Finding The Vertex: Which Equation Matches?

Hey there, math enthusiasts! Today, we're diving into a fun problem that combines algebra and geometry: finding the equation of a parabola given its vertex. We'll use our knowledge of quadratic equations to determine which one from the options provided has a vertex located at the point (-3, 2). It's like a treasure hunt, but instead of gold, we're searching for the right equation!

Understanding the Vertex Form

Before we start, let's refresh our memories about the vertex form of a quadratic equation. This form is super helpful because it directly reveals the vertex of the parabola. The vertex form looks like this: y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. The variable 'a' determines the direction the parabola opens (up or down) and how wide or narrow it is. Our goal is to manipulate the given equations into this vertex form to find the one with the vertex at (-3, 2).

Let's get cracking. The core of this problem lies in identifying the vertex of each quadratic equation. There are a few ways to do this, but the most straightforward approach, especially when you have options like these, is to convert each equation into vertex form or use the vertex formula. This will allow us to easily compare the results and find the correct one.

Analyzing the Equations and Finding the Vertex

Option A: $y=4x^2+24x+38$

To figure out the vertex, we can use a method called completing the square or the vertex formula. Let's complete the square:

1. Factor out the coefficient of the $x^2$ term (if not equal to 1): In this case, it's already 4: $y = 4(x^2 + 6x) + 38$
2. Complete the square inside the parentheses: Take half of the coefficient of the $x$ term (which is 6), square it (which is 9), and add and subtract it inside the parentheses: $y = 4(x^2 + 6x + 9 - 9) + 38$
3. Rewrite as a squared term: $y = 4((x + 3)^2 - 9) + 38$
4. Distribute and simplify: $y = 4(x + 3)^2 - 36 + 38$ which simplifies to $y = 4(x + 3)^2 + 2$

From this vertex form, we see that the vertex is at (-3, 2). This means that Option A is the correct answer.

Option B: $y=4x^2-24x+38$

Let's complete the square for this equation:

1. Factor: $y = 4(x^2 - 6x) + 38$
2. Complete the square: $y = 4(x^2 - 6x + 9 - 9) + 38$
3. Rewrite: $y = 4((x - 3)^2 - 9) + 38$
4. Simplify: $y = 4(x - 3)^2 - 36 + 38 = 4(x - 3)^2 + 2$

The vertex is at (3, 2), which doesn't match our target.

Option C: $y=4x^2+12x+2$

Let's go through the steps of completing the square:

1. Factor: $y = 4(x^2 + 3x) + 2$
2. Complete the square: $y = 4(x^2 + 3x + 9/4 - 9/4) + 2$
3. Rewrite: $y = 4((x + 3/2)^2 - 9/4) + 2$
4. Simplify: $y = 4(x + 3/2)^2 - 9 + 2 = 4(x + 3/2)^2 - 7$

The vertex is at (-3/2, -7), which is not our desired point.

Option D: $y=4x^2+16x+13$

Follow these steps:

1. Factor: $y = 4(x^2 + 4x) + 13$
2. Complete the square: $y = 4(x^2 + 4x + 4 - 4) + 13$
3. Rewrite: $y = 4((x + 2)^2 - 4) + 13$
4. Simplify: $y = 4(x + 2)^2 - 16 + 13 = 4(x + 2)^2 - 3$

The vertex is at (-2, -3), not our target.

Method 2: The Vertex Formula

An alternative method is using the vertex formula. For a quadratic equation in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by $x = -b / (2a)$.

Once you have the x-coordinate, substitute it back into the original equation to find the y-coordinate. Let's try this with the options.

• Option A: $x = -24 / (2 * 4) = -3$. Substituting -3 into the equation: $y = 4(-3)^2 + 24(-3) + 38 = 36 - 72 + 38 = 2$. Vertex: (-3, 2).
• Option B: $x = -(-24) / (2 * 4) = 3$. This confirms that B does not work, as we have already shown.
• Option C: $x = -12 / (2 * 4) = -3/2$. This also confirms that C does not work.
• Option D: $x = -16 / (2 * 4) = -2$. This confirms that D does not work.

Conclusion: The Winning Equation!

We successfully found the equation representing a graph with a vertex at (-3, 2). The vertex form helps us to get the answer. By completing the square or using the vertex formula, we are able to easily identify the vertex of each quadratic equation and locate the correct option. The key is to understand the vertex form, and complete the square. The answer is A: $y = 4x^2 + 24x + 38$. Great job, everyone!

If you want to dive deeper into quadratic equations and related concepts, here are some reliable resources:

• Khan Academy: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations-and-inequalities

These resources will help you to further practice and master your understanding. Keep up the excellent work, and always remember to enjoy the journey of learning! Don't be afraid to try, make mistakes, and learn from them. The more you practice, the more confident you'll become in solving these types of problems.