Equation of a Parabola with a Given Vertex

Oct 13, 2025 by miodeya.com 24 views

A parabola is a fundamental concept in mathematics, particularly in algebra and geometry. It is a U-shaped curve that can be represented by a quadratic equation. In this article, we will focus on identifying the equation of a parabola with a given vertex.

What is a Parabola?

A parabola is a quadratic curve that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants. The parabola can open upwards or downwards, depending on the value of a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards.

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 form, the vertex is the point (h, k), and the parabola opens upwards or downwards depending on the value of a.

Equation of a Parabola with a Given Vertex

Given that the vertex of the parabola is (5, 0), we need to find the equation of the parabola in the vertex form. Since the vertex is (5, 0), we can substitute these values into the equation y = a(x - h)^2 + k.

Substituting the Values

Substituting the values of h = 5 and k = 0 into the equation, we get:

y = a(x - 5)^2 + 0

Simplifying the equation, we get:

y = a(x - 5)^2

Comparing with the Options

Now, let's compare this equation with the options given:

A. y = x^2 - 5 B. y = x^2 + 5 C. y = (x - 5)^2 D. y = (x + 5)^2 E. y = x^2 - 25

Analyzing the Options

Option A and B are not in the vertex form, and option E is not correct because it does not have the correct vertex.

Option C is in the vertex form, and it has the correct vertex (5, 0). Therefore, option C is the correct answer.

Conclusion

In conclusion, the equation of a parabola with a vertex at (5, 0) is y = (x - 5)^2. This equation is in the vertex form, and it has the correct vertex.

Key Takeaways

A parabola is a quadratic curve that can be represented by the equation y = ax^2 + bx + c.
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.
To find the equation of a parabola with a given vertex, substitute the values of h and k into the equation y = a(x - h)^2 + k.
Compare the resulting equation with the options given to find the correct answer.

Final Answer

In our previous article, we discussed the equation of a parabola with a given vertex. In this article, we will provide a comprehensive Q&A guide to help you understand parabola equations better.

Q: What is a parabola?

A: A parabola is a quadratic curve that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants. The parabola can open upwards or downwards, depending on the value of a.

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. In this form, the vertex is the point (h, k), and the parabola opens upwards or downwards depending on the value of a.

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, substitute the values of h and k into the equation y = a(x - h)^2 + k. Then, simplify the equation to get the final answer.

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

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards.

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

A: To determine the direction of a parabola, look at the value of a in the equation y = ax^2 + bx + c. If a is positive, the parabola opens upwards. If a is negative, the parabola opens downwards.

Q: What is the difference between a parabola and a circle?

A: A parabola is a quadratic curve that can be represented by the equation y = ax^2 + bx + c, while a circle is a circular curve that can be represented by the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Q: Can a parabola have a negative value of a?

A: Yes, a parabola can have a negative value of a. In this case, the parabola will open downwards.

Q: How do I graph a parabola?

A: To graph a parabola, use the equation y = ax^2 + bx + c to find the x-intercepts and the vertex. Then, plot the points and draw a smooth curve through them.

Q: What is the equation of a parabola with a vertex at (3, 4)?

A: The equation of a parabola with a vertex at (3, 4) is y = a(x - 3)^2 + 4. To find the value of a, substitute the point (3, 4) into the equation and solve for a.

Q: Can a parabola have a horizontal axis of symmetry?

A: No, a parabola cannot have a horizontal axis of symmetry. The axis of symmetry of a parabola is always vertical.

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

A: To find the equation of a parabola with a given axis of symmetry, use the equation y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Then, substitute the values of h and k into the equation and simplify.

Conclusion

In conclusion, parabola equations are an essential part of mathematics, particularly in algebra and geometry. By understanding the concepts and formulas discussed in this article, you will be able to solve problems and graph parabolas with ease.

Key Takeaways

A parabola is a quadratic curve that can be represented by the equation y = ax^2 + bx + c.
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.
To find the equation of a parabola with a given vertex, substitute the values of h and k into the equation y = a(x - h)^2 + k.
The axis of symmetry of a parabola is always vertical.
A parabola can have a negative value of a, in which case it opens downwards.

Final Answer

The final answer is that parabola equations are a fundamental concept in mathematics, and understanding them is essential for solving problems and graphing parabolas.