Finding The Vertex Of Y=(x-1)^2-5 A Step-by-Step Guide

by ADMIN 55 views

Hey guys! In this article, we're diving into the world of quadratic equations and their graphical representations – parabolas. Specifically, we're going to tackle the problem of finding the vertex of a parabola. The vertex is a crucial point, as it represents either the minimum or maximum value of the quadratic function. We'll use a clear, step-by-step approach to make sure you understand exactly how to find it.

Understanding Quadratic Equations and Parabolas

Before we jump into the problem, let's quickly recap what quadratic equations and parabolas are all about. A quadratic equation is an equation of the form y = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. When we graph a quadratic equation, we get a U-shaped curve called a parabola. This curve can open upwards (if a > 0) or downwards (if a < 0).

The vertex of the parabola is the point where the parabola changes direction. If the parabola opens upwards, the vertex is the lowest point on the graph, representing the minimum value of the function. Conversely, if the parabola opens downwards, the vertex is the highest point, representing the maximum value. Understanding this fundamental concept is key to identifying and interpreting the vertex correctly.

Key Features of a Parabola:

  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • Vertex: The point where the parabola changes direction (minimum or maximum point).
  • X-intercepts (Roots or Zeros): The points where the parabola intersects the x-axis (where y = 0).
  • Y-intercept: The point where the parabola intersects the y-axis (where x = 0).

For many students, the vertex form of a quadratic equation, y = a(x – h)² + k, offers the most direct route to finding the vertex. This is because the vertex is simply the point (h, k). The value of ‘a’ still dictates whether the parabola opens upwards or downwards, influencing whether the vertex is a minimum or maximum point. Transforming a quadratic equation into vertex form often involves completing the square, a technique that might initially seem complex but becomes straightforward with practice.

The ability to quickly identify the vertex is not just an academic exercise; it has practical applications across various fields. In physics, for example, understanding the trajectory of a projectile involves identifying the vertex of the parabolic path to determine the maximum height reached. Similarly, in business and economics, quadratic models can be used to represent cost or profit functions, where the vertex indicates the point of minimum cost or maximum profit. Thus, mastering the techniques to find the vertex can provide valuable insights and solutions in real-world scenarios, highlighting the importance of this concept in mathematics and beyond.

Our Problem: y = (x - 1)² - 5

Okay, let's get down to the specific problem. We're given the equation y = (x - 1)² - 5. Notice that this equation is already in a special form called vertex form. This makes our job much easier! The vertex form of a quadratic equation is given by:

y = a(x - h)² + k

where (h, k) is the vertex of the parabola. Comparing our equation, y = (x - 1)² - 5, to the vertex form, we can easily identify the values of h and k.

In this form, the transformation of the standard quadratic equation is clear. The ‘h’ value represents a horizontal shift, and the ‘k’ value represents a vertical shift. The squared term (x – h)² shows how the parabola opens around the vertex, and the ‘a’ term (which is 1 in our case, implying a standard width) determines whether the parabola opens upwards or downwards. Thus, recognizing and utilizing the vertex form simplifies the process of understanding and graphing quadratic functions.

The implications of the vertex are vast, extending from theoretical mathematics to practical applications. For instance, in optimization problems, the vertex represents the maximum or minimum value of a function, a critical point in decision-making across various disciplines. Architects and engineers use parabolic shapes in bridge design and construction because of their structural properties, with the vertex playing a key role in load distribution and stability. Similarly, in the field of optics, parabolic mirrors are used to focus light to a single point, utilizing the unique properties of the parabola’s geometry. Therefore, mastering the identification and interpretation of the vertex not only enhances mathematical proficiency but also provides a foundation for problem-solving in a multitude of real-world contexts.

Identifying the Vertex

Now, let's pinpoint the vertex. By comparing y = (x - 1)² - 5 with y = a(x - h)² + k, we can see that:

  • h = 1
  • k = -5

Therefore, the vertex of the parabola is (h, k) = (1, -5).

It’s worth emphasizing how the signs in the vertex form affect the vertex’s coordinates. The equation has (x – h), implying that the x-coordinate of the vertex is the value that makes the expression inside the parenthesis zero. Thus, (x – 1) becomes zero when x is 1, which is why the x-coordinate of the vertex is 1, not -1. The ‘k’ value, on the other hand, directly translates to the y-coordinate of the vertex, maintaining its sign. Understanding this nuance is crucial for accurate identification of the vertex from the vertex form of a quadratic equation.

Furthermore, recognizing the vertex as a critical point provides additional insights into the behavior of the quadratic function. The vertex is the point where the function changes direction, marking either a minimum or a maximum value. For parabolas opening upwards (where ‘a’ is positive), the vertex is the minimum point, and for parabolas opening downwards (where ‘a’ is negative), the vertex is the maximum point. This understanding is particularly useful in practical applications, such as optimizing business processes, designing efficient structures, or predicting the trajectory of projectiles in physics. Therefore, identifying the vertex not only provides the coordinates of a specific point on the graph but also unlocks a deeper understanding of the function’s properties and its potential applications in various fields.

The Answer

So, the vertex of the graph of y = (x - 1)² - 5 is (1, -5). That's it!

Remember, the vertex form of a quadratic equation is your best friend when it comes to finding the vertex. Just plug and chug, guys!

Additional Tips and Tricks

  • Practice Makes Perfect: The best way to master finding the vertex is to practice with various quadratic equations. Try different forms and challenge yourself to convert them to vertex form.
  • Completing the Square: If your equation isn't in vertex form, you might need to use the technique of completing the square to rewrite it. This is a powerful method for transforming quadratic equations.
  • The Vertex Formula: If completing the square feels too cumbersome, you can use the vertex formula: h = -b/2a and k = f(h), where f(x) = ax² + bx + c. This formula directly gives you the coordinates of the vertex.
  • Graphical Approach: You can also use graphing software or calculators to visually identify the vertex of the parabola. This can be a helpful way to check your work.

Another handy tip for spotting potential errors is to consider the symmetry of the parabola. Once you’ve identified the vertex, the parabola should be symmetrical around the vertical line that passes through the vertex (the axis of symmetry). If the graph you’ve sketched or the function’s behavior doesn’t exhibit this symmetry, it might indicate a mistake in your calculations or understanding. This quick check can save you time and effort in the long run by helping you catch errors early on.

Furthermore, it’s beneficial to understand how the coefficients in the quadratic equation influence the shape and position of the parabola. The ‘a’ coefficient not only determines whether the parabola opens upwards or downwards but also affects its width. A larger absolute value of ‘a’ results in a narrower parabola, while a smaller value widens it. The ‘b’ coefficient, along with ‘a’, influences the x-coordinate of the vertex, and the ‘c’ coefficient determines the y-intercept of the parabola. By grasping these relationships, you can quickly anticipate the general shape and position of the parabola, which can be invaluable for problem-solving and verification.

Conclusion

Finding the vertex of a parabola doesn't have to be intimidating. By understanding the vertex form of a quadratic equation and applying the steps we've discussed, you can confidently tackle these problems. Keep practicing, and you'll become a vertex-finding pro in no time! Remember, the vertex is a key feature of the parabola, providing valuable information about the function's minimum or maximum value and its overall behavior. So go forth and conquer those parabolas!

I hope this guide has been helpful. Let me know if you have any questions, and keep exploring the fascinating world of mathematics!