College Algebra, Chapter 8, 8.1, Section 8.1, Problem 52

a.) Determine the equations for the family of parabolas with vertex at the origin, focus on the positive $y$-axis, and with focal diameters $1, 2, 4$ and $8$.

b.) Draw the graphs and state conclusions.



a.) If the focus is on positive $y$-axis, then its focus is located at $F(0, p)$. The equation $x^2 = 4py$ is a parabola with vertex at origin and opens upward with focus at $F(0, p)$ where $4p$ is the length of the focal diameters. So, if the focal diameters are $1, 2, 4$ and $8$ then the equations of the parabola are $x^2 = y, x^2 = 2y, x^2 = 4y$ and $x^2 = 8y$ respectively.



b.)







It shows from the graph that as the focal diameters increases, the graph of the parabola expands horizontally.