(x-1)^2=8y

5 min read Jun 17, 2024

Exploring the Equation: (x-1)^2 = 8y

The equation (x-1)^2 = 8y represents a parabola in the xy-plane. Let's delve into its properties and how to analyze it.

Understanding the Equation

Standard Form: The equation is already in a standard form that reveals key characteristics:
- Vertex: The vertex of the parabola is located at (1, 0). This is because the equation is in the form (x-h)^2 = 4p(y-k), where (h,k) represents the vertex. In this case, h = 1 and k = 0.
- Focus: The focus lies at a distance of p units above the vertex. In our equation, 4p = 8, so p = 2. This means the focus is located at (1, 2).
- Directrix: The directrix is a horizontal line located a distance of p units below the vertex. Therefore, the directrix is the line y = -2.
Direction: Since the equation is in the form (x-h)^2 = 4p(y-k), the parabola opens upwards.

Visualizing the Parabola

To visualize the parabola, you can:

Plot the vertex: Plot the point (1, 0).
Plot the focus: Plot the point (1, 2).
Draw the directrix: Draw the horizontal line y = -2.
Sketch the curve: The parabola will be symmetrical about the vertical line x = 1. The curve will pass through the focus and get increasingly wider as you move away from the vertex.

Key Properties

Axis of Symmetry: The vertical line x = 1 is the axis of symmetry.
Symmetry: The parabola is symmetric about the axis of symmetry.
Focal Length: The distance between the vertex and the focus (or the vertex and the directrix) is called the focal length. In this case, the focal length is 2.

Applications

Parabolas have many applications in mathematics, physics, and engineering, including:

Reflecting telescopes: The shape of a parabolic mirror is used to focus light from a distant source onto a single point.
Satellite dishes: The dish of a satellite dish is shaped like a parabola to focus radio waves from a satellite onto a receiver.
Projectile motion: The path of a projectile under the influence of gravity can be modeled as a parabola.

Further Exploration

You can further explore the equation (x-1)^2 = 8y by:

Finding the equation of the tangent line: Use the derivative of the parabola's equation to find the slope of the tangent line at any given point.
Calculating the distance between points on the parabola and the focus: You can use the distance formula to verify that the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix.
Finding the intercepts: Set y = 0 and solve for x to find the x-intercepts. Similarly, set x = 0 and solve for y to find the y-intercept.

By understanding the properties and characteristics of this equation, you can gain valuable insights into the behavior of parabolas and their diverse applications.