the vertex represents the minimum point of the function. This means that the y-coordinate of the vertex is the smallest value the function can attain.
If a < 0, the parabola opens downwards, and the vertex represents the maximum point of the function. In this case, the y-coordinate of the vertex is the largest value the function can attain.

The vertex of a parabola is a critical point that defines the symmetry and extreme values of the quadratic function. The parabola is symmetric about a vertical line passing through the vertex, known as the axis of symmetry. This means that for any two points equidistant from the axis of symmetry, the function values are the same. The vertex is also the turning point of the parabola; the function either decreases up to the vertex and then increases (for a > 0) or increases up to the vertex and then decreases (for a < 0). Understanding these properties is essential for analyzing and interpreting quadratic functions.

Methods to Find the Vertex

There are several methods to find the vertex of a quadratic function, each with its own advantages and applications. Here, we will explore three common methods:

Using the Vertex Formula: This method involves using the formula h = -b / 2a to find the x-coordinate of the vertex and then substituting this value into the function to find the y-coordinate. This is a direct and efficient method, especially when the quadratic function is given in standard form.
Completing the Square: This method involves rewriting the quadratic function in vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex. Completing the square is a useful technique for understanding the structure of the quadratic function and can be applied even when the vertex formula is not readily available.
Using Calculus (Differentiation): This method involves finding the critical points of the function by taking the derivative and setting it equal to zero. The critical points correspond to the x-coordinates of the vertices. This method is particularly useful for finding the vertices of more complex functions beyond simple quadratics.

In this article, we will primarily focus on the vertex formula and completing the square methods, as they are most commonly used and directly applicable to the given problem.

Applying the Vertex Formula to f(x) = x² + 12x

Now, let's apply the vertex formula to find the vertex of the given function, f(x) = x² + 12x. First, we need to identify the coefficients a, b, and c in the standard form f(x) = ax² + bx + c. In this case, we have:

a = 1 (the coefficient of x²)
b = 12 (the coefficient of x)
c = 0 (there is no constant term)

Next, we use the formula h = -b / 2a to find the x-coordinate of the vertex:

h = -12 / (2 * 1) = -12 / 2 = -6

Now that we have the x-coordinate, h = -6, we can find the y-coordinate, k, by substituting h back into the function:

k = f(-6) = (-6)² + 12(-6) = 36 - 72 = -36

Therefore, the vertex of the function f(x) = x² + 12x is (-6, -36). This corresponds to option A in the given choices.

Step-by-Step Calculation Using the Vertex Formula

To further clarify the process, let's break down the calculation into a step-by-step approach:

Identify the coefficients: In the function f(x) = x² + 12x, we have a = 1, b = 12, and c = 0.
Calculate the x-coordinate (h): Use the formula h = -b / 2a. Substitute the values of a and b: h = -12 / (2 * 1) = -6.
Calculate the y-coordinate (k): Substitute the value of h into the function: k = f(-6) = (-6)² + 12(-6).
Simplify the expression: k = 36 - 72 = -36.
Write the vertex coordinates: The vertex is (h, k), which is (-6, -36).

This step-by-step approach ensures clarity and minimizes the chances of making errors during the calculation. By following these steps, you can confidently find the vertex of any quadratic function using the vertex formula.

Completing the Square Method

Another powerful method for finding the vertex of a quadratic function is completing the square. This technique involves rewriting the quadratic function in vertex form, which is f(x) = a(x - h)² + k, where (h, k) represents the vertex. Let's apply this method to f(x) = x² + 12x.

Rewrite the function: Start with f(x) = x² + 12x. To complete the square, we need to add and subtract a value that will make the expression inside the parentheses a perfect square trinomial.
Find the value to add and subtract: Take half of the coefficient of the x term (which is 12), square it, and both add and subtract it within the function. Half of 12 is 6, and 6 squared is 36. So, we add and subtract 36: f(x) = x² + 12x + 36 - 36
Form the perfect square trinomial: The first three terms now form a perfect square trinomial: f(x) = (x² + 12x + 36) - 36
Factor the perfect square trinomial: The expression inside the parentheses can be factored as (x + 6)²: f(x) = (x + 6)² - 36
Rewrite in vertex form: Now the function is in vertex form, f(x) = a(x - h)² + k. Comparing this with our result, we have a = 1, h = -6, and k = -36. Note that since we have (x + 6)², this is equivalent to (x - (-6))², so h = -6.

Thus, the vertex is (-6, -36), which confirms our result from the vertex formula method.

Detailed Steps for Completing the Square

To ensure a clear understanding of the completing the square method, let's outline the steps in detail:

Start with the quadratic function: Begin with the function in the form f(x) = ax² + bx + c. In our case, f(x) = x² + 12x.
If a ≠ 1, factor out a from the x² and x terms: In our example, a = 1, so we can skip this step.
Find the value to add and subtract: Take half of the coefficient of the x term (b/2), square it ((b/2)²), and add and subtract this value inside the function. In our case, (12/2)² = 6² = 36. So, we add and subtract 36.
Rewrite the function: f(x) = x² + 12x + 36 - 36.
Form the perfect square trinomial: Group the first three terms: f(x) = (x² + 12x + 36) - 36.
Factor the perfect square trinomial: Factor the expression inside the parentheses: f(x) = (x + 6)² - 36.
Identify the vertex: The function is now in vertex form, f(x) = a(x - h)² + k. The vertex is (h, k). In our case, the vertex is (-6, -36).

By following these detailed steps, you can effectively use the completing the square method to find the vertex of any quadratic function.

