CCSS.Math.Content.HSG-GPE.A.2
The Standard
Derive the equation of a parabola given a focus and directrix.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students represent the condition that any point P(x,y) is equally distant from a fixed point and a line. They use distance formulas, then square and simplify to get a parabola equation.
What Mastery Looks Like
- A student writes the point-to-point and point-to-line distances correctly, sets them equal, and simplifies without losing terms. They check the vertex, axis, and opening direction against the given geometry.
Common Misconceptions
- Students may measure to a point on the directrix instead of using perpendicular distance to the line. They often confuse p with 4p when simplifying. Some assume the vertex is at the origin or place it at the focus.
How to Assess It
- Exit ticket: A parabola has focus (0, 2) and directrix y = -2. Derive its equation, showing the equal-distance equation before simplifying.
Lesson moves
Ways to Teach It
On coordinate paper, mark a focus and directrix, then use rulers to plot ten equidistant points and sketch the resulting curve.
Ask students to explain why squaring the two distance expressions does not change the set of points on the parabola.
Run a matching game with focus and directrix cards, distance equation cards, and simplified parabola equation cards.
Map locations equidistant from a rescue station and a straight shoreline, then write the parabola that models the boundary.
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Printable HSG-GPE.A.2 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-GPE.A.2, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.8.G.B.8
Deriving a parabola requires setting the variable point’s distance to the focus equal to its distance from the directrix.
- CCSS.Math.Content.HSG-GPE.A.1
Students reuse distance relationships and coordinate algebra from circle equations to express points equidistant from a focus and directrix.
- CCSS.Math.Content.HSA-CED.A.2
Deriving a parabola uses the same skill of turning a two-variable relationship into an equation and interpreting its coordinate graph.
Keep exploring
Related Standards
- CCSS.Math.Content.HSF-IF.C.8a
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in ...
- CCSS.Math.Content.HSG-GPE.A.3
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
- CCSS.Math.Content.HSA-REI.C.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
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