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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up. The inverse variation xy = 8 relates the constant speed x in mi/h to the time y in hours that it takes to travel 8 miles. Graph this inverse variation. . California Standards.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up The inverse variation xy = 8 relates the constant speed x in mi/h to the time y in hours that it takes to travel 8 miles. Graph this inverse variation.

  3. California Standards 13.0 Studentsadd, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptionally challenging problems by using these techniques. Also covered: 17.0

  4. Vocabulary rational function excluded value discontinuous function asymptote

  5. A rational function is a function whose rule is a quotient of polynomials. The inverse variations you studied in the previous lesson are a special type of rational function. Rational functions: For any function involving x and y, an excluded value is any x-value that makes the function value y undefined. For a rational function, an excluded value is any value that makes the denominator equal to 0.

  6. Additional Example 1: Identifying Excluded Values Identify any excluded values for each rational function. A. x = 0 Set the denominator equal to 0. The excluded value is 0. B. x– 2 = 0 Set the denominator equal to 0. Solve for x. x = 2 The excluded value is 2.

  7. Check It Out! Example 1 Identify any excluded values for each rational function. a. x = 0 Set the denominator equal to 0. The excluded value is 0. b. x– 1 = 0 Set the denominator equal to 0. x = 1 Solve for x. The excluded value is 1.

  8. Check It Out! Example 1 Identify any excluded values for each rational function. c. x + 4 = 0 Set the denominator equal to 0. Solve for x. x = –4 The excluded value is –4.

  9. Many rational functions are discontinuous functions, meaning their graphs contain one or more jumps, breaks, or holes. This occurs at an excluded value. One place that a graph of a rational function may be discontinuous is at an asymptote. An asymptote is a line that a graph gets closer to as the absolute value of a variable increases. In the graph shown, both the x- and y-axes are asymptotes. A graph will get closer and closer to but never touch its asymptotes.

  10. Writing Math Vertical lines are written in the form x = b, and horizontal lines are written in the form y = c.

  11. Look at the graph of y = The denominator is 0 when x = 0, so 0 is an excluded value. This means there is a vertical asymptote at x = 0. Notice the horizontal asymptote at y = 0. This is because there is no value of x for which y = 0.

  12. Look at the graph of y = Notice that there is a vertical asymptote at x = 3 and there is a horizontal asymptote at y = 2. This is because there is no value of x for which If is never 0, then can never be equal to 2.

  13. Step 1 Write in y = form. Additional Example 2A: Identifying Asymptotes Identify the asymptotes. Step 2 Identify the asymptotes. vertical: x = –7 horizontal: y = 0

  14. +3 +3 2x = 3 Solve for x. Is an excluded value. Additional Example 2B: Identifying Asymptotes Identify the asymptotes. Step 1 Identify the vertical asymptote. 2x– 3 = 0 Find the excluded value. Set the denominator equal to 0. Add 3 to both sides.

  15. Vertical asymptote: x = ; horizontal asymptote: y = 8 Additional Example 2B Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = 8 y = 8 y = c

  16. +5 +5 x = 5 Check It Out! Example 2a Identify the asymptotes. Step 1 Identify the vertical asymptote. x– 5 = 0 Find the excluded value. Set the denominator equal to 0. Add 5 to both sides. x = 5 Solve for x. 5 is an excluded value.

  17. Check It Out! Example 2a Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = 0 y = 0 y = c Vertical asymptote: x = 5; horizontal asymptote: y = 0

  18. –16 –16 4x = –16 Check It Out! Example 2b Identify the asymptotes. Step 1 Identify the vertical asymptote. 4x + 16 = 0 Find the excluded value. Set the denominator equal to 0. Subtract 16 from both sides. x = –4 Solve for x. –4 is an excluded value.

  19. Check It Out! Example 2b Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = 5 y = 5 y = c Vertical asymptote: x = –4; horizontal asymptote: y = 5

  20. –77 –77 x = –77 Check It Out! Example 2c Identify the asymptotes. Step 1 Identify the vertical asymptote. Find the excluded value. Set the denominator equal to 0. x + 77 = 0 Subtract 77 from both sides. Solve for x. –77 is an excluded value.

