Find an equation of the parabola with vertex (-3,4) and focus (1,4).

Name: Problem 105, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-05-14T04:25:10-08:00
Duration: 2 min 10 s
Channel: Julie Silva
Description: Find an equation of the parabola with vertex (-3,4) and focus (1,4).

step 1 We're being asked to write an equation for the parabola whose vertex is at the point negative 3

Find an equation of the parabola with vertex (-3,4) and...

Key Concepts

Parabola

A parabola is a set of points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. This unique geometric property is the basis for deriving its equation and understanding its shape. The parabola is a fundamental conic section that appears in various applications, including physics, engineering, and architecture.

Vertex

The vertex of a parabola is the point where the curve changes direction, often considered the turning point. It is located exactly midway between the focus and the directrix. In the standard equation of a parabola, the vertex serves as the reference point from which other parts of the curve are defined via translations along the coordinate axes.

Focus

The focus is a fixed point inside the parabola that, along with the directrix, defines the set of all points on the curve. Every point on the parabola is equidistant from the focus and the directrix. The relative position of the focus with respect to the vertex determines the orientation and the 'width' of the parabola.

Directrix

The directrix is a line used in the geometric definition of a parabola, where every point on the parabola is equidistant from this line and the focus. The equation of the parabola is derived based on this equal distance property, making the directrix a key element in understanding the structure and orientation of the parabola.

Standard Form Equation of a Parabola

The standard form equation of a parabola is typically expressed in a way that incorporates the vertex and the distance from the vertex to the focus (denoted by p). For a parabola that opens horizontally or vertically, the equation takes the form (x - h)^2 = 4p(y - k) or (y - k)^2 = 4p(x - h), where (h, k) is the vertex. This form concisely represents the geometric characteristics of the parabola and provides a direct method for graphing it.

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