Conic Sections

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Conic Sections

  1. 1. Conic Sections Chapter 10: Section 1
  2. 2. Types of Conic Sections <ul><li>Parabola </li></ul><ul><li>Circle </li></ul><ul><li>Ellipse </li></ul><ul><li>Hyperbola </li></ul>
  3. 3. Parabola <ul><li>Standard Formulas </li></ul><ul><ul><li>y = ax 2 + bx + c </li></ul></ul><ul><ul><li>Or </li></ul></ul><ul><ul><li>x = ay 2 + by + c </li></ul></ul>
  4. 4. Circle <ul><li>Standard Formula </li></ul><ul><li>(x-h) 2 + (y-k) 2 = r 2 </li></ul><ul><li>where the center of the circle is (h,k) and radius r </li></ul>
  5. 5. Ellipse <ul><li>Standard Formulas </li></ul><ul><li>x 2 + y 2 = 1 </li></ul><ul><li>a 2 b 2 </li></ul><ul><li>or </li></ul><ul><li>x 2 + y 2 = 1 </li></ul><ul><li>b 2 a 2 </li></ul>
  6. 6. Hyperbola <ul><li>Standard Formulas </li></ul><ul><li>x 2 - y 2 = 1 </li></ul><ul><li>a 2 b 2 </li></ul><ul><li>or </li></ul><ul><li>y 2 - x 2 = 1 </li></ul><ul><li>a 2 b 2 </li></ul>
  7. 7. X-intercepts <ul><li>The x-intercepts are the point(s) where the line crosses the x-axis </li></ul>
  8. 8. How do I find the x-intercept? <ul><li>Look at the graph if one is provided. Where ever the line crosses the x-axis, that is an x-intercept. </li></ul><ul><li>X-intercept = 0 </li></ul>
  9. 9. What are the x-intercepts? x-intercept = +3 and -3 No x-intercepts
  10. 10. Find the x-intercepts X-intercepts = +3 and -3 X-intercepts = +2 and -2
  11. 11. How to find the x-intercept if given the equation <ul><li>At all points on the x-axis, the value of y is zero. </li></ul><ul><li>So, if zero is substituted for y in the equation it will yield the x-intercept(s), IF it exists. </li></ul><ul><li>y = 2x 2 </li></ul><ul><li>0 = 2x 2 Substitute 0 for y </li></ul><ul><li>0/2 = x 2 Divide both sides by 2 </li></ul><ul><li>0 = x 2 </li></ul><ul><li>√ 0 = √x 2 Take the square root of both sides </li></ul><ul><li>0 = x x-intercept = 0 </li></ul>
  12. 12. Example #1 <ul><li>y = 3x 2 + 2 </li></ul><ul><li>0 = 3x 2 + 2 Substitute 0 for y </li></ul><ul><li> -2 = 3x 2 Subtract 2 from both sides </li></ul><ul><li> -2/3 = x 2 Divide both sides by 3 </li></ul><ul><li>√ -2/3 = √x 2 Take the square root of both sides </li></ul><ul><li>You cannot take the square root of a negative number…so this equation has no x-intercepts . </li></ul>
  13. 13. Example #2 <ul><li>x 2 + y 2 = 25 </li></ul><ul><li>x 2 + (0) 2 = 25 Substitute 0 for y </li></ul><ul><li>x 2 = 25 </li></ul><ul><li>√ x 2 = √25 Take the square root of both sides </li></ul><ul><li>x = ±5 x-intercept = +5 and -5 </li></ul>
  14. 14. How do I find the domain? <ul><li>The domain is all the </li></ul><ul><li>x-values on the graph. </li></ul><ul><li>x ≥ -6 </li></ul><ul><li>x ≤ 6 </li></ul>-6 0 6
  15. 15. Example #1 <ul><li>The domain is all the </li></ul><ul><li>x-values on the graph. </li></ul><ul><li>Domain: all real numbers </li></ul>
  16. 16. Example #2 <ul><li>Domain: x ≥ -3 </li></ul><ul><li>x ≤ 3 </li></ul>
  17. 17. Example #3 <ul><li>Domain: x ≤ -2 </li></ul><ul><li> x ≥ 2 </li></ul>

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