Finding The Vertex Of A Quadratic Function By Completing The Square

Hey guys! Today, we're diving into how to find the vertex of a quadratic function using the completing-the-square method. This is super useful because the vertex tells us where the function hits its maximum or minimum point. We’ll break down each step so it’s crystal clear. Our example function is:

$$f(x) = 5x^2 + 10x + 8$$

We want to rewrite this function in vertex form, which is:

$$f(x) = a(x - h)^2 + k$$

Where (h, k) is the vertex of the parabola. Let's get started!

Step-by-Step Guide to Completing the Square

1. Factor out the Leading Coefficient

First off, let's focus on the terms with x. We need to factor out the coefficient of the x^2 term, which in our case is 5. This step is crucial because it sets us up to complete the square properly. Factoring out the 5 from the first two terms, we get:

$$f(x) = 5(x^2 + 2x) + 8$$

Notice how we only factored out from the terms involving x. The constant term, 8, stays outside the parentheses for now. This is super important to keep in mind as we move forward. Factoring out the leading coefficient allows us to focus on the quadratic and linear terms inside the parenthesis, making it easier to manipulate and complete the square.

2. Completing the Square

Now comes the fun part – completing the square! Inside the parentheses, we have x^2 + 2x. To complete the square, we need to add and subtract a value that will turn this into a perfect square trinomial. This value is calculated by taking half of the coefficient of the x term (which is 2), squaring it, and then adding and subtracting it inside the parentheses. Half of 2 is 1, and 1 squared is 1. So we add and subtract 1:

$$f(x) = 5(x^2 + 2x + 1 - 1) + 8$$

We've essentially added 0 (since we're adding and subtracting the same number), so we haven't changed the equation's value. But now, the first three terms inside the parentheses (x^2 + 2x + 1) form a perfect square trinomial. This is exactly what we wanted! We can rewrite this as (x + 1)^2. Now our equation looks like this:

$$f(x) = 5((x + 1)^2 - 1) + 8$$

Remember, the goal here is to get our quadratic function into vertex form. By completing the square, we transform the quadratic expression into a form that includes a squared term, which directly relates to the vertex coordinates. It might seem a bit tricky at first, but with practice, it becomes second nature. Just remember to focus on creating that perfect square trinomial!

3. Distribute and Simplify

Alright, let’s keep things moving! The next step is to distribute the 5 back into the parentheses. Make sure you only distribute to the terms inside the parentheses – the constant term outside is waiting for its turn. So, distributing the 5, we get:

$$f(x) = 5(x + 1)^2 - 5 + 8$$

Now, we just need to simplify by combining the constant terms. We have -5 and +8, which combine to give us +3. So our equation now looks like this:

$$f(x) = 5(x + 1)^2 + 3$$

And just like that, we’ve transformed our original function into vertex form! This form is super helpful because it directly reveals the vertex of the parabola. The key here is to be careful with your distribution and make sure you combine the constants correctly. This step brings us closer to easily identifying the vertex coordinates. Distributing and simplifying is a crucial algebraic manipulation that helps us clearly see the vertex form of the quadratic equation.

4. Identify the Vertex

Okay, guys, the moment we’ve been working towards is here! Now that we have our function in vertex form:

$$f(x) = 5(x + 1)^2 + 3$$

We can easily identify the vertex. Remember, the vertex form is f(x) = a(x - h)^2 + k, where (h, k) is the vertex. In our case, we have (x + 1), which can be rewritten as (x - (-1)). So, h is -1, and k is 3. Therefore, the vertex of our parabola is (-1, 3). Woohoo! We found it!

Now, to determine if this vertex is a minimum or a maximum, we look at the coefficient a in our vertex form. In our case, a is 5, which is positive. When a is positive, the parabola opens upwards, meaning the vertex is the lowest point on the graph. So, our vertex is a minimum. Identifying the vertex is straightforward once you have the equation in vertex form. Just remember to pay attention to the signs and the values of h and k. And checking the sign of a tells us whether we’re looking at a minimum or a maximum point.

5. Final Answer

Alright, let’s wrap this up! We started with the quadratic function:

$$f(x) = 5x^2 + 10x + 8$$

And after going through the steps of completing the square, we found the vertex form:

$$f(x) = 5(x + 1)^2 + 3$$

From this, we identified the vertex as (-1, 3). And because the coefficient a (which is 5) is positive, we know that this vertex is a minimum point. So, the final answer is:

D. Minimum at (-1, 3)

Why Completing the Square Matters

You might be wondering, “Why go through all this trouble?” Well, completing the square is super useful for a few reasons:

• Finding the Vertex: As we just saw, it gives us the vertex form of the quadratic equation, making the vertex easy to spot.
• Solving Quadratic Equations: It’s a method we can use to solve quadratic equations, especially when factoring isn’t straightforward.
• Graphing Parabolas: Knowing the vertex and whether it’s a minimum or maximum helps us sketch the graph of the parabola.

Tips and Tricks for Mastering Completing the Square

• Practice, Practice, Practice: Like any math skill, completing the square gets easier with practice. Work through lots of examples!
• Be Careful with Signs: Pay close attention to the signs, especially when adding and subtracting inside the parentheses.
• Double-Check Your Work: It’s always a good idea to double-check your steps to avoid mistakes.
• Understand the Goal: Remember that the goal is to create a perfect square trinomial. Keep that in mind as you work through the steps.

Conclusion

And there you have it, guys! We’ve walked through how to find the vertex of a quadratic function using the completing-the-square method. It might seem like a lot of steps at first, but with a little practice, you’ll become a pro. Remember, the key is to break it down step by step, stay organized, and don’t be afraid to make mistakes – that’s how we learn! Keep practicing, and you'll master this technique in no time. Happy math-ing!