(x-1)^2+8(y+2)=0

3 min read Jun 17, 2024

Exploring the Equation: (x-1)^2 + 8(y+2) = 0

This equation represents a parabola in the standard form. Let's break down its characteristics and how to interpret it.

Understanding the Standard Form

The general standard form of a parabola with a vertical axis of symmetry is:

(x - h)^2 = 4p(y - k)

Where:

(h, k) is the vertex of the parabola.
p is the distance between the vertex and the focus, and also the distance between the vertex and the directrix.

Analyzing the Equation

Our equation, (x-1)^2 + 8(y+2) = 0, can be rewritten to match the standard form:

(x - 1)^2 = -8(y + 2)

Now, we can identify the key features:

Vertex: The vertex is located at (1, -2).
4p: -8. Therefore, p = -2. This indicates the parabola opens downward.

Interpreting the Results

Focus: Since 'p' is negative, the focus lies below the vertex. The focus is located at (1, -4).
Directrix: The directrix is a horizontal line located 'p' units above the vertex. Therefore, the directrix is the line y = 0.

Visualizing the Parabola

To visualize the parabola, plot the vertex, focus, and directrix. The parabola will be symmetrical about the vertical line passing through the vertex, with the focus as a point that is equidistant to all points on the parabola and the directrix.

Summary

The equation (x-1)^2 + 8(y+2) = 0 represents a parabola with a vertex at (1, -2), a focus at (1, -4), and a directrix at y = 0. The parabola opens downward. Understanding the standard form and key parameters helps in accurately interpreting and visualizing the curve represented by this equation.