find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curv

step 1 First, this parabola is not in the standard position since the vertex is 3 -2, which means the p

Find a Cartesian equation for the conic section satisfying...

Key Concepts

Parabola

A parabola is a type of conic section defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This geometric property gives the parabola its characteristic symmetrical, U-shaped curve, which can open upward, downward, leftward, or rightward, depending on its orientation.

Vertex

The vertex of a parabola is the point where the curve changes direction. It lies exactly midway between the focus and the directrix and acts as the point of symmetry for the parabola. The vertex is fundamental in determining the parabola's standard form equation and its overall orientation in the coordinate plane.

Focus

The focus is a fixed point used in the definition of a parabola, playing a key role in its geometric construction. Every point on the parabola is equidistant from the focus and the directrix, making the focus essential in deriving the parabola’s equation and understanding its properties, such as the distance that defines the 'steepness' or 'width' of the curve.

Directrix

The directrix is a fixed line in the plane used in conjunction with the focus to define a parabola. By ensuring that each point on the parabola is equidistant from the focus and the directrix, the directrix helps establish the curve’s unique geometric properties. It is crucial for formulating the distance relationship that leads to the parabola's Cartesian equation.

Cartesian Equation of a Parabola

The Cartesian equation of a parabola is formulated using the distance definition of the parabola. By setting the distance from a generic point (x, y) to the focus equal to its distance to the directrix, one obtains an equation that can be simplified into forms such as (x - h)^2 = 4p(y - k) or (y - k)^2 = 4p(x - h), where (h, k) is the vertex of the parabola and p denotes the distance from the vertex to the focus. This equation is instrumental in graphing and analyzing the parabola's properties.

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