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Parabola
78 topics across 6 chapters
Chapter 1
Core meaning (geometry of a parabola)
1
Conic sections overview (where parabolas fit)
2
Focus–directrix definition
2 subtopics
3
Derive an equation from a focus and directrix (worked setup)
4
Find focus and directrix from a given equation (practice patterns)
5
Vertex & axis of symmetry (geometric view)
2 subtopics
6
Compute the vertex from an equation (vertex form and formulas)
7
Use symmetry to generate points: if (x,y) then (2h−x,y)
8
Focal length & latus rectum
2 subtopics
9
Compute latus rectum endpoints from p and the vertex
10
Relate latus rectum length to p (length = 4p) and focal width idea
11
Reflective property (rays reflect through the focus)
2 subtopics
12
Reflective property: angle argument / proof sketch
13
Ray-tracing exercises (incoming parallel rays reflect through focus)
Chapter 2
Equations & algebraic forms
14
Common equation forms of a parabola
4 subtopics
15
Vertex form: y=a(x−h)²+k (read vertex, direction, width)
16
Standard form: y=ax²+bx+c (coefficients and shape)
17
Factored form: y=a(x−r₁)(x−r₂) (read roots)
18
Sideways parabolas: x=a(y−k)²+h and y²=4px style
19
Convert between forms (rewrite the same parabola)
3 subtopics
20
Completing the square (step-by-step technique)
21
Expanding and factoring quadratics (algebra fluency)
22
Match features to forms (which form makes which feature obvious?)
23
Parameters & what they mean (a, p, opening, width)
3 subtopics
24
Focus/directrix parameter p: y²=4px and (x−h)²=4p(y−k)
25
Interpret the coefficient a (opening direction and “width”)
26
Axis of symmetry formula x=−b/(2a) and vertex from (a,b,c)
Chapter 3
Graphing & transformations
27
Sketch from key features (vertex, symmetry, intercepts)
2 subtopics
28
Pick symmetric x-values around the axis to build a point set
29
Fast sketch checklist: vertex, opening, 2–4 points, intercepts (if any)
30
Transformations of y=x² (shifts, stretches, reflections)
4 subtopics
↗ Interpret the coefficient a (opening direction and “width”) (see Chapter 2)
31
Translations: effects of (x−h) and +k on the graph
32
Stretches/compressions: how |a| changes curvature
33
Reflections: a<0 flips vertical parabolas; sign effects in sideways forms
34
Use a table / graphing tool to verify shape & features
35
Domain, range, and intervals (increasing/decreasing)
2 subtopics
36
Find domain/range directly from vertex form (and opening direction)
37
Intervals of increase/decrease using the vertex as the turning point
Chapter 4
Solving parabola problems (roots, intersections, optimization)
38
Find the vertex by completing the square (problem workflow)
2 subtopics
↗ Completing the square (step-by-step technique) (see Chapter 2)
39
Drill set: rewrite 10 quadratics into vertex form and verify by graphing
40
Roots / x-intercepts of a parabola
3 subtopics
↗ Expanding and factoring quadratics (algebra fluency) (see Chapter 2)
41
Solve roots by factoring (when possible) and check solutions
42
Quadratic formula + discriminant (number of real intersections)
43
Intersection with lines (solve systems)
2 subtopics
44
Line–parabola intersections by substitution (set up and solve)
↗ Quadratic formula + discriminant (number of real intersections) (see Chapter 4)
45
Max/min and optimization with quadratics
2 subtopics
46
Max/min via the vertex (complete the square or x=−b/2a)
47
Optimization word problems (revenue, area, constraints)
Chapter 5
Applications & modeling
48
Quadratic function modeling from constraints/data points
2 subtopics
49
Solve for a,h,k (or a,b,c) from given points/conditions
50
Model sanity checks: units, domain restrictions, and interpreting parameters
51
Projectile motion (basic quadratic model)
2 subtopics
52
Build a projectile model y(x) from kinematics assumptions
53
Compute max height and range from the quadratic model (interpret results)
54
Parabolic mirrors, headlights, and satellite dishes
3 subtopics
↗ Reflective property (rays reflect through the focus) (see Chapter 1)
55
Dish/headlight problems: compute focal length and placement of the source
56
Engineering constraints (approximation, scaling, and tolerance) in parabolic designs
57
Quadratic regression / curve fitting (intro)
2 subtopics
58
Quadratic regression with a calculator/spreadsheet (run and read output)
59
Evaluate fit: residuals, overfitting intuition, and extrapolation caution
Chapter 6
Advanced extensions
60
Parametric descriptions of parabolas
2 subtopics
61
Simple parametrizations: x=t, y=at²+bt+c; identify direction and vertex
62
Eliminate a parameter to recover the Cartesian equation
63
Calculus connections (slopes and areas)
2 subtopics
64
Derivative of a quadratic; tangent line slope at a point
65
Area under a parabola (basic definite integrals / geometric comparisons)
66
Distance-formula derivations (locus viewpoint)
3 subtopics
67
Distance formula refresher (2D coordinate geometry)
↗ Derive an equation from a focus and directrix (worked setup) (see Chapter 1)
68
Practice: set up “distance to focus = distance to directrix” and simplify cleanly
69
Rotated parabola & quadratic forms (beyond standard orientation)
2 subtopics
70
Recognize rotation (xy term) and why standard parabola formulas don’t apply directly
71
Complete-the-square with matrices / quadratic forms (high-level workflow)

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