Finding The Vertex Coordinates Of F(x) = -2x² + 4x + 9 A Comprehensive Guide

Hey guys! Let's dive into a super important concept in math: finding the vertex of a quadratic function. Specifically, we're going to break down how to determine the vertex coordinates of the function f(x) = -2x² + 4x + 9. This is a classic problem, and mastering it will seriously level up your algebra game. So, grab your pencils, and let's get started!

Understanding Quadratic Functions and the Vertex

Before we jump into the nitty-gritty, let’s make sure we're all on the same page about quadratic functions. A quadratic function is basically a polynomial function with the highest degree of 2. It generally looks like this: f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' isn't zero (otherwise, it wouldn't be quadratic, would it?). When you graph a quadratic function, you get a parabola – a U-shaped curve. This parabola can open upwards (if 'a' is positive) or downwards (if 'a' is negative).

Now, the vertex is a crucial point on this parabola. It's the point where the parabola changes direction. Think of it as the parabola's turning point. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, the vertex is the maximum point. Understanding the vertex helps us understand a lot about the behavior of the quadratic function, like its maximum or minimum value and its axis of symmetry.

The vertex is super useful in real-world applications too! Imagine you're trying to figure out the optimal launch angle for a projectile to reach the maximum distance, or you want to maximize the profit of your business given a cost function. These scenarios often involve quadratic functions, and finding the vertex helps you solve these problems. So, yeah, the vertex is kind of a big deal.

The coordinates of the vertex are given as (h, k), where 'h' represents the x-coordinate and 'k' represents the y-coordinate. There are a couple of ways to find these coordinates, and we'll explore them in detail. The vertex form of a quadratic equation is f(x) = a(x - h)² + k, where (h, k) is directly visible. Transforming our given equation into this form is one way to find the vertex. Another method involves using a formula derived from the standard form of the quadratic equation, which we’ll also discuss. Both methods are powerful tools, and knowing both gives you flexibility in problem-solving. We need to find the x-coordinate (h) and y-coordinate (k) of the vertex.

Method 1: Completing the Square

Okay, let's get our hands dirty with our first method: completing the square. This technique is a classic for transforming a quadratic function from standard form (f(x) = ax² + bx + c) to vertex form (f(x) = a(x - h)² + k). The vertex form is super handy because, as we mentioned, the vertex coordinates (h, k) are right there for you to see. So, how do we do it?

Let’s start with our function: f(x) = -2x² + 4x + 9. The first step is to factor out the coefficient of the x² term (which is -2 in our case) from the first two terms. This gives us: f(x) = -2(x² - 2x) + 9. Notice that we've only factored out from the x² and x terms, leaving the constant term (+9) outside the parentheses. This is important because we want to manipulate the expression inside the parentheses to form a perfect square trinomial.

Now, here's the key part: to complete the square, we need to add and subtract a value inside the parentheses that will make the expression a perfect square. The value we need is (b/2)², where 'b' is the coefficient of the x term inside the parentheses. In our case, 'b' is -2, so (b/2)² = (-2/2)² = (-1)² = 1. So, we add and subtract 1 inside the parentheses: f(x) = -2(x² - 2x + 1 - 1) + 9. It looks a bit messy now, but trust me, it's about to get cleaner.

Next, we rewrite the perfect square trinomial (x² - 2x + 1) as a squared binomial: f(x) = -2((x - 1)² - 1) + 9. See how the x² - 2x + 1 neatly transformed into (x - 1)²? That’s the magic of completing the square! Now, we distribute the -2 to both terms inside the parentheses: f(x) = -2(x - 1)² + 2 + 9. Finally, we simplify the expression by combining the constant terms: f(x) = -2(x - 1)² + 11.

Boom! We've done it! Our function is now in vertex form: f(x) = -2(x - 1)² + 11. Comparing this to the vertex form f(x) = a(x - h)² + k, we can easily identify the vertex coordinates. The x-coordinate, h, is 1, and the y-coordinate, k, is 11. So, the vertex of our parabola is (1, 11). That wasn't so bad, was it? Completing the square might seem a bit tricky at first, but with practice, you'll become a pro at it.

