CCSS HSG-GPE.A.2 Worksheets
Algebra 2 · Expressing Geometric Properties with Equations
Derive the equation of a parabola given a focus and directrix.
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Teaching guide
What students learn
Derive the equation of a parabola given a focus and directrix.
Worked example
For the parabola (x - 1)² = -8(y + 1), state the vertex, the direction it opens, the focus, and the directrix.
Answer: Vertex (1, -1); opens down; focus (1, -3); directrix y = 1
Why: Match (x - h)² = 4p(y - k): h = 1, k = -1, and 4p = -8 so p = -2. The vertex is (1, -1). Since p < 0 the parabola opens down. The focus is p = -2 units from the vertex along the axis x = 1: (1, -3). The directrix is the same distance on the other side: y = 1.
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