A circle is the set of points a fixed distance from a centre; a parabola is the set of points the same distance from a POINT and a LINE. These free printable worksheets give the focus and the directrix and ask for the equation — and a companion setting reads the focus and directrix back off an equation.
If they get stuck
The answer key prints on a separate page.
The method is two observations and a substitution. Every point on the curve is equally far from the focus and the directrix, so the vertex — the point on the curve closest to both — sits exactly halfway between them. The distance from the vertex to either one is p, and the equation is (x − h)² = 4p(y − k). Nothing else is needed, and a student who can find the halfway point can do the whole page.
p carries a sign and that is where the marks are lost. A directrix ABOVE the focus makes p negative and the parabola opens downwards; a student who reads the two numbers without noticing which is higher writes the curve upside down. Both cases are dealt on every sheet on purpose, and the worked answer says which way the curve opens rather than leaving it inside the arithmetic.
The equation of a parabola
