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

In the realm of conic sections, understanding the standard equation of a parabola is crucial. This article delves into the process of finding the equation of a parabola given its focus and vertex. We will go through each step in detail, ensuring clarity and a comprehensive understanding for both beginners and advanced learners.

Step-by-Step Guide to Finding the Parabola’s Equation

To determine the standard equation of a parabola given its focus and vertex, we must follow a systematic approach:

Identify the Vertex and Focus

Vertex: V (1, 3) Focus: F (1, 4)

Determine the Orientation of the Parabola

The orientation of a parabola is crucial for writing its standard equation. Here, we analyze the orientation based on the x-coordinates of the vertex and focus:

Since the x-coordinates of both the vertex and the focus are the same (both are 1), the parabola opens vertically. As the focus is above the vertex, the parabola opens upward.

Find the Distance p

The distance ( p ) is the distance from the vertex to the focus. We calculate ( p 4 - 3 1 ).

Write the Standard Form of the Equation

The standard form of a vertically oriented parabola is derived from the vertex form equation:

[x - h^2 4py - k]

Where ((h, k)) is the vertex. Substituting the values, we get:

[1 - (x - 1)^2 4(1)(y - 3)]

Upon simplification, the equation becomes:

[x - 1 4y - 12]

Further simplification yields:

[x - 1 4y - 12]

or, equivalently:

[x^2 - 2x - 4y 13 0]

Finally, we express y in terms of x:

Alternative Expressions

Another form of the standard equation can be written as:

[y frac{1}{4}x^2 - frac{1}{2}x - frac{13}{4}]

Key Concepts and Related Information

Axis of Symmetry

The axis of symmetry for this parabola is vertical at ( x 1 ).

Directrix

The directrix is always perpendicular to the axis of symmetry. Since the parabola opens upwards, the directrix is located at ( y 2 ).

Summary of the Findings

In conclusion, the standard equation of the parabola given the focus at (1, 4) and the vertex at (1, 3) is:

[x - 1^2 4y - 3]

This can also be expressed as:

[y frac{1}{4}x^2 - frac{1}{2}x - frac{13}{4}]

Further, the parabola opens upward, has a vertex at (1, 3), and its directrix is at ( y 2 ).

Understanding these calculations and equations is essential for anyone interested in conic sections and their real-world applications.