Finding the Standard Equation of a Parabola with a Given Vertex and Focus

When analyzing a parabola, understanding how to derive its standard equation given its vertex and focus is a fundamental skill. This article will guide you through a step-by-step process to find the standard equation of a parabola with a vertex at -32 and a focus at -22.

Understanding the Basics

A parabola is a conic section characterized by its symmetry about the axis of symmetry. The standard form of a parabola's equation is given by y - k2 4p(x - h), where:

(h, k) represents the vertex of the parabola. 4p determines the directrix and the distance from the vertex to the focus. The focus of the parabola is at (h p, k).

Given Data and Formulas

The parabola considered has a vertex at (-32, -22) and a focus at (-22, -22). Let's break down the steps to find the standard equation:

Step 1: Extracting the Vertex and Focus

The vertex (h, k) (-32, -22).

Step 2: Determining the Value of p

The focus is 10 units away from the vertex along the x-axis. Since the focus is at (-22, -22), and the vertex is at (-32, -22), we calculate p as:

p -22 - (-32) 10

Step 3: Using the Standard Equation

The standard form of the parabola's equation is:

y - k2 4p(x - h)

Substituting the values h -32, k -22, and p 10:

y 222 40(x 32)

Rewriting the equation for clarity:

y 484 40(x 32)

Alternative Method

A more elegant approach is to shift the origin to the vertex and then derive the equation in this new coordinate system.

Step 4: Shifting the Origin

Let's shift the origin from (0, 0) to (-32, -22). In this new coordinate system, the vertex is at the origin (0, 0), and the focus is at (0 10, 0) (10, 0).

The standard form of the parabola in the new coordinate system is:

y2 4ax

Here, a is the focal distance from the origin to the focus, which is 10.

So, the equation in the new coordinate system is:

y2 4

Step 5: Reversing the Shift

Now, we need to reverse the shift of the origin to find the equation of the parabola in the original coordinate system (0, 0).

The origin shift transformation is:

x' x 32 y' y 22

The equation in the new (x', y') coordinate system is:

(y' - 22)2 40(x' 32)

Re-substituting x' and y':

(y - 22)2 40(x 32)

Thus, the standard equation of the parabola in the original coordinate system is:

(y 22)2 40(x 32)

Conclusion

The equation of the parabola with a vertex at -32 and a focus at -22 is given by: