Conic Sections Study Guide

Overview of Conic Sections and Assignments

This study guide covers conic sections and their properties, which will be explored in M308 Algebra 2, Chapter 6. The following sections outline the assignments, key concepts, examples, and formulas related to each conic section.

Part 1: Assignment Schedule

Week of March 30 - April 15

  • March 30 (Mon): Introduction to Conics - Media Center Lab Day.
  • March 31 (Tue): Late Start - Conic Sections Intro Video. Watch Day 2 Notes Video on Circles.
  • April 1 (Wed): Study Circles (Section 2.2). Complete Pg. 198-200, problems #1-3, 19-24 (standard form), and #41, 42, 46, 48.
  • April 2 (Thu): Graphing Ellipses (Section 6.2). Complete Pg. 624-625, #3-13 odds, and #29, 31, 37, 39.
  • April 3 (Fri): No School. Catch up with Classkick Assignments.
  • April 6 (Mon): Graphing Parabolas (Section 6.1). Complete Pg. 613-614, #19-29 (graph all).
  • April 7 (Tue): Graphing Hyperbolas (Section 6.3). Complete Pg. 633-634, #1-4, 6, 7, 9, 12, 17, 19, 22.
  • April 8 (Wed): Standard Form for Conics. Complete Worksheet 1.
  • April 9 (Thu): Combination of Conics (Section 6.4). Complete Pg. 641, #2, 9, 10, 12; Graph #13, 14, 16-17 and #20, 22, 24, 29, 36-38. Revise #34 for equation = -1, not -6.
  • April 10 (Fri): Quiz #1 - Study for quiz.
  • April 13 (Mon): Parabolas (Writing Equations, Section 6.2). Complete Pg. 614, #31-33, 37-42 and Pg. 646, #51.
  • April 14 (Tue): Ellipses (Writing Equations, Section 6.1). Complete Pg. 624, #15-23, 25, 26.
  • April 15 (Wed): ACT for Juniors. No School for Freshmen, Sophomores, and Seniors.

Part 2: Important Definitions and Concepts

General Definitions

  • Conic Sections: The set of curves obtained by intersecting a right circular cone with a plane. Types include circles, ellipses, parabolas, and hyperbolas.
  • Standard Form: The form in which equations of conic sections are expressed for easier identification of their key features.

Specific Conic Definitions

1. Circle
  • Definition: The set of points that are equidistant from a fixed point (the center) in a plane.
  • Equation: The general equation is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 where (h, k) is the center and r is the radius.
2. Ellipse
  • Definition: The set of points in a plane where the sum of the distances to two fixed points (foci) is constant.
  • Standard Form:
      - Centered at origin: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (if a > b it's a horizontal ellipse; if b > a it's a vertical ellipse).
      - If centered at (h, k): (x−h)2a2+(y−k)2b2=1\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
3. Parabola
  • Definition: Set of points that are equidistant from a point (focus) and a line (directrix).
  • Standard Form:
      - Vertical: y=a(x−h)2+ky = a(x - h)^2 + k (opens up or down).
      - Horizontal: x=a(y−k)2+hx = a(y - k)^2 + h (opens left or right).
4. Hyperbola
  • Definition: The set of points where the absolute difference in distances to two fixed points (foci) is constant.
  • Standard Form:
      - Centered at origin: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (opens left and right) or y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 (opens up and down).

Part 3: Key Properties for Graphing Conics

Graphing Ellipses

  1. Center: Identify the center point (h, k) of the ellipse.
  2. Axes: Determine lengths a (semi-major) and b (semi-minor).
  3. Vertices: Calculate endpoints of the major axis and minor axis as $(h ext{± } a, k)$ and $(h, k ext{± }b)$ respectively.
  4. Foci: Calculate foci coordinates using c=extsqrt(a2−b2)c = ext{sqrt}(a^2 - b^2) where c is the distance from the center to each focus.
  5. Eccentricity: It’s defined as e=cae = \frac{c}{a}; it measures how much the ellipse deviates from being circular.

Graphing Parabolas

  1. Vertex: Identify vertex at (h, k).
  2. Focus: Locate the focus point.
  3. Directrix: Equation of the directrix defined by the distance from the vertex.
  4. Axis of Symmetry: Vertical or horizontal axis passing through the vertex.
  5. Length of Latus Rectum: The width across at the focus, extLatusRectum=4p∣a∣ext{Latus Rectum} = \frac{4p}{|a|} for standard form.

Graphing Hyperbolas

  1. Center: Determine the center point (h, k).
  2. Transverse Axis: Length is 2a; vertices are found as (h±a, k) or (h, k±a).
  3. Conjugate Axis: Length is 2b; co-vertices are computed as (h±b, k) or (h, k±b).
  4. Foci: Calculate positions using c=extsqrt(a2+b2)c = ext{sqrt}(a^2 + b^2).
  5. Asymptotes: Found equations are y−k=ba(x−h)y-k = \frac{b}{a}(x-h) and y−k=−ba(x−h)y-k = -\frac{b}{a}(x-h) for sideways hyperbolas.

Part 4: Additional Practice Problems

Practice solving the following conic sections:

Circles

  1. Identify the center and radius: (x−3)2+(y+1)2=16(x - 3)^2 + (y + 1)^2 = 16.
  2. Write the equation of a circle with center (5, -4) and radius 9.

Ellipses

  1. Find vertices and foci of the ellipse from standard form: (x−2)216+(y−3)29=1\frac{(x-2)^2}{16} + \frac{(y-3)^2}{9} = 1.
  2. Graph the ellipse and calculate the eccentricity.

Parabolas

  1. Given focus (2, 0) and directrix x = 0, write the equation of the parabola.
  2. Identify the vertex and axis of symmetry for 2y=x22y = x^2.

Hyperbolas

  1. Write the equation for a hyperbola given foci (0, ±5) and vertices (0, ±3).
  2. Identify the asymptotes for the hyperbola given by x225−y29=1\frac{x^2}{25} - \frac{y^2}{9} = 1.

Exam Preparation Guidelines

  • Ensure to review definitions and properties for each type of conic section thoroughly.
  • Work through practice problems, focusing on transforming standard equations and identifying key features.
  • Memorize important formulas for distance between points, eccentricity, and standard forms.
  • Utilize any quizzes and notes from class to reinforce learning and prepare for the upcoming test.