Finding The Vertex: A Guide To The Function F(x) = X² + 4x + 3

by ADMIN 63 views

Hey guys! Let's dive into finding the vertex of a quadratic function. In math, specifically algebra, the vertex is super important. It's the highest or lowest point on the graph of a parabola, which is the U-shaped curve that represents a quadratic function. Knowing how to find the vertex is like having a key to unlock a lot of information about the function, such as its maximum or minimum value and the axis of symmetry. Today, we're going to break down how to find the vertex for the function f(x) = x² + 4x + 3. It's easier than you might think, and once you get the hang of it, you'll be able to find the vertex of any quadratic function.

First off, let's talk about the standard form of a quadratic equation. This is written as f(x) = ax² + bx + c. In our example, f(x) = x² + 4x + 3, we can see that a = 1, b = 4, and c = 3. The value of 'a' tells us whether the parabola opens upwards (if a > 0) or downwards (if a < 0). In our case, since a = 1, which is greater than 0, our parabola opens upwards, meaning the vertex will be the minimum point on the graph. The coefficients b and c also play a role in determining the position of the vertex. Understanding these basics is crucial because they set the foundation for understanding what the vertex means within the context of the function's behavior. We'll use the values of a, b, and c to find the vertex coordinates which will provide us with a complete description of the parabola's behavior. The vertex is not only a point on the graph; it is a center of symmetry, indicating the spot where the function changes direction. Mastering these concepts will allow you to analyze a variety of functions, making complex problems feel much more manageable. Trust me, once you grasp this, it will become second nature! So, let's get started. Get ready to have fun with math!

Finding the vertex is a crucial step in understanding the behavior of the quadratic function. The vertex gives us key insights, such as where the function reaches its minimum or maximum value. This information is incredibly useful in various real-world applications, such as in physics, engineering, and economics. For instance, in physics, it can help determine the maximum height of a projectile. In economics, it can help find the point where a cost function is minimized or a revenue function is maximized. Let's start with a couple of different methods to get you to the vertex. The first method is using the vertex formula. This formula gives us a direct way to find the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it back into the original equation to find the y-coordinate. Another method is completing the square, which involves rewriting the quadratic function in vertex form. Let's try the vertex formula first. The formula to find the x-coordinate of the vertex is x = -b / 2a. In our function, f(x) = x² + 4x + 3, a = 1 and b = 4. So, x = -4 / (2 * 1) = -2. This means that the x-coordinate of the vertex is -2. Now, to find the y-coordinate, we'll substitute x = -2 back into the original function: f(-2) = (-2)² + 4(-2) + 3 = 4 - 8 + 3 = -1. Therefore, the vertex of the function f(x) = x² + 4x + 3 is (-2, -1). Awesome, right? Let's also go over the second method, completing the square, because it will help you understand the concept even better!

Method 1: Using the Vertex Formula

Alright, let's get into the specifics of finding the vertex of the function f(x) = x² + 4x + 3 using the vertex formula. This method is pretty straightforward and gets you the answer directly. As we mentioned earlier, the vertex formula gives us a way to find the x-coordinate of the vertex. The formula is x = -b / 2a. Remember, in our equation, a is the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term. In our case, a = 1, b = 4, and c = 3. This is essential for the function f(x) = x² + 4x + 3. To use the formula, we substitute the values of a and b into it. So we get x = -4 / (2 * 1). Doing the math, we find that x = -2. This tells us that the x-coordinate of the vertex is -2. But we're not done yet; we still need to find the y-coordinate. To do this, we plug the x-coordinate (-2) back into the original equation, f(x) = x² + 4x + 3. So, we calculate f(-2) = (-2)² + 4(-2) + 3. That's f(-2) = 4 - 8 + 3, which simplifies to f(-2) = -1. So, the y-coordinate of the vertex is -1. Putting it all together, the vertex of the function f(x) = x² + 4x + 3 is at the point (-2, -1). This means the parabola's lowest point is at (-2, -1). Now, the vertex formula is super helpful because it's a quick and efficient way to find the vertex. But, you know, it’s not the only way. There is another method, which will give us a better understanding of quadratic functions.

