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Understanding Rational Functions: Graphs and Variations

Learn about direct, inverse, and joint variation in rational functions. Explore graphing techniques, asymptotes, and function simplification. Practice adding, subtracting, and solving rational expressions and equations.

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Understanding Rational Functions: Graphs and Variations

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  1. Chapter 9 Rational Functions

  2. Review: Direct Variation

  3. 9-1 Inverse Variation

  4. Joint Variation

  5. The reciprocal function • Functions that model inverse variations have a parent function that is called the reciprocal function and look like: • The value x=0 must be excluded from the domain of this function, because it causes the value of the function to be undefined. • Using Desmos, graph the function. What feature do you see on the graph at x=0? • This feature is a vertical asymptote. • What do you see on the graph at y=0? Can you explain this feature? • This feature is a horizontal asymptote.

  6. Homework: • Page 499 (1-27) odd

  7. Direct, Inverse, or neither?

  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

  13. 9-3 Rational Functions and their Graphs

  14. 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.

  15. Groups • Hannah H, Devon, Kirsten • Elisa, Sean, Haylee • Madeline Hein, Madison, Barry • Noelle M., Ryan, Austin • Madeline Huttie, Chesna, Noelle H. • Alicia, Chloe, Ashlyn • Allyssa, Lindsey, Eric • Gary, Isaac, Emily • Megan, Cici, Rachel • Hannah L, Isabella, Gurmaan

  16. Graphing Activity • DO NOT use technology to draw the graph • Fill out the worksheet first • When the worksheet is complete you can begin to draw your graph • When you finish drawing show your graph to me, then you can use Desmos to verify your result. • Turn in worksheet and graph

  17. 9-3 Rational Functions and their Graphs

  18. Simplify • (

  19. 9-4 Rational Expressions

  20. 9-4 Rational Expressions

  21. 9-4 Rational Expressions

  22. 9-4 Rational Expressions

  23. 9-4 Rational Expressions

  24. 9-4 Rational Expressions

  25. 9-4 Rational Expressions

  26. Complete these addition problems. Think about what you have to do to the fractions to be able to add them.

  27. 9-5 Adding and Subtracting Rational Expressions

  28. To add or subtract rational expressions First, determine the least common multiple of the denominators. Then multiply the numerator and denominator of each term by the factors missing from the LCM. Add or subtract numerators as indicated.

  29. Complex Fractions

  30. 9-6 Solving Rational Equations

  31. 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.

  32. 9-6 Solving Rational Equations

  33. 9-6 Solving Rational Equations • Homework: page 532 (1-21) odd • Chapter 9 test Tuesday 4/9 or Wednesday 4/10

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