Parabola Patterns Slides Focus Point Parabolas
Understanding the geometry behind the curve: Focus, Directrix, and the Distance Rule.
The Definition
A parabola is the set of all points that are the same distance from:
A fixed point called the Focus.
A fixed line called the Directrix.
Key Rule:
Distance to Focus = Distance to Directrix
Directrix (y = -p) Focus (0, p) Point (x, y) d1 d2 d1 = d2
Meeting "p"
The variable p represents the distance from the Vertex to the Focus.
1
The Focus is at (0, p)
2
The Directrix is at y = -p
3
The Vertex is exactly in the middle at (0, 0)
Focus (0, p) y = -p Vertex (0,0) p p
Symmetry is Key
The Formula Bridge
To find the equation, we set the distances equal:
Distance to Focus
\[ \sqrt{(x-0)^2 + (y-p)^2} \]
=
Distance to Directrix
\[ \sqrt{(y - (-p))^2} \]
The simplified result?
x² = 4py
Real-World Focus
Parabolas have a unique Reflective Property. Any ray coming into the parabola parallel to its axis is reflected directly to the Focus.
Satellite Dishes
Flashlights
Radio Telescopes
Solar Cookers
All lines lead to the Focus
Focus Finder Worksheet Focus Finder Worksheet
Geometry Tier 2 Intervention: Standard HS.G-GPE.A.2
Student:
Date:
Mission Objective
Prove that a parabola is "equidistant" by graphing a set of points and measuring the distances to the focus and directrix.
1
The Coordinate Plane Setup
Follow these steps to set up your "Focus Point" graph:
Plot the Focus (F) at (0, 2). Label it.
Draw the Directrix (d) as a dashed horizontal line at y = -2.
Plot the Vertex (V) at (0, 0).
What is the value of p (the distance from Vertex to Focus)?
p = _________
x y
2
The Equidistant Test
Plot the point P(4, 2) on the graph above. Now, let's check the distances:
Distance to Focus (0, 2)
Count the horizontal units from (4,2) to (0,2):
Distance = ________ units
Distance to Directrix (y = -2)
Count the vertical units from (4,2) down to the line y = -2:
Distance = ________ units
Are the distances equal? (Circle one): YES NO
3
Connecting the Dots
Plot the point (-4, 2) . Use the symmetry of the parabola to sketch the smooth curve that passes through V(0,0) , P(4,2) , and (-4,2) .
Observation Challenge:
As the Focus gets further from the directrix, does the parabola get wider or narrower ?
Equation Builder Handout Equation Builder
Scaffolded Derivation: From Geometry to Algebra
Student:
The Setup
Focus: (0, p)
Directrix: y = -p
Point: (x, y)
Rule: Dist to Focus = Dist to Directrix
Step 1
Use the Distance Formula
Substitute Focus (0, p) and Point (x, y) into the left side. Substitute Point (x, y) and Directrix line into the right side.
Distance to Focus
\[ \sqrt{(x - 0)^2 + (y - p)^2} \]
=
Distance to Directrix
\[ \sqrt{(y - (-p))^2} \]
Step 2
Square Both Sides
This removes the radical signs. Fill in the missing parts below.
(x - 0)² + (y - p)² = (y + p)²
Simplify (x - 0)² to just x²
Step 3
Expand the Squares
Use FOIL to expand (y - p)² and (y + p)².
Expand (y - p)²
Hint: y² - 2py + p²
Expand (y + p)²
Hint: y² + 2py + p²
Step 4
Simplify to Final Form
Combine the expanded terms. Cross out terms that appear on both sides.
x² + y² - 2py + p² = y² + 2py + p²
Subtract y² and p² from both sides.
Add 2py to both sides.
Final Standard Form Equation:
x² = _________
Parabola Checkup Exit Ticket Parabola Checkup
Exit Ticket • Progress Monitoring
Student:
1
Vocabulary Link
Draw a line to match the term with its visual description.
2
Identify the Value
A parabola has a focus at (0, 3) and a directrix at y = -3 .
A) What is the value of p?
B) Write the equation in standard form \(x^2 = 4py\):
x² =
Quick Sketch for Reference
3
Reflect and Explain
In your own words, what does it mean for a parabola to be "equidistant"?
Mastered
Progressing
Needs Support
Checkup Score: ____ / 10
Intervention Facilitator Guide Intervention Facilitator Guide
Topic: Parabolas as Equidistant Point Sets (HS.G-GPE.A.2)
Lesson Snapshot
Target Audience
Tier 2 Small Group (3-5 Students)
Duration
45 - 60 Minutes
Prior Knowledge
Distance Formula, Basic Graphing, Squaring Binomials
Success Criteria
Students derive and apply \(x^2 = 4py\)
Instructional Delivery
1
The "Aha!" Moment (15 mins)
Use the Parabola Patterns Slides . Emphasize that the parabola isn't just a "u-shape"—it is a collection of points following a strict rule. Use physical string or a ruler on the Focus Finder worksheet to show that distance \(d_1 = d_2\).
Check for Understanding: "If I move the focus further up, what happens to the point directly between the focus and the directrix?"
2
Guided Graphing (15 mins)
Work through the Focus Finder Worksheet together. Many Tier 2 learners struggle with vertical/horizontal counting when measuring distances to lines. Explicitly model counting units from the point (4,2) down to the line \(y = -2\).
3
The Algebra Bridge (15 mins)
The Equation Builder Handout is heavily scaffolded. Don't let students get bogged down in the FOIL process—the goal is to see how the geometry (distances) creates the algebra (\(x^2 = 4py\)). If a student struggles, focus on the "p" substitution step.
Differentiation
For Visual Learners
Use different colored highlighters for Focus-to-Point and Directrix-to-Point distances.
For Algebraic Struggle
Provide the expanded terms (\(y^2 + 2py + p^2\)) as pre-printed stickers they can place on the worksheet.
Extension
Ask: "What would change if the directrix was vertical (\(x = -p\))?"
Common Hurdles
• Confusing \(p\) with the y-coordinate of the Focus. (It is the distance!)
• Mixing up signs on the directrix (using \(y=p\) instead of \(y=-p\)).
• Forgetting to square the binomials properly in the derivation.
Standards Alignment
Colorado HS.G-GPE.A.2: Derive the equation of a parabola given a focus and directrix.
