Find the vertex, focus, and directrix of each parabola; find...

Find the vertex, focus, and directrix of each parabola; find the center, vertices, and foci of each ellipse; and find the center, vertices, foci, and asymptotes of each hyperbola. Graph each conic.
$$\frac{(x-3)^{2}}{16}-\frac{(y+4)^{2}}{9}=1$$

Key Concepts

Asymptotes

Asymptotes are straight lines that a hyperbola approaches but never touches. They indicate the direction in which the branches of the hyperbola extend towards infinity and are key for graphing the overall shape of the hyperbola accurately.

Directrix

A directrix is a line used in the definition of a parabola such that every point on the parabola is equidistant from the focus and the directrix. Although not used in the definitions of ellipses and hyperbolas, the concept of a directrix is essential in understanding the geometric properties of parabolas.

Focus

A focus (or foci in plural) is a special point used to define conic sections. For ellipses and hyperbolas, the relative distances of any point on the curve to the foci determine its location. In parabolas, the focus, along with the directrix, is fundamental in defining the curve's reflective properties.

Vertex

A vertex is a key point where the curve either attains a maximum or minimum or where the conic changes direction. In ellipses and hyperbolas, the vertices lie on the principal axis of symmetry, while in parabolas the vertex serves as the turning point of the curve.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types of conic sections include circles, ellipses, parabolas, and hyperbolas, each defined by specific geometric properties and standard equations. Understanding their definitions and properties provides the foundation for analyzing and graphing these curves.

Hyperbola

A hyperbola is defined as the set of all points for which the absolute difference of the distances to two fixed points (the foci) is constant. It consists of two separate branches and is characterized by a center, vertices, and foci, along with asymptotes that describe the direction in which the branches extend.

Standard Form Equations

Each type of conic section has a standard form equation that makes it easier to identify its key components such as vertices, foci, centers, and asymptotes (for hyperbolas). The standard form is crucial for systematic problem solving and graphing, as it highlights the geometric features directly from the algebraic expression.

Center

The center of a conic section, particularly in ellipses and hyperbolas, represents the midpoint between key features (such as the vertices or foci). It serves as a reference point for the symmetry of the graph and is essential in rewriting the conic's equation into its standard form.

Transcript

Please give Ace some feedback