Finding The Vertex: Quadratic Equations Explained

by Alex Johnson 50 views

Hey there, math enthusiasts! Let's dive into the world of quadratic equations and uncover a neat trick for identifying the vertex of a parabola. The question we're tackling is: Which equation has a vertex at (1, -6)? We'll break down the concepts, explore the given options, and arrive at the correct answer. Get ready to flex those math muscles!

Understanding the Vertex: The Heart of a Parabola

First things first, what exactly is a vertex? Imagine a parabola, that graceful U-shaped curve. The vertex is the most crucial point on that curve – it's the tip of the U, the point where the curve changes direction. Think of it as the peak (if the parabola opens downwards) or the valley (if it opens upwards). Knowing the vertex gives us valuable information about the parabola's position and its minimum or maximum value.

Now, let's talk about the standard form of a quadratic equation: y = ax² + bx + c. The vertex form is even more helpful for locating the vertex: y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. See how convenient that is? The vertex is practically spelled out for you!

To find the vertex using the standard form, we can use a handy formula: h = -b / 2a. Once we find 'h', we can substitute it back into the equation to find 'k'. The (h, k) values give us the vertex coordinates. Remember, the 'a' value determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and how 'wide' or 'narrow' the parabola is.

The Vertex and Its Significance

The vertex is more than just a point on a graph; it unlocks a deeper understanding of quadratic functions. Let's consider some key aspects of the vertex:

  • Minimum or Maximum Value: The y-coordinate of the vertex represents either the minimum or maximum value of the function. If the parabola opens upward, the vertex is the minimum point; if it opens downward, the vertex is the maximum point.
  • Axis of Symmetry: The vertical line passing through the vertex is the axis of symmetry. The parabola is symmetrical about this line.
  • Real-world Applications: Quadratic functions and their vertices model many real-world phenomena, from the trajectory of a ball thrown in the air to the design of suspension bridges. Understanding the vertex helps us analyze and optimize these situations.

Analyzing the Given Equations

Alright, let's get our hands dirty and examine the provided equations. We're looking for the equation whose graph has a vertex at (1, -6). We'll use the vertex formula or complete the square to identify the correct equation. Remember, our goal is to match the vertex's coordinates.

Option A: y = 3x² + 6x - 3

To find the vertex, we'll use the formula h = -b / 2a. In this equation, a = 3, b = 6, and c = -3. So, h = -6 / (2 * 3) = -1. Now, substitute x = -1 back into the equation to find k: y = 3(-1)² + 6(-1) - 3 = 3 - 6 - 3 = -6. Thus, the vertex for option A is (-1, -6).

Option B: y = 3x² - 6x - 3

Here, a = 3, b = -6, and c = -3. Calculating 'h', we get h = -(-6) / (2 * 3) = 1. Now, substitute x = 1 into the equation: y = 3(1)² - 6(1) - 3 = 3 - 6 - 3 = -6. The vertex for option B is (1, -6).

Option C: y = 3x² - 8x - 1

For this equation, a = 3, b = -8, and c = -1. Calculating 'h', we have h = -(-8) / (2 * 3) = 4/3. We can see immediately that the x-coordinate of the vertex is not 1, so we can stop here. This is not the correct option.

Option D: y = 3x² - 3x - 6

Here, a = 3, b = -3, and c = -6. Calculating 'h', we get h = -(-3) / (2 * 3) = 1/2. Again, we can see that the x-coordinate is not 1, so this is not the right choice.

The Answer and the Reasoning

After analyzing each equation, we found that only option B: y = 3x² - 6x - 3 has a vertex at (1, -6). Therefore, that's our answer! We used the vertex formula (h = -b / 2a) to find the x-coordinate of the vertex and then substituted this value back into the equation to find the y-coordinate. Another way is by completing the square to change to vertex form. By finding the x and y coordinates, we can determine the exact vertex location.

By following these steps, you can confidently identify the vertex of any quadratic equation! Remember, practice makes perfect. Keep exploring these concepts, and you'll become a vertex-finding pro in no time.

Key Takeaways and Tips

Let's summarize the key takeaways:

  • Vertex Importance: The vertex is the most crucial point on a parabola.
  • Standard Form vs. Vertex Form: Understand the standard form (y = ax² + bx + c) and the vertex form (y = a(x - h)² + k).
  • Vertex Formula: Use the formula h = -b / 2a to find the x-coordinate of the vertex.
  • Substitute to Find k: Substitute the 'h' value back into the equation to find the y-coordinate ('k') of the vertex.
  • Check the Options: Systematically analyze each equation to find the one with the correct vertex coordinates.

Final Thoughts

I hope this explanation has clarified the process of finding the vertex of a quadratic equation. Keep practicing, and you'll become proficient in no time. Mathematics is all about understanding the concepts and applying them step by step. Good luck, and happy solving!

For further exploration, you might find the following resources helpful:

Keep up the great work, and happy learning!"