Understanding the Vertex Form of a Parabola

Oct 13, 2025 by miodeya.com 28 views

Which Equation Has a Graph That Is a Parabola with a Vertex at (-1, -1)?

A parabola is a type of quadratic equation that has a U-shaped graph. The vertex of a parabola is the point at which the graph changes direction, and it is typically represented by the coordinates (h, k). In this case, we are looking for an equation that has a vertex at (-1, -1).

Understanding the Vertex Form of a Parabola

The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, we know that the vertex is at (-1, -1), so we can substitute these values into the equation.

Substituting the Vertex Values into the Equation

To find the equation of the parabola, we need to substitute the vertex values into the equation y = a(x - h)^2 + k. We know that h = -1 and k = -1, so we can substitute these values into the equation.

y = a(x - (-1))^2 + (-1) y = a(x + 1)^2 - 1

Comparing the Equation to the Options

Now that we have the equation y = a(x + 1)^2 - 1, we can compare it to the options given in the problem.

Option A: y = (x + 1)^2 - 1
Option B: y = (x - 1)^2 + 1
Option C: y = (x - 1)^2 - 1
Option D: y = (x + 1)^2 + 1

Analyzing Option A

Option A is y = (x + 1)^2 - 1. This equation matches the equation we derived earlier, y = a(x + 1)^2 - 1. Since the vertex is at (-1, -1), we can conclude that option A is the correct answer.

Analyzing Option B

Option B is y = (x - 1)^2 + 1. This equation does not match the equation we derived earlier, y = a(x + 1)^2 - 1. The vertex of this parabola is at (1, 1), not (-1, -1), so option B is not the correct answer.

Analyzing Option C

Option C is y = (x - 1)^2 - 1. This equation does not match the equation we derived earlier, y = a(x + 1)^2 - 1. The vertex of this parabola is at (1, -1), not (-1, -1), so option C is not the correct answer.

Analyzing Option D

Option D is y = (x + 1)^2 + 1. This equation does not match the equation we derived earlier, y = a(x + 1)^2 - 1. The vertex of this parabola is at (-1, 1), not (-1, -1), so option D is not the correct answer.

Conclusion

Based on our analysis, we can conclude that the correct answer is option A, y = (x + 1)^2 - 1. This equation has a graph that is a parabola with a vertex at (-1, -1).

Key Takeaways

The vertex form of a parabola is given by the equation y = a(x - h)^2 + k.
To find the equation of a parabola, we need to substitute the vertex values into the equation.
The equation y = a(x + 1)^2 - 1 has a graph that is a parabola with a vertex at (-1, -1).

Additional Examples

Find the equation of a parabola with a vertex at (2, 3).
Find the equation of a parabola with a vertex at (-2, 1).
Find the equation of a parabola with a vertex at (0, 0).

Solutions to Additional Examples

The equation of a parabola with a vertex at (2, 3) is y = a(x - 2)^2 + 3.
The equation of a parabola with a vertex at (-2, 1) is y = a(x + 2)^2 + 1.
The equation of a parabola with a vertex at (0, 0) is y = ax^2.

Conclusion

In this article, we discussed how to find the equation of a parabola with a given vertex. We used the vertex form of a parabola and substituted the vertex values into the equation to find the correct answer. We also analyzed the options given in the problem and concluded that option A is the correct answer.
Frequently Asked Questions (FAQs) About Parabolas

In this article, we will answer some frequently asked questions about parabolas. We will cover topics such as the vertex form of a parabola, how to find the equation of a parabola, and how to analyze the options given in a problem.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How do I find the equation of a parabola with a given vertex?

A: To find the equation of a parabola with a given vertex, you need to substitute the vertex values into the equation y = a(x - h)^2 + k.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the point at which the graph changes direction. It is typically represented by the coordinates (h, k).

Q: How do I analyze the options given in a problem?

A: To analyze the options given in a problem, you need to compare the equation of the parabola to the options given. You can use the vertex form of a parabola to determine which option is correct.

Q: What is the difference between a parabola and a quadratic equation?

A: A parabola is a type of quadratic equation that has a U-shaped graph. A quadratic equation is a polynomial equation of degree two, but it does not necessarily have a U-shaped graph.

Q: Can a parabola have a vertex at any point on the coordinate plane?

A: No, a parabola can only have a vertex at a point on the coordinate plane where the x-coordinate is a real number.

Q: How do I determine the value of a in the equation y = a(x - h)^2 + k?

A: To determine the value of a in the equation y = a(x - h)^2 + k, you need to use the given information about the parabola. For example, if you know the y-intercept of the parabola, you can use it to determine the value of a.

Q: Can a parabola have a vertex at the origin (0, 0)?

A: Yes, a parabola can have a vertex at the origin (0, 0). In this case, the equation of the parabola is y = ax^2.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the vertex form of the equation and plot the vertex on the coordinate plane. Then, you can use the equation to determine the direction of the parabola and plot additional points.

Q: Can a parabola have a vertex at a point with a negative x-coordinate?

A: Yes, a parabola can have a vertex at a point with a negative x-coordinate. In this case, the equation of the parabola will have a negative value for h.

Q: How do I determine the direction of a parabola?

A: To determine the direction of a parabola, you can use the equation y = a(x - h)^2 + k. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward.

Conclusion

In this article, we answered some frequently asked questions about parabolas. We covered topics such as the vertex form of a parabola, how to find the equation of a parabola, and how to analyze the options given in a problem. We hope that this article has been helpful in answering your questions about parabolas.