Finding The Vertex: Equation Exploration

Hey guys! Let's dive into a fun math problem today. We're going to explore how to identify the equation that represents a graph with a vertex at a specific point. This is super helpful when you're working with parabolas (those U-shaped curves) and want to quickly figure out their key features. Specifically, we're looking for the equation whose graph has its vertex at the point (-3, 2). Don't worry, it's not as scary as it sounds! We'll break it down step by step and make sure you understand the concepts. So, grab your pencils, and let's get started!

Understanding the Vertex Form

Alright, before we jump into the equations, let's chat about something called the vertex form of a quadratic equation. This form is your best friend when dealing with vertices. The vertex form of a quadratic equation is written as: y = a(x - h)^2 + k, where:

• (h, k) represents the coordinates of the vertex.
• a determines the direction of the parabola's opening (upward if a > 0, downward if a < 0) and its width.

See? It's all about the vertex! If we know the vertex, we can plug its coordinates (h, k) directly into this form. Knowing this form is like having a secret weapon. It allows you to swiftly identify the vertex of a parabola without having to go through a bunch of calculations. This is because the values of 'h' and 'k' are explicitly given in the equation.

For our problem, the vertex is at (-3, 2). This means h = -3 and k = 2. Let's plug those values into the vertex form: y = a(x - (-3))^2 + 2, which simplifies to y = a(x + 3)^2 + 2. Notice that the 'a' value is still unknown, but the rest of the vertex form is there. The value of 'a' changes how wide or narrow the parabola is, but doesn't change the vertex's location. This simplified form is very important, because it allows you to get closer to the answer. Now, all we have to do is check the given equations to see which one is in, or can be converted to, this form!

Analyzing the Given Equations

Now, let's take a look at the answer choices provided. We need to figure out which of them has a vertex at (-3, 2). The answer choices are in the standard form: y = ax^2 + bx + c. We'll need to convert these into vertex form to determine the vertex. Here are the given equations:

A. y = 4x^2 + 24x + 38 B. y = 4x^2 - 24x + 38 C. y = 4x^2 + 12x + 2 D. y = 4x^2 + 16x + 13

To convert these standard-form equations into vertex form, we can use a method called completing the square. It's a bit like a mathematical magic trick that lets us rewrite the equation into our desired form. Let's try to complete the square, and convert the equations into vertex form! We'll work through each option to find the one that matches our vertex of (-3, 2). This conversion process is very important, because it allows to find a specific vertex easily.

Analyzing Equation A

Let's start with equation A: y = 4x^2 + 24x + 38. To complete the square, we first factor out the coefficient of the x^2 term (which is 4) from the first two terms:

y = 4(x^2 + 6x) + 38

Next, we need to add and subtract a value inside the parentheses to complete the square. We take half of the coefficient of the x term (which is 6), square it ((6/2)^2 = 9), and add and subtract it inside the parentheses:

y = 4(x^2 + 6x + 9 - 9) + 38

Now, rewrite the first three terms inside the parentheses as a perfect square and simplify:

y = 4((x + 3)^2 - 9) + 38 y = 4(x + 3)^2 - 36 + 38 y = 4(x + 3)^2 + 2

Hey, look at that! The equation is now in vertex form. We can see that the vertex is at (-3, 2). So, equation A is a potential answer.

Analyzing Equation B

Now let's analyze equation B: y = 4x^2 - 24x + 38. Factor out 4:

y = 4(x^2 - 6x) + 38

Complete the square (half of -6 is -3, and (-3)^2 = 9):

y = 4(x^2 - 6x + 9 - 9) + 38

Rewrite and simplify:

y = 4((x - 3)^2 - 9) + 38 y = 4(x - 3)^2 - 36 + 38 y = 4(x - 3)^2 + 2

This equation has a vertex at (3, 2), not (-3, 2). So, equation B is not the answer.

Analyzing Equation C

Let's analyze equation C: y = 4x^2 + 12x + 2. Factor out 4:

y = 4(x^2 + 3x) + 2

Complete the square (half of 3 is 3/2, and (3/2)^2 = 9/4):

y = 4(x^2 + 3x + 9/4 - 9/4) + 2

Rewrite and simplify:

y = 4((x + 3/2)^2 - 9/4) + 2 y = 4(x + 3/2)^2 - 9 + 2 y = 4(x + 3/2)^2 - 7

This equation has a vertex at (-3/2, -7), not (-3, 2). So, equation C is not the answer.

Analyzing Equation D

Finally, let's analyze equation D: y = 4x^2 + 16x + 13. Factor out 4:

y = 4(x^2 + 4x) + 13

Complete the square (half of 4 is 2, and 2^2 = 4):

y = 4(x^2 + 4x + 4 - 4) + 13

Rewrite and simplify:

y = 4((x + 2)^2 - 4) + 13 y = 4(x + 2)^2 - 16 + 13 y = 4(x + 2)^2 - 3

This equation has a vertex at (-2, -3), not (-3, 2). So, equation D is not the answer.

Conclusion: The Right Choice

Alright, after analyzing all the equations, we found that only one of them has a vertex at (-3, 2). The correct equation is A. y = 4x^2 + 24x + 38. By converting the equations to vertex form, we were able to quickly and accurately identify the one with the desired vertex. This demonstrates the power of the vertex form! It streamlines the process and lets you easily extract key information about a parabola's graph. Remember, guys, practice makes perfect. Keep working on these types of problems, and you'll become a vertex-finding pro in no time! Keep practicing the completing the square method, and soon you'll be converting quadratic equations like a boss. That is all. See you next time!