Ellipses - Formulas and Graphs

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Ellipses - Formulas and Graphs

  1. 1. Algebra II Equations of Ellipses
  2. 2. Basic Equations <ul><li>Center is at ( h , k ) </li></ul><ul><ul><li>Remember to change signs </li></ul></ul><ul><li>Major axis is determined by a </li></ul><ul><ul><li>Always the larger denominator </li></ul></ul><ul><ul><li>Association with x or y determines direction </li></ul></ul><ul><ul><li>Length is 2 a or | a | units each direction from the center </li></ul></ul><ul><li>The points on the major axis are vertices </li></ul>
  3. 3. Basic Equations (cont.) <ul><li>Minor axis is determined by b </li></ul><ul><ul><li>Perpendicular to major axis </li></ul></ul><ul><ul><li>Length is 2 b or | b | units each direction from the center </li></ul></ul><ul><li>The points on the minor axis are CO-vertices </li></ul>
  4. 4. Basic Equations (cont.) <ul><li>Foci are |c| units each direction from the center on the major axis </li></ul><ul><li>Foci are determined by the equation above </li></ul>
  5. 5. Features of an Ellipse <ul><li>Include sketch of graph for all! </li></ul><ul><li>Put in standard form by dividing to get “=1” </li></ul>
  6. 6. Features of an Ellipse (cont.) <ul><li>Since nothing is with the x or y </li></ul><ul><ul><li>h = 0 </li></ul></ul><ul><ul><li>k = 0 </li></ul></ul><ul><ul><li>the center is at the origin </li></ul></ul>
  7. 7. Features of an Ellipse (cont.) <ul><li>larger denominator determines the equation </li></ul><ul><ul><li>larger denominator is always a 2 </li></ul></ul><ul><ul><li>a 2 = 49 </li></ul></ul><ul><ul><li>a = ±7 </li></ul></ul><ul><ul><li>a is with y  major axis is in y -direction </li></ul></ul><ul><ul><li>b = ±2 </li></ul></ul>
  8. 8. Features of an Ellipse (cont.) <ul><li>major axis is in y -direction </li></ul><ul><ul><li>Measure a = ±7 from center in y -direction </li></ul></ul><ul><ul><li>Measure b = ±2 from center in x -direction </li></ul></ul>b a
  9. 9. Features of an Ellipse (cont.) <ul><li>Sketch the graph </li></ul><ul><li>Calculate the foci </li></ul><ul><ul><li>a 2 – b 2 = c 2 </li></ul></ul><ul><ul><li>49 – 4 = c 2 </li></ul></ul><ul><ul><li>45 = c 2 </li></ul></ul><ul><ul><li>±6.7  c </li></ul></ul><ul><li>Plot foci on major axis </li></ul>c
  10. 10. Developing Equation for Ellipse <ul><li>Center (0, 0) </li></ul><ul><li>Co-vertex (0, 4) </li></ul><ul><li>Vertex (10, 0) </li></ul><ul><ul><li>Must be two vertices  also (–10, 0) </li></ul></ul><ul><ul><li>Point is on x -axis means this is the major axis </li></ul></ul><ul><li>Determines which formula to use </li></ul><ul><ul><li>a must be with the x </li></ul></ul>
  11. 11. Developing Equation for Ellipse (cont) <ul><li>Vertex (10, 0) & (–10, 0) </li></ul><ul><ul><li>a is distance from center to vertex </li></ul></ul><ul><ul><li>a = 10 </li></ul></ul><ul><ul><li>a 2 = 100 </li></ul></ul><ul><li>Co-vertex (0, 4) & (0, -4) </li></ul><ul><ul><li>b is distance from center to co-vertex </li></ul></ul><ul><ul><li>b = 4 </li></ul></ul><ul><ul><li>b 2 = 16 </li></ul></ul>
  12. 12. Developing Equation for Ellipse (cont) <ul><li>Center (0, 0) </li></ul><ul><ul><li>h = 0 </li></ul></ul><ul><ul><li>k = 0 </li></ul></ul><ul><li>Plug-in a 2 & b 2 </li></ul><ul><ul><li>a 2 = 100 </li></ul></ul><ul><ul><li>b 2 = 16 </li></ul></ul>
  13. 13. Developing Equation for Ellipse (cont) <ul><li>For problems giving focus </li></ul><ul><ul><li>Use: a 2 – b 2 = c 2 to solve for the missing value </li></ul></ul><ul><ul><li>Remember the focus is c </li></ul></ul>

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