College Algebra Module 15 Conic Sections Module Learning
College Algebra Module 15: Conic Sections
Module Learning Outcomes: Conic Sections • • • Introduction to The Ellipse Introduction to The Hyperbola Introduction to The Parabola
The Ellipse
Learning Outcomes: The Ellipse Introduction to The Ellipse • Write equations of ellipses in standard form • Graph ellipses centered at the origin • Graph ellipses not centered at the origin • Solve applied problems involving ellipses
Conic Sections • A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape. Ellipse Hyperbola Parabola
Understanding Ellipses •
Understanding Ellipses (cont. )
Equation of a Horizontal Ellipse •
Equation of a Vertical Ellipse •
Finding the Equation of an Ellipse •
Finding the Equation of an Ellipse (cont. ) •
Graphs of Ellipses •
Equations of an Ellipse •
The Hyperbola
Learning Outcomes: The Hyperbola Introduction to the Hyperbola • Locate a hyperbola’s vertices and foci • Write equations of hyperbolas in standard form • Graph hyperbolas centered at the origin • Graph hyperbolas not centered at the origin • Solve applied problems involving hyperbolas
Understanding Hyperbolas •
Understanding Hyperbolas (cont. ) • Every hyperbola has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. • The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex.
Equation of a Horizontal Hyperbola •
Equation of a Vertical Hyperbola •
Asymptotes of a Hyperbola •
Hyperbolas Not Centered at the Origin
Finding the Equation of a Hyperbola •
Finding the Equation of a Hyperbola (cont. ) •
Graphs of Hyperbolas •
Desmos Interactive 1 • • Topic: transformations of hyperbolas https: //www. desmos. com/calculator/jilkpkpse 1
The Parabola
Learning Outcomes: The Parabola Introduction to the Parabola • Graph parabolas with vertices at the origin • Write equations of parabolas in standard form • Graph parabolas with vertices not at the origin • Solve applied problems involving parabolas
Parabolas •
Parabolas with Vertex at the Origin •
Graphs of Parabolas with Vertex at the Origin •
Desmos Interactive 2 • Topic: find the focus, directrix, and endpoints of the focal diameter of a parabola whose vertex is at the origin • https: //www. desmos. com/calculator/wunbnybenw
Parabolas with Vertices Not at the Origin • Horizontal Parabola Axis of Symmetry Equation Focus Directrix Endpoints of Focal Diameter Vertical Parabola
Desmos Interactive 3 • Topic: find the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter of a parabola whose vertex is not at the origin • https: //www. desmos. com/calculator/gvpu 51 ti 7 n
Quick Review • • What are three types of conic sections? From the standard form equation of an ellipse, how can you tell if the major axis is a vertical line or a horizontal line? What is the transverse axis of a hyperbola? From the standard form equation of a hyperbola, how do you find the line equations for the asymptotes? What is the focal diameter? What are the coordinates of the vertices and co-vertices of a hyperbola? What is a directrix? How can you tell in which direction a parabola opens?
- Slides: 34