10.3Exploring Parabolas
Write parabola equations x² = 4py (vertical) and y² = 4px (horizontal) using the focus-directrix definition. Identify vertex, focus, directrix, and axis of symmetry. Apply the reflective property.
The parabola's reflective property — all rays parallel to the axis reflect through the focus — is used in satellite dishes, headlights, and solar collectors. The focus-directrix definition unifies the conic section family.
Essential Question
How does the focus-directrix definition of a parabola lead to the standard form equation, and how does the value of p determine the shape and direction of the parabola?
Lesson Overview
Definition
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
Standard Forms — Vertex at Origin
| Direction | Equation | Focus | Directrix |
|---|---|---|---|
| Opens up (p > 0) | x² = 4py | (0, p) | y = −p |
| Opens down (p > 0) | x² = −4py | (0, −p) | y = p |
| Opens right (p > 0) | y² = 4px | (p, 0) | x = −p |
| Opens left (p > 0) | y² = −4px | (−p, 0) | x = p |
Standard Forms — Vertex at (h, k)
| Type | Equation | Focus | Directrix |
|---|---|---|---|
| Vertical | (x−h)² = 4p(y−k) | (h, k+p) | y = k−p |
| Horizontal | (y−k)² = 4p(x−h) | (h+p, k) | x = h−p |
- Latus rectum: chord through focus perpendicular to axis; length = |4p|
- p > 0: opens up/right; p < 0: opens down/left
- Axis of symmetry: line through vertex and focus
Worked Examples
Identify all features of x² = 12y.
Form: x² = 4py → 4p = 12 → p = 3
Opens upward (p > 0)
Vertex: (0,0); Focus: (0,3); Directrix: y = −3
Axis of symmetry: x = 0; Latus rectum length: |4p| = 12
Write the equation of a parabola with vertex (0,0) and focus (−4, 0).
Focus on x-axis → horizontal parabola
p = −4 (negative → opens left)
Form: y² = 4px = 4(−4)x = −16x
Find all features of (x−2)² = 8(y+1).
Vertex: (2, −1); 4p = 8 → p = 2
Opens upward
Focus: (2, −1+2) = (2, 1)
Directrix: y = −1−2 = −3
Axis of symmetry: x = 2
Write the equation of a parabola with vertex (3, −2) and directrix x = 1.
Directrix is vertical → horizontal parabola
Distance from vertex to directrix: |3−1| = 2 → p = 2 (opens right, since focus is to the right)
Form: (y−(−2))² = 4(2)(x−3) → (y+2)² = 8(x−3)
Convert y² − 6y − 8x + 1 = 0 to standard form.
Isolate y terms: y² − 6y = 8x − 1
Complete the square: (y−3)² − 9 = 8x − 1
(y−3)² = 8x + 8 = 8(x+1)
Vertex: (−1, 3); 4p = 8 → p = 2
Opens right; Focus: (1, 3); Directrix: x = −3
Guided Practice
Identify all features of y² = −20x.
Hint: Form y² = 4px with 4p = −20. Negative p means opens left.
Write the equation of a parabola with vertex (0,0) and directrix y = 5.
Hint: Directrix y = 5 is above the vertex → parabola opens downward. p = −5.
Find all features of (y+4)² = −12(x−1).
Hint: Horizontal parabola. 4p = −12 → p = −3. Opens left.
Write the equation of a parabola with focus (2, 3) and directrix y = −1.
Hint: Vertex is midpoint between focus and directrix: y-coordinate = (3+(−1))/2 = 1. p = 3−1 = 2.
Convert x² + 4x − 8y + 20 = 0 to standard form.
Hint: Complete the square for x: (x+2)² = 8y − 16 = 8(y−2).
Key Vocabulary
Parabola
Set of all points equidistant from the focus and the directrix.
Focus
The fixed point inside the parabola; distance p from vertex.
Directrix
The fixed line outside the parabola; distance p from vertex on the opposite side.
Vertex
The point on the parabola closest to the directrix; midpoint between focus and directrix.
Axis of symmetry
The line through the vertex and focus; the parabola is symmetric about this line.
p value
The directed distance from vertex to focus; p > 0 opens up/right, p < 0 opens down/left.
Latus rectum
The chord through the focus perpendicular to the axis; length = |4p|.
Standard form
(x−h)² = 4p(y−k) (vertical) or (y−k)² = 4p(x−h) (horizontal).
Practice Quiz
Interactive Practice — 5 Questions
For x² = −8y, the focus is at:
A parabola has vertex (0,0) and focus (0, 5). Its equation is:
For (x−3)² = −12(y+1), the directrix is:
The latus rectum length of y² = 16x is:
Which parabola opens to the left?
Independent Practice
Independent Practice
Find all features of x² = −24y.
Write the equation: vertex (0,0), focus (6, 0).
Find all features of (x+3)² = 16(y−2).
Write the equation: vertex (−1, 4), directrix x = −4.
Convert x² − 10x − 4y + 29 = 0 to standard form.
Common Mistakes
Confusing the sign of p with the direction of opening
p > 0 opens up (vertical) or right (horizontal). p < 0 opens down (vertical) or left (horizontal). The sign of p tells you the direction.
Placing the focus and directrix on the same side of the vertex
The focus is INSIDE the parabola; the directrix is OUTSIDE. They are on OPPOSITE sides of the vertex, each at distance |p|.
Using the wrong form for horizontal vs. vertical parabolas
If x is squared → vertical parabola (opens up/down). If y is squared → horizontal parabola (opens left/right).
Forgetting to divide by 4 to find p from the equation x² = 4py
Always write the equation as x² = 4py first, then p = (coefficient)/4. Don't confuse 4p with p.
Math Tips
Quick identification: x² → vertical parabola; y² → horizontal parabola.
The vertex is always the midpoint between the focus and the directrix.
Latus rectum = |4p|. A wider parabola has a larger |p|; a narrower one has a smaller |p|.
To find p from standard form: 4p = coefficient of the linear term. Then p = coefficient/4.
When completing the square for a parabola, move the linear term to the right side FIRST, then complete the square on the quadratic side.