KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 11Course HomeTeacher Key

Unit 11 · Lesson 05 · Week 32 · Matrices, Systems & Conic Sections

Ellipses and Their Geometry

Recognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. In this lesson, the central focus is ellipses and their geometry. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in ellipses and their geometry.
  • Construct or analyze at least two representations related to ellipses and their geometry.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How do matrices, systems, and conic forms organize transformations, intersections, and geometric models?

Multiple-representation model

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy

Recognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms.

(x-h)^2=4p(y-k)\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1

Worked reasoning

Standard form(x−2)²/25+(y+1)²/9=1 is an ellipse centered at (2,−1) with horizontal semi-major axis 5.
Geometric meaningFor a parabola, every point is equidistant from the focus and directrix; the parameter p determines both orientation and focal distance.

Misconception check: The denominator under each squared term identifies axis length, while the signs distinguish ellipses from hyperbolas.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Ellipses and Their Geometry

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U11 · L05
Name
Date
1

Identify (x−2)²=8(y+1).

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
2

Find the center of (x+3)²/16+(y−2)²/9=1.

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
3

Find the semi-major axis of x²/25+y²/9=1.

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
4

Identify x²/9−y²/4=1.

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
5

State the asymptotes of x²/9−y²/4=1.

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
6

For y²=12x, find p.

Conics visualRecognize, graph, and model parabolas, ellipses, and hyperbolas from standard, translated, and contextual forms. xy
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