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Chapter 9

Chapter 9. Rational Functions. Review: Direct Variation. 9-1 Inverse Variation. warm up. Joint Variation. 9-3 Rational Functions and their Graphs. 9-3 Rational Functions and their Graphs. 9-3 Rational Functions and their Graphs. 9-3 Rational Functions and their Graphs.

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Chapter 9

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  1. Chapter 9 Rational Functions

  2. Review: Direct Variation

  3. 9-1 Inverse Variation

  4. warm up

  5. Joint Variation

  6. 9-3 Rational Functions and their Graphs

  7. 9-3 Rational Functions and their Graphs

  8. 9-3 Rational Functions and their Graphs

  9. 9-3 Rational Functions and their Graphs

  10. 9-3 Rational Functions and their Graphs

  11. 9-3 Rational Functions and their Graphs

  12. 9-3 Rational Functions and their Graphs • To sketch the graph of a rational function: • Determine if the function points of discontinuity for the denominator and if they are holes or vertical asymptotes. Sketch in any vertical asymptotes. • Determine if the function has a horizontal asymptote. As x gets larger (positive or negative) the graph will approach this line. • Calculate values of y for x values that are near the asymptotes. Plot these points and sketch the graph.

  13. 9-3 Rational Functions and their Graphs

  14. 9-4 Rational Expressions

  15. 9-4 Rational Expressions

  16. 9-4 Rational Expressions

  17. 9-4 Rational Expressions

  18. 9-4 Rational Expressions

  19. 9-4 Rational Expressions

  20. 9-4 Rational Expressions

  21. 9-5 Adding and Subtracting Rational Expressions

  22. Complex Fractions

  23. 9-6 Solving Rational Equations

  24. 9-6 Solving Rational Equations When a rational equation has a sum or difference of two rational expressions, you can use the LCD to simplify.

  25. 9-6 Solving Rational Equations

  26. 9-6 Solving Rational Equations • Homework: page 532 (1-21) odd • Chapter 9 test Tuesday 4/9 or Wednesday 4/10

  27. Review: • Direct and Inverse variation: • If y/x is always equal to the same number, then x and y represent a direct variation. • y=kx • If xy is always the same value then x and y vary inversely • y = k/x

  28. Review: • Discontinuities • In rational functions discontinuities occur where values of the variable make the denominator equal to zero. • If this value makes the numerator zero there will be a hole in the graph. • If the value does not make the numerator zero there will be a vertical asymptote in the graph.

  29. Review: • Horizontal Asymptotes • Horizontal Asymptotes describe end behavior of graph. • Determined by the degree of the functions in the numerator and denominator. • If degree in denominator is higher, horizontal asymptote at y=0 (X axis) • If degree in numerator is higher there is no horizontal asymptote • If degree is the same, horizontal asymptote occurs at the ratio of the leading coefficients of the numerator and denominator.

  30. Review: • Simplify Rational Expressions • Factor all parts of rational expression completely. • Cancel factors that appear in both numerator and denominator. • To multiply: factor and simplify before multiplying. • To divide: Factor, flip second function, simplify and multiply.

  31. Review: • Adding or Subtracting Rational Expressions • Must find a common denominator before you can add or subtract. • Complex Rationals: • Multiply top and bottom of rational expression by the Least Common Multiple of all complex denominators

  32. Review: • Solving Rational Equations • If possible cross multiply to solve equations. • Determine Least Common Multiple of all rationals and multiply all terms by the LCM. • Always check all of your solutions.

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