The Parabola Lab
Unit 12: Conic Sections -- Parabolas
Discover
Step 1 of 7: Equidistant Points
A point (the focus, in orange) and a line (the directrix, dashed). Click on the canvas to place points. The tool measures both distances. Find every point that is exactly the same distance from the focus and the directrix.
Equidistant points found: 0 of 6
Step 2 of 7: The Curve Emerges
Look at the points you placed. They trace a curve. That is a parabola. Not "a u-shape." Not "the graph of x squared." It is the set of all points equidistant from a point and a line.
The points you found trace a parabola.
That is the definition. Every point on this curve is exactly the same distance from the focus as from the directrix. The U-shape is a consequence of that rule, not the definition itself.
Step 3 of 7: The Distance p
Drag the focus away from the directrix. What happens to the parabola? Now drag it closer. The distance from the vertex to the focus has a name: \(p\). The vertex sits exactly halfway between focus and directrix, because that is the only point on the axis that is equidistant.
With \(p = 2\), the focus is at \((0, 2)\) and the directrix is at \(y = -2\). The vertex is at the origin, exactly halfway.
Step 4 of 7: Deriving the Standard Form
The vertex is at the origin, the focus at \((0, p)\), the directrix at \(y = -p\). For any point \((x, y)\) on the curve, the distance to the focus equals the distance to the directrix. What equation does this give?
Step 5 of 7: The Reflecting Property
Parallel rays come in from above and hit the parabola. Watch where every single one goes after it bounces. Before you hit Play, predict: will the rays scatter, roughly converge, or hit a single exact point?
Step 6 of 7: Reverse It
Now a light source sits at the focus. Watch every ray bounce out. Predict: what direction will the rays go after bouncing?
Step 7 of 7: Off-Center and General Form
When the vertex moves to \((h, k)\), the equation becomes \((x - h)^2 = 4p(y - k)\). A general-form equation might hide this. Unlike an ellipse or hyperbola (which have two squared variables), a parabola has only one. So you complete the square once, on that one variable. Try it: which variable gets completed below?
Which variable do you complete the square on?
Explore the Parabola
Standard form:
Vertex form:
General form:
Focus: Directrix:
Adjust the controls to explore how changing the vertex position, p value, and orientation affects the parabola and its equation.
Completing the Square Workbench
Enter coefficients for a general-form equation and watch the conversion step by step.
Format: \(Ax^2 + Cy^2 + Dx + Ey + F = 0\). For a parabola, exactly one of A or C must be 0.
Try This
Starter: Satellite Dish Receiver
A satellite dish is 4 feet across and 1.5 feet deep at the center. Where should the receiver go? Give the distance from the vertex of the parabola.
The dish is a parabola opening upward with vertex at the origin. The rim of the dish is at \((\pm 2, 1.5)\). Use \(x^2 = 4py\) to find \(p\).
Answer within 0.05 feet.
Stretch: Bridge Clearance
A bridge arch is parabolic, 120 feet wide at the base and 40 feet tall at the center. A truck 30 feet wide and 15 feet tall needs to pass through, centered. Does it fit? What is the clearance at the truck's edge, not at the center?
Place the vertex at \((0, 40)\) and the base at \(y = 0\). The arch passes through \((\pm 60, 0)\), giving the equation \(x^2 = -90(y - 40)\).
Answer within 0.5 feet.
Challenge: Rooftop Ball Throw
A ball is thrown from a rooftop. Its path is \(x^2 = -12(y - 9)\), with \(x\) in feet horizontally from the release point and \(y\) in feet vertically.
Part A: What is the maximum height?
Part B: Where does it land (y = 0)? Give the positive horizontal distance.
Answer within 0.5 feet.
Part C: Which direction does the parabola open?
- Define a parabola as the set of points equidistant from a focus and a directrix.
- Derive and use the standard form \(x^2 = 4py\) (and its rotated variants).
- Identify the vertex, focus, directrix, and axis of symmetry from an equation.
- Convert between standard form, vertex form, and general form using completing the square.
- Explain and apply the reflecting property of parabolas.
- Solve real-world problems involving parabolic shapes.
Quick Check
A parabola has its vertex at the origin and its focus at \((0, 3)\). What is the equation?
Instructor Notes
Teaching Notes
- Start with the locus definition before showing any formula. The definition is the entire foundation: if students do not see the parabola as "equidistant from a point and a line," the formulas become arbitrary.
- Step 5 (the reflecting property) is the payoff. The rays converging on the focus is not a demonstration of an application -- it is a consequence of the definition. Make that link explicit: the definition creates the geometry, and the geometry creates the engineering.
- Step 7 connects to simulations 27 (ellipse/hyperbola): the key difference is one squared variable vs. two. Students who completed the square on both variables for an ellipse need to understand why you only do it once here.
- The workbench in Explore mode lets students practice completing the square with instant feedback, which is harder to replicate on a whiteboard.
Common Student Errors
- Confusing \(p\) with the focal width \(4p\). Students see \(x^2 = 4py\) and read the coefficient as \(p\) instead of \(4p\). Have them extract \(p\) from a given equation immediately after derivation.
- Trying to complete the square on both variables. Students who just finished the ellipse and hyperbola units carry the habit forward. The one-squared-variable observation (Step 7) is the cure.
- Forgetting which way the parabola opens. Positive \(p\) opens toward the focus; the sign of the coefficient tells you the direction.
- Placing the directrix on the wrong side of the vertex. The directrix is always on the opposite side of the vertex from the focus, distance \(p\) away.
Discussion Questions
- Why does every parabolic dish have its receiver at the focus and not somewhere else? What would happen if the receiver were moved slightly?
- An ellipse also has a reflecting property (reflecting between foci). How does the parabola's version differ, and why does that make it useful for different applications?
- If you see an equation with one squared term and one linear term, what can you say immediately about the shape?
Exam Connection
- Given a general-form equation, identify the conic, complete the square, and extract vertex, focus, and directrix. This is a staple exam question across 12.7-12.9.
- Word problems: satellite dish, headlight reflector, bridge arch. Students must set up coordinates, write the equation, and solve for the requested quantity.
- Distinguishing parabola from ellipse/hyperbola by looking at which variables are squared.