  21. Check It Out! Example 2c Continued Identify the asymptotes. Step 2 Identify the horizontal asymptote. c = –15 y = –15 y = c Vertical asymptote: x = –77; horizontal asymptote: y = –15

  22. To graph a rational function in the form y = , you can use the asymptotes and a table of values.

  23. Additional Example 3A: Graphing Rational Functions Using Asymptotes Graph the function. Step 1 Identify the asymptotes. Use x = b. x – 3 = 0, so b = 3. vertical: x = 3 horizontal: y = 0 Use y = c. c = 0

  24. x = 3 ● ● ● ● ● ● x 0 1 2 4 5 6 y = 0 y Additional Example 3A Continued Step 2 Graph the asymptotes using dashed lines. Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote. Step 4 Plot the points and connect them with smooth curves. The curves should not touch the asymptotes.

  25. Additional Example 3B: Graphing Rational Functions Using Asymptotes Graph the function. Step 1 Identify the asymptotes. Use x = b. b = –4 vertical: x = –4 Use y = c. c = –2 horizontal: y = –2

  26.     x –6 –5 –4 –3 –2 –1 y –2.5 –3 –1 –1.5 –1.6 Undef. Additional Example 3B Continued Step 2 Graph the asymptotes using dashed lines. Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote. Step 4 Plot the points and connect them with smooth curves. The curves should not touch the asymptotes.

  27. Check It Out! Example 3a Graph each function. Step 1 Identify the asymptotes. Use x = b. b = –7 vertical: x = –7 Use y = c. c = 3 horizontal: y = 3

  28.      x –10 –8 –6 –4 –2 –1 y 4 2 3.3 3.2 3.16 –2.6 Check It Out! Example 3a Continued Graph each function. Step 2 Graph the asymptotes using dashed lines. Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote. Step 4 Draw smooth curves to show the translation.

  29. Check It Out! Example 3b Graph each function. Step 1 Identify the asymptotes. vertical: x = 3 Use x = b. x – 3 = 0, so b = 3. horizontal: y = 2 Use y = c. c = 2

  30.     x 0 1 2 4 5 6  y 4 Check It Out! Example 3b Step 2 Graph the asymptotes using dashed lines. Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote. Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them.

  31. Additional Example 4: Application Your club has $75 with which to purchase snacks to sell at an afterschool game. The number of snacks y that you can buy, if the average price of the snacks is x-dollars, is given by y = a. Describe the reasonable domain and range values. Both the number of snacks purchased and their cost will be nonnegative values so nonnegative values are reasonable for both domain and range.

  32. Additional Example 4 Continued b. Graph the function. Step 1 Identify the vertical and horizontal asymptotes. vertical: x = 0 horizontal: y = 0 Use x = b. b = 0 Use y = c. c = 0 Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.

  33. Additional Example 4 Continued Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.

  34. Additional Example 4 Continued Step 4 Plot the points and connect them with a smooth curve.

  35. A librarian has a budget of $500 to buy copies of a software program. She will receive 10 free copies when she sets up an account with the supplier. The number of copies y of the program that she can buy is given by y = + 10, where x is the price per copy. Check It Out! Example 4 a. Describe the reasonable domain and range values. The domain would be all values greater than 0 up to $500 dollars and the range would be all natural numbers greater than 10.

  36. Check It Out! Example 4 Continued b. Graph the function. Step 1 Identify the vertical and horizontal asymptotes. Use x = b. b = 0 vertical: x = 0 horizontal: y = 10 Use y = c. c = 10 Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.

  37. Check It Out! Example 4 Continued Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.

  38. Check It Out! Example 4 Continued Step 4 Plot the points and connect them with a smooth curve.

  39. The table shows some of the properties of the three types of functions you have studied and their graphs.

  40. Lesson Quiz: Part I Identify the exceeded value for each rational function. 5 2. 1. 0 3. Identify the asymptotes of and then graph the function. x = –4; y = 0

  41. Lesson Quiz: Part II 4. You have $100 to spend on CDs. A CD club advertises 6 free CDs for anyone who becomes a member. The number of CDs y that you can receive is given by y = , where x is the average price per CD. a. Describe the reasonable domain and range values. D: x > 0 R: natural numbers > 6

  42. Lesson Quiz: Part III b. Graph the function.

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