Classifying Conic Sections (Graphs)

A conic section is a curve obtained by intersecting a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas.

The general equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

The standard form of an ellipse centered at (h, k) is (x - h)²/a² + (y - k)²/b² = 1, where a and b are the semi-major and semi-minor axes.

The standard form of a hyperbola centered at (h, k) is (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1.

The equation of a parabola that opens upwards with vertex (h, k) is (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus.

A circle can be identified from its equation if it is in the form (x - h)² + (y - k)² = r², where r is a positive number.

An ellipse is distinguished from a circle by having two different axes (major and minor), while a circle has a constant radius from the center.

In a hyperbola, the distance between the foci is greater than the distance between the vertices, and the foci lie outside the vertices.

A parabola is a conic section that is formed by the intersection of a cone with a plane parallel to its side. It has a single focus and directrix.

A conic section is a hyperbola if its equation can be rearranged into the form (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1.