Finding The Vertex Of F(x) = X^2 + 8x - 2: A Step-by-Step Guide

Hey guys! Let's dive into a common math problem: finding the vertex of a quadratic function. Specifically, we're going to tackle the function f(x) = x² + 8x - 2. This is a classic question you might see in algebra, and knowing how to solve it is super useful. So, let's break it down and make sure you understand every step.

Understanding the Vertex

Before we jump into the calculations, let's quickly recap what the vertex actually is. In simple terms, the vertex is the highest or lowest point on the parabola (the U-shaped curve) that represents our quadratic function. If the parabola opens upwards (like in our case, because the coefficient of x² is positive), the vertex is the minimum point. If it opens downwards, the vertex is the maximum point. Identifying the vertex helps us understand the behavior of the quadratic function, including its range and axis of symmetry. Think of it as the turning point of the parabola – the spot where the function changes direction.

Methods to Find the Vertex

There are a couple of ways we can find the vertex, and we'll explore two popular methods here:

1. Completing the Square
2. Using the Vertex Formula

Let's start with completing the square, as this method gives us a deeper understanding of why the vertex is where it is.

1. Completing the Square

Completing the square is a technique that allows us to rewrite the quadratic function in vertex form. The vertex form of a quadratic equation is given by: f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it incredibly easy to spot the vertex! So, our goal is to transform f(x) = x² + 8x - 2 into this form.

Here's how we do it, step by step:

Step 1: Group the x terms

First, we group the terms containing x: f(x) = (x² + 8x) - 2. We're focusing on the x² + 8x part for now.

Step 2: Complete the square inside the parentheses

To complete the square, we need to add and subtract a value inside the parentheses that will allow us to form a perfect square trinomial. Remember, a perfect square trinomial can be factored into the form (x + a)² or (x - a)². The value we need to add and subtract is *(b/2)², where b is the coefficient of our x term. In our case, b = 8, so (b/2)² = (8/2)² = 4² = 16. We'll add and subtract 16 inside the parentheses:

f(x) = (x² + 8x + 16 - 16) - 2

Step 3: Factor the perfect square trinomial

Now, we can factor the perfect square trinomial x² + 8x + 16. This factors nicely into (x + 4)². So our equation becomes:

f(x) = ((x + 4)²) - 16 - 2

Step 4: Simplify

Finally, we combine the constant terms outside the parentheses: f(x) = (x + 4)² - 18. Now, we have our function in vertex form! Notice how we've transformed the original equation into a form that clearly shows the vertex.

Step 5: Identify the vertex

Comparing our equation f(x) = (x + 4)² - 18 with the vertex form f(x) = a(x - h)² + k, we can see that h = -4 (because we have (x + 4), which is the same as (x - (-4))) and k = -18. Therefore, the vertex of the graph is (-4, -18). Completing the square not only gives us the vertex but also reveals the transformation of the basic x² parabola.

2. Using the Vertex Formula

The vertex formula is a shortcut method to find the vertex directly from the standard form of the quadratic equation, which is f(x) = ax² + bx + c. The x-coordinate of the vertex, often denoted as h, is given by the formula: h = -b / 2a. Once we find the x-coordinate, we can plug it back into the original function to find the y-coordinate, often denoted as k. So, the vertex is (h, k).

Let's apply this formula to our function, f(x) = x² + 8x - 2.

Step 1: Identify a, b, and c

In our equation, a = 1 (the coefficient of x²), b = 8 (the coefficient of x), and c = -2 (the constant term).

Step 2: Calculate the x-coordinate (h)

Using the vertex formula h = -b / 2a, we get: h = -8 / (2 * 1) = -8 / 2 = -4. So, the x-coordinate of the vertex is -4.

Step 3: Calculate the y-coordinate (k)

Now, we plug x = -4 back into our original function to find the y-coordinate, k: k = f(-4) = (-4)² + 8(-4) - 2 = 16 - 32 - 2 = -18. Therefore, the y-coordinate of the vertex is -18.

Step 4: Identify the vertex

Putting it all together, the vertex of the graph is (-4, -18). See? The vertex formula provides a quick and direct way to find the vertex, especially when you just need the coordinates and not the vertex form of the equation.

Conclusion

So, guys, we've successfully found the vertex of the graph of the function f(x) = x² + 8x - 2 using two different methods: completing the square and the vertex formula. Both methods led us to the same answer: (-4, -18). Understanding both methods is beneficial because completing the square gives you the vertex form of the equation, which is useful for other analyses, while the vertex formula provides a quicker calculation.

Therefore, the correct answer is D. (-4, -18).

I hope this explanation was helpful! Remember, practice makes perfect. Try working through similar problems to solidify your understanding. Keep up the great work, and you'll master these concepts in no time! Understanding how to find the vertex is a foundational skill in algebra and is essential for further studies in mathematics and related fields. Keep practicing, and you'll be solving these problems like a pro in no time! Good luck!