Find The Vertex Of A Parabola: $y=3x^2+2x+1$

Hey math whizzes! Ever wondered how to pinpoint the exact lowest or highest point on a U-shaped graph, known as a parabola? That magical point is called the vertex, and today, we're diving deep into how to find it for the equation $y = 3x^2 + 2x + 1$. This stuff might seem a bit tricky at first, but trust me, once you get the hang of it, you'll be spotting vertices like a pro. We'll break down the process step-by-step, making sure you understand every single bit. So, grab your notebooks, get comfy, and let's tackle this parabola puzzle together! We'll be looking at the equation $y = ax^2 + bx + c$, which is the standard form for a quadratic equation. In our case, $a=3$, $b=2$, and $c=1$. The value of 'a' tells us whether our parabola opens upwards (if 'a' is positive, like in our case where $a=3$) or downwards (if 'a' is negative). A positive 'a' means we're looking for the minimum point, while a negative 'a' would mean we're looking for the maximum point. The 'x'-coordinate of the vertex is super important because it helps us find the axis of symmetry, which is the vertical line that cuts the parabola perfectly in half. This coordinate is calculated using a handy formula that involves 'a' and 'b'. Once we have the 'x'-coordinate, we can plug it back into our original equation to find the corresponding 'y'-coordinate, which gives us the complete location of the vertex. It's like putting together a treasure map where the vertex is the X that marks the spot! Let's get into the nitty-gritty of how to calculate these coordinates. We'll explore the formula and work through our specific example, $y = 3x^2 + 2x + 1$, to make sure everything clicks. Remember, practice makes perfect, and the more you work through these types of problems, the more confident you'll become. So, let's get started on unraveling the mystery of the vertex!

Understanding the Vertex Formula

Alright guys, let's talk about the vertex formula. For any quadratic equation in the standard form $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by the formula: $x = -\frac{b}{2a}$ This little gem is derived from calculus (finding the minimum or maximum using derivatives), but you don't need to be a calculus expert to use it! Think of it as a shortcut for finding that crucial $x$-value. The 'a' and 'b' are just the coefficients from your quadratic equation. For our equation, $y = 3x^2 + 2x + 1$, we have $a=3$ and $b=2$. So, let's plug these values into the formula:

$$x = -\frac{2}{2 \times 3}$$

$$x = -\frac{2}{6}$$

$$x = -\frac{1}{3}$$

Boom! We've just found the $x$-coordinate of our vertex. It's $-1/3$. This tells us that the axis of symmetry for our parabola is the vertical line $x = -1/3$. Pretty neat, right? This $x$-value is essential because it's the horizontal position where the parabola reaches its lowest point (since $a=3$ is positive). Now, this formula is a lifesaver, especially when you don't want to deal with completing the square, which is another method to find the vertex. The reason this formula works is tied to the symmetry of the parabola. The vertex is exactly in the middle of any two points on the parabola that have the same $y$-value. The formula $x = -b/(2a)$ effectively finds that midpoint for us. It's a fundamental piece of knowledge when working with quadratic functions and their graphical representations. Keep this formula handy, as it's a cornerstone for many parabola-related problems. We'll use this $x$-coordinate in the next step to find the $y$-coordinate, completing our vertex coordinates.

Calculating the Y-Coordinate of the Vertex

So, we've got the $x$-coordinate of the vertex for $y = 3x^2 + 2x + 1$, which is $x = -1/3$. Now, how do we find the corresponding $y$-coordinate? Easy peasy! We just take this $x$-value and plug it back into our original equation. Think of it as finding the height of the parabola at that specific horizontal position. Here’s how it works:

Original equation: $y = 3x^2 + 2x + 1$

Substitute $x = -1/3$:

$$y = 3\left(-\frac{1}{3}\right)^2 + 2\left(-\frac{1}{3}\right) + 1$$

First, let's square the $(-1/3)$: $\left(-\frac{1}{3}\right)^2 = \frac{1}{9}$.

Now, substitute that back:

$$y = 3\left(\frac{1}{9}\right) + 2\left(-\frac{1}{3}\right) + 1$$

Next, multiply:

$$y = \frac{3}{9} - \frac{2}{3} + 1$$

Simplify the fraction $3/9$ to $1/3$:

$$y = \frac{1}{3} - \frac{2}{3} + 1$$

Now, combine the fractions: $1/3 - 2/3 = -1/3$.

$$y = -\frac{1}{3} + 1$$

Finally, add 1 (which is $3/3$) to $-1/3$:

$$y = -\frac{1}{3} + \frac{3}{3}$$

$$y = \frac{2}{3}$$

And there you have it! The $y$-coordinate of the vertex is $2/3$. So, the vertex of the graph of $y = 3x^2 + 2x + 1$ is the point $\left(-\frac{1}{3}, \frac{2}{3}\right)$. This is the lowest point on our parabola because the coefficient 'a' (which is 3) is positive. This means the parabola opens upwards. Finding the $y$-coordinate by substitution is a universal method for any vertex calculation once you have the $x$-coordinate. It's the final step in pinpointing that crucial turning point of the parabola. This process reinforces the relationship between the equation and its graphical representation, showing you exactly where the function reaches its extreme value.

Connecting to the Options

Alright folks, we've done the hard work and found the vertex of our parabola $y = 3x^2 + 2x + 1$ to be $\left(-\frac{1}{3}, \frac{2}{3}\right)$. Now, let's look back at the options provided:

A. $\left(-\frac{1}{3}, 0\right)$ B. $\left(-\frac{1}{3}, \frac{2}{3}\right)$ C. $\left(\frac{1}{3}, 2\right)$ D. $\left(\frac{1}{3}, 0\right)$

Compare our calculated vertex with these options. It's a perfect match with Option B! This confirms that our calculations are correct and we've successfully identified the vertex. It's always a good idea to double-check your work, especially when you have multiple-choice options. Sometimes, a small arithmetic error can lead you to the wrong answer, so going through the steps again, or using a different method like completing the square, can be very beneficial. In this case, our direct calculation using the vertex formula $x = -b/(2a)$ and then substituting that $x$ back into the equation yielded the exact coordinates given in option B. This consistency is a strong indicator of correctness. Remember that the vertex represents a key feature of the quadratic graph – the point where the function's behavior changes from decreasing to increasing (or vice versa). Understanding how to find it is fundamental to analyzing quadratic functions.

Conclusion: Mastering the Vertex

So there you have it, math adventurers! We've successfully navigated the world of quadratic equations and pinpointed the vertex of the graph $y = 3x^2 + 2x + 1$. By using the formula $x = -b/(2a)$ to find the $x$-coordinate and then substituting that value back into the equation to find the $y$-coordinate, we arrived at the correct vertex: $\left(-\frac{1}{3}, \frac{2}{3}\right)$. This point is the absolute lowest point on this particular parabola because the leading coefficient (3) is positive, meaning the parabola opens upwards. Mastering how to find the vertex is a super valuable skill in mathematics. It helps us understand the shape and position of parabolas, which appear in many real-world scenarios, from projectile motion to the design of satellite dishes. Don't get discouraged if it takes a few tries to get the hang of it. Keep practicing with different equations, and you'll soon be a vertex-finding champion! The key takeaways are the formula for the $x$-coordinate and the substitution method for the $y$-coordinate. Keep these in your mathematical toolkit, and you'll be well-equipped to handle any quadratic function that comes your way. Happy graphing!