Steps To Convert F(x) = 18x + 3x^2 Into Vertex Form

Introduction

Hey guys! Today, we are going to dive deep into transforming a quadratic function into its vertex form. You know, that form that tells you exactly where the peak or valley of your parabola is? We'll be walking through the steps of converting the function $f(x) = 18x + 3x^2$ into vertex form. The problem statement already gives us a head start, showing some initial steps, but we've got to complete the process by finding a missing value in the final step. So, let’s buckle up and get started! We’ll break it down bit by bit, so it’s super clear and easy to follow. Remember, mastering the vertex form is crucial for understanding quadratic functions and their graphs. This skill pops up everywhere, from simple algebra problems to more complex calculus questions. So, pay close attention, and let’s get this bread!

Writing the Function in Standard Form

Alright, first things first, we need to get our function into the standard form. Why? Because it makes the whole process smoother and easier to handle. The standard form of a quadratic function is generally written as $f(x) = ax^2 + bx + c$. Looking at our given function, $f(x) = 18x + 3x^2$, it’s clear that the terms are just a little out of order. No biggie, we can fix that! To get it into standard form, we simply rearrange the terms so that the term with $x^2$ comes first, followed by the term with $x$, and then the constant term (if any). In our case, we just need to switch the positions of $18x$ and $3x^2$. So, let’s do it! By rearranging the terms, we get $f(x) = 3x^2 + 18x$. See? That’s much more like it. Now it’s in the familiar standard form, where $a = 3$, $b = 18$, and $c = 0$ (since there’s no constant term explicitly added or subtracted). This might seem like a small step, but trust me, it’s a fundamental one. Getting the function into standard form sets us up perfectly for the next steps in converting it to vertex form. We need to correctly identify these coefficients (a, b, and c) because they play a vital role in the subsequent calculations. So, now that we’ve nailed the standard form, we’re ready to move on to the next part of our journey. Let’s keep this momentum going, guys!

Factoring 'a' Out of the First Two Terms

Okay, now that we’ve got our function in standard form ($f(x) = 3x^2 + 18x$), it’s time for the next step: factoring out the coefficient 'a' from the first two terms. Remember, 'a' is the coefficient of the $x^2$ term, which in our case is 3. Factoring out 'a' is a crucial step because it helps us to complete the square, which is the key to getting to vertex form. So, what does it mean to factor out 3 from the first two terms? It means we're going to divide both $3x^2$ and $18x$ by 3 and then write 3 outside a set of parentheses. Let’s do the math: when we divide $3x^2$ by 3, we get $x^2$. And when we divide $18x$ by 3, we get $6x$. So, inside the parentheses, we'll have $x^2 + 6x$. Now, we put it all together with the 3 we factored out in front, and we get: $f(x) = 3(x^2 + 6x)$. See how we’ve essentially pulled out the 3, leaving us with a simpler quadratic expression inside the parentheses? This is going to make our lives much easier when we complete the square. Factoring out the leading coefficient is like setting the stage for the main act. It prepares the expression for the manipulation needed to reveal the vertex form. By doing this, we ensure that the coefficient of the $x^2$ term inside the parentheses is 1, which is what we want for completing the square. This step is essential, guys, so make sure you’ve got it down. Next up, we’re diving into the heart of the matter: completing the square!

Repair Input Keyword

What is the missing value in the last step of writing the function $f(x) = 18x + 3x^2$ in vertex form?

Title

Vertex Form Steps for f(x) = 18x + 3x^2 A Mathematics Guide