Finding The Vertex Of H(x) = X² + 6x + 5 A Step By Step Guide

by ADMIN 62 views

In the realm of mathematics, quadratic functions hold a significant position, particularly in algebra and calculus. Understanding their properties and characteristics is crucial for solving various mathematical problems. Among these characteristics, the vertex stands out as a key feature, representing the function's minimum or maximum point. This article delves into the process of finding the vertex of a quadratic function, focusing on the specific example of h(x) = x² + 6x + 5. We will explore different methods, providing a comprehensive guide for students and enthusiasts alike.

Understanding Quadratic Functions and the Vertex

Before diving into the calculation, let's establish a clear understanding of quadratic functions and the significance 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 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. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point.

The vertex of a parabola is the point where the function changes direction. It is the extremum (minimum or maximum) of the function and plays a vital role in analyzing the behavior of the quadratic function. The vertex is represented as a coordinate pair (h, k), where h is the x-coordinate (also known as the axis of symmetry) and k is the y-coordinate, representing the minimum or maximum value of the function.

For our specific function, h(x) = x² + 6x + 5, we can identify the coefficients as follows:

  • a = 1
  • b = 6
  • c = 5

Since a = 1, which is greater than 0, the parabola opens upwards, indicating that the vertex represents the minimum point of the function. Now, let's explore the methods to find this vertex.

Method 1: Completing the Square

Completing the square is a powerful algebraic technique used to rewrite a quadratic expression in a form that reveals the vertex directly. The goal is to transform the expression into the vertex form:

f(x) = a(x - h)² + k

Where (h, k) represents the vertex.

Let's apply this method to our function, h(x) = x² + 6x + 5:

  1. Isolate the x² and x terms:

    h(x) = (x² + 6x) + 5

  2. Complete the square inside the parentheses:

    To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of x is 6, so half of it is 3, and the square of 3 is 9.

    h(x) = (x² + 6x + 9 - 9) + 5

  3. Rewrite the expression inside the parentheses as a perfect square:

    h(x) = (x + 3)² - 9 + 5

  4. Simplify the expression:

    h(x) = (x + 3)² - 4

Now, the function is in vertex form. By comparing this to the general vertex form, we can identify the vertex:

  • h = -3
  • k = -4

Therefore, the vertex of the function h(x) = x² + 6x + 5 is (-3, -4).

Completing the square provides a clear and systematic way to find the vertex by transforming the quadratic function into its vertex form. This method is particularly useful when a deep understanding of the algebraic manipulation is desired.

Method 2: Using the Vertex Formula

An alternative and often quicker method to find the vertex is by using the vertex formula. This formula directly calculates the coordinates of the vertex (h, k) based on the coefficients a, b, and c of the quadratic function.

The vertex formula is given by:

  • h = -b / 2a
  • k = f(h)

Where h is the x-coordinate of the vertex, and k is the y-coordinate, which is found by substituting h back into the original function.

Let's apply this formula to our function, h(x) = x² + 6x + 5:

  1. Identify the coefficients:

    • a = 1
    • b = 6
    • c = 5

  2. Calculate the x-coordinate (h) of the vertex:

    h = -b / 2a = -6 / (2 * 1) = -3

  3. Calculate the y-coordinate (k) of the vertex:

    Substitute h = -3 into the function:

    k = h(-3) = (-3)² + 6(-3) + 5 = 9 - 18 + 5 = -4

Therefore, the vertex of the function h(x) = x² + 6x + 5 is (-3, -4), which aligns with the result obtained using the completing the square method.

The vertex formula provides a direct and efficient way to calculate the vertex, particularly when the primary goal is to find the coordinates without delving into the algebraic manipulation of completing the square. This method is widely used in various mathematical contexts due to its simplicity and speed.

Visualizing the Vertex

To further solidify our understanding, let's visualize the vertex on the graph of the function h(x) = x² + 6x + 5. As we determined earlier, the parabola opens upwards because the coefficient a is positive. The vertex (-3, -4) represents the minimum point on the parabola. The parabola is symmetric about the vertical line passing through the vertex, which is called the axis of symmetry. In this case, the axis of symmetry is the line x = -3.

By plotting the vertex and a few other points on the graph, we can sketch the parabola and visually confirm that the vertex is indeed the minimum point. This visualization reinforces the concept of the vertex as a key characteristic of the quadratic function, influencing its shape and behavior.

Applications of Finding the Vertex

Finding the vertex of a quadratic function is not just a mathematical exercise; it has practical applications in various fields, including:

  • Physics: Determining the maximum height of a projectile's trajectory.
  • Engineering: Optimizing the design of parabolic structures like bridges and antennas.
  • Economics: Finding the maximum profit or minimum cost in business models.
  • Computer Graphics: Creating smooth curves and shapes.

The ability to find the vertex allows us to solve optimization problems, analyze the behavior of quadratic systems, and make informed decisions in real-world scenarios. Understanding the significance of the vertex extends beyond the mathematical realm and empowers us to apply these concepts in diverse fields.

Conclusion

In this comprehensive guide, we explored the process of finding the vertex of the quadratic function h(x) = x² + 6x + 5. We discussed two primary methods: completing the square and using the vertex formula. Both methods led us to the same result: the vertex is located at (-3, -4). We also visualized the vertex on the graph of the parabola, reinforcing its significance as the minimum point of the function. Furthermore, we highlighted the practical applications of finding the vertex in various fields.

Mastering the techniques to find the vertex of a quadratic function is a fundamental skill in mathematics. It not only enhances our understanding of quadratic functions but also equips us with the tools to solve optimization problems and analyze real-world phenomena. Whether you prefer the algebraic approach of completing the square or the directness of the vertex formula, the ability to find the vertex empowers you to unlock the secrets of quadratic functions and their applications.

This article serves as a valuable resource for students, educators, and anyone seeking a deeper understanding of quadratic functions and the significance of the vertex. By mastering these concepts, you can confidently tackle a wide range of mathematical challenges and appreciate the elegance and power of quadratic functions in the world around us.