Conic Sections

What AP Precalc asks here

Four families of conic sections show up in standard form: circles, ellipses, parabolas, and hyperbolas. Recognizing them from an implicit equation is mostly pattern matching: both squared terms with the same sign → ellipse (or circle if coefficients match); both squared but opposite signs → hyperbola; only one squared term → parabola. Completing the square on a general quadratic reveals the center and standard form.

Look at the squared terms: matching coefficients and same sign → circle; different but same sign → ellipse; opposite signs → hyperbola; one squared → parabola.

$h, k$ from the shifts; semi-axes from the denominators (square root for ellipse/hyperbola).

Complete the square on the $x$ terms and on the $y$ terms.