Method 2: Using the Vertex Formula

Alright, let’s explore another way to pinpoint the vertex coordinates: the vertex formula. This method is a direct approach and can be a real time-saver, especially if you’re comfortable with formulas. The vertex formula is derived from the standard form of a quadratic equation (f(x) = ax² + bx + c), and it gives us a straightforward way to calculate the x and y coordinates of the vertex.

The formula for the x-coordinate (h) of the vertex is: h = -b / 2a. Remember, 'a' and 'b' are the coefficients from our quadratic equation. For our function, f(x) = -2x² + 4x + 9, 'a' is -2 and 'b' is 4. Let's plug these values into the formula: h = -4 / (2 * -2) = -4 / -4 = 1. Voila! We've found the x-coordinate of the vertex: h = 1. Easy peasy, right?

Now, to find the y-coordinate (k) of the vertex, we simply substitute the x-coordinate (h) back into our original function. So, we need to calculate f(1). Let's do it: f(1) = -2(1)² + 4(1) + 9 = -2 + 4 + 9 = 11. And there you have it! The y-coordinate of the vertex is k = 11.

Putting it all together, the vertex coordinates are (1, 11). Notice that we got the same result as we did using the completing the square method. This is a good check to ensure our calculations are correct. The vertex formula is a powerful tool, and it's definitely worth memorizing. It's quick, efficient, and gets the job done. Plus, it reinforces the connection between the coefficients of the quadratic equation and the location of its vertex. So, whether you prefer completing the square or using the vertex formula, you now have two solid methods for finding the vertex of a quadratic function.

Applying the Vertex Coordinates

Now that we've successfully determined the vertex coordinates of f(x) = -2x² + 4x + 9, which are (1, 11), let's talk about what this actually means and how we can use this information. Knowing the vertex is incredibly useful for understanding the behavior and properties of the quadratic function and its corresponding parabola.

Firstly, let's consider the shape of the parabola. Since the coefficient 'a' in our function is -2 (which is negative), the parabola opens downwards. This means that the vertex (1, 11) represents the maximum point of the function. In other words, the highest value that f(x) can reach is 11, and it occurs when x = 1. This is a crucial piece of information if we're trying to maximize something in a real-world scenario, like the height of a projectile or the profit of a business.

The vertex also tells us about the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In our case, the axis of symmetry is x = 1. This means that the parabola is perfectly symmetrical around the vertical line x = 1. This symmetry is a key characteristic of parabolas and can be helpful in graphing the function and understanding its behavior.

Graphing the function becomes much easier once we know the vertex and the direction the parabola opens. We know the highest point is (1, 11), and the parabola opens downwards. We can also find other points on the parabola by plugging in different x-values into the function. For example, we can find the y-intercept by setting x = 0: f(0) = -2(0)² + 4(0) + 9 = 9. So, the y-intercept is (0, 9). Knowing the vertex and the y-intercept gives us a good starting point for sketching the graph. Because of the symmetry, we also know there's a point on the parabola that's symmetrical to (0,9) across the line x=1, which would be (2,9).

In real-world applications, the vertex can represent various things depending on the context. For instance, if f(x) represents the height of a projectile as a function of time (x), then the vertex represents the maximum height the projectile reaches and the time at which it reaches that height. If f(x) represents the profit of a business as a function of the number of units sold (x), then the vertex represents the number of units that should be sold to maximize profit and the maximum profit itself. These applications highlight the practical significance of understanding and finding the vertex of a quadratic function.

Conclusion

So there you have it, folks! We've explored how to determine the vertex coordinates of the quadratic function f(x) = -2x² + 4x + 9 using two different methods: completing the square and the vertex formula. We found that the vertex is (1, 11), and we discussed how this point represents the maximum value of the function and the axis of symmetry of the parabola. Remember, the vertex is a key feature of quadratic functions, and understanding how to find it opens the door to solving a wide range of problems, both in math and in real-world scenarios.

Both completing the square and using the vertex formula are valuable techniques, and choosing the one that works best for you depends on your personal preference and the specific problem you're facing. Completing the square is a more fundamental method that helps you understand the structure of quadratic functions, while the vertex formula is a quicker and more direct approach. The important thing is to practice both methods so you can become confident in your ability to find the vertex of any quadratic function. Keep practicing, and you'll be a vertex-finding master in no time! You've got this!