Step-by-Step Guide for the Vertex Formula

  1. Identify a, b, and c: First, recognize the coefficients from the quadratic equation f(x) = ax² + bx + c. For f(x) = x² + 4x + 3, we have a = 1, b = 4, and c = 3. This is the first and most important step in finding the vertex of the function. Correctly identifying these values is crucial, because any error will impact the final result.
  2. Calculate the x-coordinate: Use the vertex formula: x = -b / 2a. Substitute the values of a and b that you just found. For our function, x = -4 / (2 * 1) = -2. This x value is the x-coordinate of the vertex. It tells us where the parabola's peak or valley is located on the x-axis. Keep this value, as we'll need it soon to find the y-coordinate.
  3. Calculate the y-coordinate: Substitute the x-coordinate you found in Step 2 back into the original function. For f(x) = x² + 4x + 3, calculate f(-2) = (-2)² + 4(-2) + 3 = -1. This gives you the y-coordinate of the vertex. It represents the height or depth of the parabola at its peak or valley. This calculation provides the other half of the vertex coordinates.
  4. Write the Vertex as a Coordinate: The vertex is written as a point (x, y). In our example, the vertex is (-2, -1). This coordinate point is the key to understanding the parabola's position and orientation on the graph. It also provides important information, such as the minimum or maximum value of the function.

Method 2: Completing the Square

Let's switch gears and learn a different approach to find the vertex: completing the square. This method not only helps you find the vertex but also transforms the quadratic equation into vertex form, which is super useful. The vertex form of a quadratic equation is f(x) = a(x - h)² + k, where (h, k) is the vertex. So, by completing the square, we are essentially rewriting our equation into this form, making it easy to spot the vertex directly. Starting with our original function, f(x) = x² + 4x + 3, let's walk through the steps to complete the square. First, we'll focus on the x² + 4x part of the equation and leave the constant term (+3) aside for a moment. To complete the square, we need to add and subtract a specific number to make the x² + 4x part a perfect square trinomial. To find this number, we take the coefficient of the x term (which is 4), divide it by 2 (giving us 2), and then square the result (2² = 4). So, we'll add and subtract 4 inside the equation. So, adding and subtracting 4 doesn't change the equation; it just changes the way we write it.

Now our equation looks like this: f(x) = x² + 4x + 4 - 4 + 3. Notice that x² + 4x + 4 is a perfect square trinomial because it can be factored into (x + 2)². Rewriting the equation, we get f(x) = (x + 2)² - 4 + 3. Simplifying the constants, we get f(x) = (x + 2)² - 1. Now, the equation is in vertex form, f(x) = a(x - h)² + k. Comparing this to our equation, we can see that h = -2 and k = -1. Therefore, the vertex of the function is (-2, -1). See? The answer matches with what we found using the vertex formula. Completing the square is awesome because it shows you how the quadratic equation is built, and it directly gives you the vertex. Let's go through the steps.

Step-by-Step Guide for Completing the Square

  1. Isolate the x² and x terms: Start with the function f(x) = x² + 4x + 3. Our goal is to work with the terms involving x to create a perfect square trinomial. So we focus on the part x² + 4x. Leave the constant term (+3) on the side for now. This step prepares the function for the transformation by isolating the variables we need to manipulate.
  2. Complete the square: To complete the square, take the coefficient of the x term, divide it by 2, and square the result. The coefficient of our x term is 4. Divide by 2 to get 2, and then square to get 4. Add and subtract this value inside the equation: f(x) = x² + 4x + 4 - 4 + 3. This crucial step ensures that we create a perfect square trinomial without changing the equation's value.
  3. Factor the perfect square trinomial: The expression x² + 4x + 4 is a perfect square trinomial, which can be factored into (x + 2)². Rewrite the equation as f(x) = (x + 2)² - 4 + 3. This simplification reveals the squared term, which is the heart of the vertex form.
  4. Simplify and write in vertex form: Simplify the remaining constant terms: -4 + 3 = -1. This gives us the vertex form of the equation: f(x) = (x + 2)² - 1. Compare this to the standard vertex form f(x) = a(x - h)² + k. From this, you can identify the vertex as (-2, -1), because h = -2 and k = -1. The vertex form directly reveals the vertex coordinates.

Conclusion

Alright, guys! We've successfully found the vertex of the quadratic function f(x) = x² + 4x + 3 using two different methods: the vertex formula and completing the square. Both methods lead us to the same answer: the vertex is at the point (-2, -1). Remember, the vertex formula is super handy for a quick solution, while completing the square gives you a deeper understanding of the equation. So, now you're well-equipped to tackle any quadratic function! You can calculate the vertex and understand its importance in understanding the behavior of quadratic functions. Remember that practice is key, so keep working through different examples and applying these methods. Have fun exploring the wonderful world of math, and remember: it is all about having fun. See you next time, and happy calculating!