Introduction to Parabolas

A parabola is a conic section that can be defined as the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix). In this article, we will explore how to find the equation of a parabola given its vertex and a point it passes through. We will also discuss the relationship between the vertex, the directrix, and the focus.

Vertex Form of a Parabola

The vertex form of a parabola's equation is given by:

y a(x - h)^2 k

where (h, k) is the vertex of the parabola.

Given Vertex and a Point

Suppose we are given a parabola with vertex at (-6, -6) and it passes through the point (3, -10). We need to find the equation of this parabola.

Step 1: Substitute the Vertex into the Vertex Form

Since the vertex is (-6, -6), the equation becomes:

y a(x 6)^2 - 6

Step 2: Find the Value of (a)

Since the parabola passes through the point (3, -10), we substitute (x 3) and (y -10) into the equation:

-10 a(3 6)^2 - 6

Simplifying, we get:

-10 a(9) - 6

-4 9a

a -frac{4}{9}

Step 3: Substitute (a) Back into the Equation

Substituting (a -frac{4}{9}) into the vertex form equation, we get:

y -frac{4}{9}(x 6)^2 - 6

Understanding the Parabola

The equation of the parabola is now:

y -frac{4}{9}(x 6)^2 - 6

Alternative Method Using Directrix and Focus

We can also find the equation of the parabola by using the directrix and focus. If the vertex is (-6, -6), the directrix and focus can be determined.

The general form of a parabola's equation when the vertex is at (h, k) and the directrix is y k - p (for a vertical parabola) or x h - p (for a horizontal parabola) is:

(y - k)^2 4p(x - h) (for a vertical parabola)

(x - h)^2 4p(y - k) (for a horizontal parabola)

Example: Horizontal Parabola

Consider the parabola with vertex at (-6, -6) and a point it passes through (3, -10). Given that it is a horizontal parabola, we can use the formula:

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

Substituting the vertex and the point:

(3 6)^2 4p(-10 6)

81 4p(-4)

p -frac{81}{16}

The equation of the parabola is then:

(x 6)^2 -frac{81}{16}(y 6)

Conclusion

In this article, we have explored how to find the equation of a parabola given its vertex and a point it passes through. We used the vertex form of the parabola and also discussed the alternative method using the directrix and focus. Understanding these concepts is crucial for solving more complex problems involving parabolas.