Students will learn how to derive the equation of a hyperbola in standard form from the following definition of a hyperbola: Let \(P\) be any point on the hyperbola. If \(d_1\) and \(d_2\) are the distances between \(P\) and each focus, then \(\lvert d_2 - d_1 \rvert\) is some constant. Here's a video. Then students will learn how to derive the equations for the slant asymptotes from standard form. Finally, students will learn how to convert between equations in standard form and graphs of ellipses centered at the origin.