Graphical Interpretation of the Vertex

The graphical interpretation of the vertex provides a visual understanding of its significance. The function f(x) = x² + 12x represents a parabola that opens upwards because the coefficient of x² (which is 1) is positive. The vertex, (-6, -36), is the lowest point on this parabola, representing the minimum value of the function.

When we plot the graph of f(x) = x² + 12x, we observe the following:

The parabola is symmetric about the vertical line x = -6, which is the axis of symmetry.
The vertex (-6, -36) lies on the axis of symmetry.
The function values decrease as x approaches -6 from the left and increase as x moves away from -6 to the right.
The y-coordinate of the vertex, -36, is the minimum value of the function. There are no function values lower than -36.

This graphical representation reinforces the concept that the vertex is a critical point that defines the behavior of the quadratic function. Understanding the graphical interpretation helps in visualizing the function's properties and its applications in various contexts.

Visualizing the Parabola and its Vertex

Visualizing the parabola and its vertex is crucial for understanding quadratic functions. Imagine a U-shaped curve that opens upwards. The vertex is the very bottom point of this curve. For the function f(x) = x² + 12x, this bottom point is at (-6, -36). This means that the function reaches its lowest value, -36, when x is -6. As x moves away from -6 in either direction, the function values increase, creating the upward-opening parabolic shape.

The axis of symmetry, the vertical line that passes through the vertex, divides the parabola into two mirror-image halves. In this case, the axis of symmetry is the line x = -6. This symmetry is a key characteristic of parabolas and highlights the importance of the vertex in understanding the function's behavior.

Significance of the Vertex in Real-World Applications

The vertex of a quadratic function is not just a mathematical concept; it has significant implications in various real-world applications. Quadratic functions are used to model a wide range of phenomena, from projectile motion in physics to optimization problems in economics and engineering. The vertex often represents the optimal solution or the extreme point in these scenarios.

Here are some examples of how the vertex is used in real-world applications:

Projectile Motion: In physics, the trajectory of a projectile (e.g., a ball thrown in the air) can be modeled by a quadratic function. The vertex represents the highest point the projectile reaches before descending. Knowing the vertex helps determine the maximum height and range of the projectile.
Optimization Problems: In business and economics, quadratic functions are used to model cost, revenue, and profit functions. The vertex represents the point where the function reaches its maximum or minimum value. For example, a company might use a quadratic function to model its profit as a function of production level. The vertex would then represent the production level that maximizes profit.
Engineering Design: Engineers use quadratic functions to design various structures, such as bridges and arches. The vertex plays a crucial role in determining the stability and load-bearing capacity of these structures. For instance, the shape of a parabolic arch is designed to distribute weight evenly, and the vertex represents the point of maximum stress.
Curve Fitting and Regression Analysis: In statistics and data analysis, quadratic functions are used to fit curves to data points. The vertex can provide valuable information about the trend and behavior of the data. For example, a quadratic regression model might be used to analyze the relationship between advertising expenditure and sales revenue. The vertex would then represent the point where the sales revenue is maximized.

Practical Examples of Vertex Applications

To further illustrate the significance of the vertex, let's consider some practical examples:

Throwing a ball: When you throw a ball, its path follows a parabolic trajectory. The vertex represents the highest point the ball reaches before it starts to descend. Knowing the vertex can help you determine how far the ball will travel and how high it will go.
Maximizing profit: A company can use a quadratic function to model its profit as a function of the price of its product. The vertex represents the price that will generate the maximum profit. By finding the vertex, the company can set the optimal price to maximize its earnings.
Designing a bridge: Engineers use parabolic shapes in bridge design because they distribute weight evenly. The vertex of the parabola is a critical point that determines the bridge's stability. The location and load-bearing capacity of the bridge are directly related to the vertex of the parabolic arch.

In each of these examples, the vertex provides valuable information that helps us understand and optimize the situation. By understanding the properties of quadratic functions and the significance of the vertex, we can apply these concepts to solve a wide range of real-world problems.

Conclusion: Mastering the Vertex of Quadratic Functions

In conclusion, finding the vertex of the quadratic function f(x) = x² + 12x is a fundamental mathematical exercise with far-reaching implications. We have demonstrated two primary methods for determining the vertex: the vertex formula and completing the square. Both methods lead us to the same result: the vertex is located at (-6, -36). This point represents the minimum value of the function and is a crucial feature for understanding the behavior of the parabola.

Throughout this article, we have explored the following key concepts:

The definition and properties of quadratic functions.
The significance of the vertex as the minimum or maximum point of a parabola.
The vertex formula (h = -b / 2a) and its application.
The method of completing the square and its connection to the vertex form of a quadratic function.
The graphical interpretation of the vertex and the parabola.
Real-world applications of the vertex in fields such as physics, economics, and engineering.

By mastering these concepts, you will be well-equipped to analyze and solve a wide range of problems involving quadratic functions. The ability to find the vertex is not only essential for mathematical problem-solving but also provides valuable insights into real-world phenomena modeled by quadratic functions. Whether you are a student, engineer, or data analyst, understanding the vertex will enhance your problem-solving capabilities and deepen your understanding of mathematical principles.

Final Thoughts on Vertex Applications

The journey to understanding and applying the vertex of quadratic functions is a testament to the power and elegance of mathematics. As we've explored, the vertex is more than just a point on a graph; it's a key to unlocking the behavior of quadratic functions and their applications in the real world. From modeling projectile motion to optimizing business profits, the vertex provides critical insights that can inform decisions and drive innovation.

By continuing to explore and apply these concepts, you'll not only strengthen your mathematical skills but also gain a deeper appreciation for the role of mathematics in shaping our understanding of the world around us. So, embrace the challenge, delve into the intricacies of quadratic functions, and unlock the power of the vertex in your own pursuits.