Back to Derive parabola equation — Problem 4 · Task Set 31

Exercises: Derive the Equation of a Parabola Given Focus and Directrix

Show your work for each problem. Unless a problem says otherwise, assume the axis of symmetry is horizontal or vertical, and give equations in standard form.

Grade 10·23 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-gpe-a-2

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$Coordinate plane with a parabola opening left from the origin, a marked point at (-3, 0), and a dashed vertical line at x = 3$

In which direction does the parabola $y^2 = -12x$ open, and where is its focus?

Left, focus $(-3, 0)$ — the squared variable is $y$ , so the axis is horizontal, and $4p = -12$ gives $p = -3$ .

Right, focus $(3, 0)$ — the axis is horizontal, and the focus is taken $3$ units from the vertex on the positive side.

Down, focus $(0, -3)$ — the negative coefficient means the parabola opens downward from the origin.

Left, focus $(-12, 0)$ — the axis is horizontal and the coefficient $-12$ locates the focus directly.