Solve the given problems. The equation of a parabola with...

Solve the given problems.
The equation of a parabola with vertex $(h, k)$ and axis parallel to the $x$ -axis is $(y-k)^{2}=4 p(x-h) .$ (This is shown in Section 21.7.) Sketch the parabola for which $(h, k)$ is (2,-3) and $p=2.$

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Key Concepts

Standard Form of a Parabola

The standard form of a parabola that opens either to the left or right is given by (y - k)² = 4p (x - h). This form clearly identifies the vertex of the parabola, (h, k), and emphasizes the role of the parameter p in determining the shape and orientation of the parabola.

Vertex of a Parabola

The vertex of a parabola, represented as (h, k) in its equation, is the point where the parabola changes direction. It is a key reference point for sketching the graph and understanding the geometric properties of the parabola.

Parameter p

The parameter p in the parabolic equation determines the distance between the vertex and the focus, as well as the directrix. The sign of p indicates the direction in which the parabola opens, while its absolute value affects the 'width' or 'steepness' of the parabola.

Orientation of the Parabola

When the axis of the parabola is parallel to the x-axis, as indicated by the equation (y - k)² = 4p (x - h), the parabola opens either to the right (if p is positive) or to the left (if p is negative). This orientation influences how the vertex, focus, and directrix are positioned relative to one another.

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