Comparison of Parabola A and Parabola B

Oct 13, 2025 by miodeya.com 10 views

**Compare and Contrast the Parabolas: A Comprehensive Analysis**

What are Parabolas?

A parabola is a type of quadratic curve that is defined as the set of all points that are equidistant from a fixed point (known as the focus) and a fixed line (known as the directrix). In this article, we will compare and contrast two parabolas with different definitions and find their center and radius.

Parabola A: Points that are the same distance from (0, 4) and the x-axis

The equation of this parabola can be written as:

y = (1/4)x^2 + 4

This parabola opens upwards, and its vertex is at the point (0, 4). The focus of this parabola is at the point (0, 4), and the directrix is the x-axis.

Parabola B: Points that are the same distance from (0, -6) and the x-axis

The equation of this parabola can be written as:

y = -(1/4)x^2 - 6

This parabola also opens upwards, but its vertex is at the point (0, -6). The focus of this parabola is at the point (0, -6), and the directrix is the x-axis.

Comparison of Parabola A and Parabola B

	Parabola A	Parabola B
Equation	y = (1/4)x^2 + 4	y = -(1/4)x^2 - 6
Vertex	(0, 4)	(0, -6)
Focus	(0, 4)	(0, -6)
Directrix	x-axis	x-axis
Opening	Upwards	Upwards

As we can see from the table above, both parabolas have the same equation form, but with different constants. Parabola A has a vertex at (0, 4), while Parabola B has a vertex at (0, -6). The focus and directrix of both parabolas are the same, and they both open upwards.

Contrast of Parabola A and Parabola B

	Parabola A	Parabola B
Distance from Focus	y = (1/4)x^2 + 4	y = -(1/4)x^2 - 6
Distance from Directrix	y = (1/4)x^2 + 4	y = -(1/4)x^2 - 6
Shape	U-shaped	U-shaped
Axis of Symmetry	y = 4	y = -6

As we can see from the table above, both parabolas have the same shape, which is U-shaped. However, the distance from the focus and directrix is different for both parabolas. Parabola A has a distance of y = (1/4)x^2 + 4 from the focus, while Parabola B has a distance of y = -(1/4)x^2 - 6 from the focus.

Finding the Center and Radius of Parabola A and Parabola B

The center of a parabola is the midpoint between the focus and the directrix. Since the focus and directrix of both parabolas are the same, the center of both parabolas is the same.

The radius of a parabola is the distance from the center to the focus. Since the center of both parabolas is the same, the radius of both parabolas is the same.

Conclusion

In this article, we compared and contrasted two parabolas with different definitions. We found that both parabolas have the same equation form, but with different constants. We also found that the focus and directrix of both parabolas are the same, and they both open upwards. Additionally, we found that the center and radius of both parabolas are the same.

Frequently Asked Questions

Q: What is a parabola?

A: A parabola is a type of quadratic curve that is defined as the set of all points that are equidistant from a fixed point (known as the focus) and a fixed line (known as the directrix).

Q: What is the equation of a parabola?

A: The equation of a parabola can be written in the form y = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola.

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is equidistant from all points on the parabola.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a fixed line that is equidistant from all points on the parabola.

Q: What is the center of a parabola?

A: The center of a parabola is the midpoint between the focus and the directrix.

Q: What is the radius of a parabola?

A: The radius of a parabola is the distance from the center to the focus.

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

A: To find the equation of a parabola, you need to know the coordinates of the vertex, the focus, and the directrix. You can then use the equation y = ax^2 + bx + c to write the equation of the parabola.

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

A: To find the vertex of a parabola, you need to know the coordinates of the focus and the directrix. You can then use the equation y = ax^2 + bx + c to find the vertex.

Q: How do I find the focus of a parabola?

A: To find the focus of a parabola, you need to know the coordinates of the vertex and the directrix. You can then use the equation y = ax^2 + bx + c to find the focus.

Q: How do I find the directrix of a parabola?

A: To find the directrix of a parabola, you need to know the coordinates of the vertex and the focus. You can then use the equation y = ax^2 + bx + c to find the directrix.

Q: How do I find the center of a parabola?

A: To find the center of a parabola, you need to know the coordinates of the focus and the directrix. You can then use the equation y = ax^2 + bx + c to find the center.

Q: How do I find the radius of a parabola?

A: To find the radius of a parabola, you need to know the coordinates of the center and the focus. You can then use the equation y = ax^2 + bx + c to find the radius.