Finding The Parabola's Vertex: A Step-by-Step Guide

by Alex Johnson 52 views

Hey there, math enthusiasts! Today, we're diving into the exciting world of parabolas and their vertices. We'll be answering the question, "Which equation has a graph that is a parabola with a vertex at (5,3)?" This is a fundamental concept in algebra, and understanding it will give you a solid foundation for tackling more complex problems. So, buckle up, and let's unravel the secrets of the parabola! We'll explore the vertex form of a quadratic equation, dissect each answer choice, and understand how to identify the correct equation.

Understanding Parabolas and Their Vertex Form

Let's start with the basics. A parabola is a U-shaped curve that's formed by graphing a quadratic equation. The most important point on a parabola is its vertex. The vertex is the point where the parabola changes direction. It's either the lowest point (the minimum) if the parabola opens upwards, or the highest point (the maximum) if the parabola opens downwards. Knowing the vertex is crucial because it tells us a lot about the parabola's position and its extreme value.

Now, let's talk about the vertex form of a quadratic equation. This form is incredibly useful because it directly reveals the vertex of the parabola. The vertex form is expressed as:

y = a(x - h)^2 + k

In this equation:

  • (h, k) represents the coordinates of the vertex.
  • a determines the direction the parabola opens (upward if a > 0, downward if a < 0) and how wide or narrow it is.

So, if we have an equation in vertex form, we can immediately identify the vertex by simply looking at the values of h and k. Remember, the h value is subtracted from x inside the parentheses, so be careful with the signs!

For example, if we have the equation y = 2(x - 3)^2 + 4, the vertex is at (3, 4). Notice how the h value is 3, even though it appears as -3 in the equation. That's because the vertex form has a minus sign built-in: (x - h). If the equation was y = 2(x + 3)^2 + 4, it's the same as y = 2(x - (-3))^2 + 4, so the vertex would be at (-3, 4). Understanding the vertex form is the key to solving our problem.

Now, let's look at the given options and see how we can use this knowledge to find the correct answer.

Analyzing the Answer Choices

We are given four equations and asked to find the one whose graph is a parabola with a vertex at (5, 3). Let's examine each option using what we've learned about the vertex form of a quadratic equation.

Option A: y = (x - 5)^2 + 3

This equation is already in vertex form! Comparing it to y = a(x - h)^2 + k, we can see that:

  • h = 5
  • k = 3

Therefore, the vertex of this parabola is at (5, 3). This is exactly what we're looking for! But, let's keep going and check the other options to make sure.

Option B: y = (x + 5)^2 + 3

This equation can be rewritten as y = (x - (-5))^2 + 3. Comparing it to the vertex form, we see that:

  • h = -5
  • k = 3

So, the vertex of this parabola is at (-5, 3). This is not the vertex we are looking for, so we can eliminate this option.

Option C: y = (x - 3)^2 + 5

In this equation:

  • h = 3
  • k = 5

This means the vertex of this parabola is at (3, 5). Again, not the correct vertex, so we can eliminate this option as well.

Option D: y = (x + 3)^2 + 5

We can rewrite this equation as y = (x - (-3))^2 + 5. So:

  • h = -3
  • k = 5

The vertex of this parabola is at (-3, 5). This is also not the vertex we're looking for, so we can eliminate this option.

The Solution

After analyzing all four options, we found that only Option A: y = (x - 5)^2 + 3 has a vertex at (5, 3). This is because the equation is already in vertex form, and we can directly identify the vertex coordinates by comparing it to the standard form y = a(x - h)^2 + k. Therefore, the correct answer is A. Understanding the vertex form of a quadratic equation is key to quickly solving this kind of problem. Make sure to always double-check the signs and remember that the vertex is always represented as (h, k) in the vertex form.

Conclusion

Congratulations, math explorers! You've successfully navigated the world of parabolas and vertices. We've learned about the vertex form, analyzed different equations, and identified the correct one. Remember, the vertex form is a powerful tool for understanding parabolas. Keep practicing, and you'll become a vertex-finding expert in no time! Always remember to pay close attention to the signs and understand how the values of h and k directly relate to the vertex's coordinates.

By carefully examining each equation and applying our knowledge of the vertex form, we were able to pinpoint the correct answer. This skill is crucial in algebra and forms the basis for understanding more advanced concepts. Keep up the great work, and happy math-ing!

For further exploration, you might find these external resources helpful:

This link provides a comprehensive overview of parabolas, the vertex form, and related concepts. It includes video lessons, practice exercises, and articles to deepen your understanding.