Introduction to Finding Vertex, Focus, and Focal Length for Parabolas

Understanding the Equation

To find the vertex, focus, and focal length for the given equation x^2 4y 8x -4, we start by rewriting it in a standard form. This process involves completing the square. Here's a step-by-step guide to help you understand the solution more thoroughly.

Step-by-Step Solution

1. Rewrite the Equation in a Standard Form

First, let's rewrite the given equation in a more manageable form: [x^2 8x 4y -4] To complete the square, we analyze the terms involving (x): [x^2 8x 4y -4] Isolate the (x) terms on one side and the constant on the other: [x^2 8x -4y - 4] Now, complete the square for the (x) terms. We take half of the coefficient of (x) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives 16. Adding 16 to both sides: [x^2 8x 16 -4y - 4 16] Simplify the right side of the equation: [x^2 8x 16 -4y 12] Now we can write the left side as a perfect square: [(x 4)^2 -4y 12] Rearrange this equation to isolate the (y) term on the right side: [(x 4)^2 -4(y - 3)] This is the standard form of a parabola: [(x - h)^2 4a(y - k)] where ((h, k)) is the vertex and (a) is the coefficient of the (y) term.

2. Identify the Vertex, Focus, and Focal Length

From the standard form, we can determine the vertex, focus, and focal length of the parabola.

Vertex ((h, k)):

From the equation ((x 4)^2 -4(y - 3)), the vertex is at ((-4, 3)).

Focus:

The focus of the parabola is given by ((h, k a)), where (a -1). Thus, the focus is at ((-4, 3 - 1) (-4, 2)).

Focal Length:

The focal length is the distance between the vertex and the focus, which is 1 unit.

Conclusion and Summary

In summary, for the given equation (x^2 4y 8x -4), the vertex is ((-4, 3)), the focus is ((-4, 2)), and the focal length is 1 unit. The steps involved in finding these elements included rewriting the equation in standard form, completing the square, and identifying key components of the parabola's equation.

Frequently Asked Questions

Q: How do I identify the vertex of a parabola from its equation?

A: To identify the vertex, rewrite the equation in the standard form ((x - h)^2 4a(y - k)). The vertex can be read directly from the equation as ((h, k)).

Q: What is the formula for the focus of a parabola?

A: The focus of a parabola in the form ((x - h)^2 4a(y - k)) is at ((h, k a)).

Q: How do I calculate the focal length?

A: The focal length of a parabola is the absolute value of (a). In our example, since (a -1), the focal length is 1 unit.

References

[1] Stewart, J. (2016). Precalculus: Mathematics for Calculus (7th ed.). Cengage Learning. [2] Larson, R., Edwards, B. H. (2016). Calculus: Early Transcendental Functions (6th ed.). Cengage Learning.