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MAC 1140

MAC 1140. Module 6 Nonlinear Functions and Equations II. Learning Objectives. Upon completing this module, you should be able to identify a rational function and state its domain. find and interpret vertical asymptotes. find and interpret horizontal asymptotes. solve rational equations.

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MAC 1140

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  1. MAC 1140 Module 6 Nonlinear Functions and Equations II

  2. Learning Objectives Upon completing this module, you should be able to • identify a rational function and state its domain. • find and interpret vertical asymptotes. • find and interpret horizontal asymptotes. • solve rational equations. • solve applications involving rational equations. • solve applications involving variations. • solve polynomial inequalities. • solve rational inequalities. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  3. Learning Objectives (Cont.) 9. learn properties of rational exponents. • learn radical notation. • use power functions to model data. • solve equations involving rational exponents. • solve equations involving radical expressions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  4. Nonlinear Functions and Equations II There are three sections in this module: 4.6 Rational Functions and Models 4.7 More Equations and Inequalities 4.8 Radical Equations and Power Functions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  5. What is a Rational Function? • A rational function is a nonlinear function. The domain of a rational function includes all real numbers except the zeros of the denominator q(x). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  6. What is a Vertical Asymptote? In this graph, the line x = 2 is a vertical asymptote. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  7. What is a Horizontal Asymptote? In this graph, the line y = 25 is a horizontal asymptote. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  8. Let’s Look at an Example Use the graph of to sketch the graph of Include all asymptotes in your graph. Write g(x) in terms of f(x). Solution g(x) is a translation of f(x) left one unit and down 2 units. The vertical asymptote is x = 1 The horizontal asymptote is y = 2 g(x) = f(x + 1)  2 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  9. How to Find Vertical and Horizontal Asymptotes? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  10. More Examples For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Solution To identify horizontal asymptote, look at the leading coefficient of the highest power term in both numerator and denominator. Horizontal Asymptote: If the Degree of numerator equals the degree of the denominator, y = a/b is asymptote, so y = 2/4 = 1/2 To identify vertical asymptote, set the denominator to 0. Vertical Asymptote: 4x 8 = 0, x = 2 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  11. Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Solution (Cont.) b) Horizontal Asymptote: Degree: numerator < denominator y = 0 is the horizontal asymptote. Vertical Asymptote: x2 9 = 0 x =  3 are the vertical asymptotes. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  12. Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Solution (Cont.) c) Horizontal Asymptote: Degree: numerator > denominator no horizontal asymptotes Vertical Asymptote: no vertical asymptotes (There will be a “hole” in the graph.) The graph is the line y = x + 2 with the point (2, 4) missing. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  13. What is a Slant/Oblique Asymptote? A third type of asymptote is neither horizontal or vertical. Occurs when the numerator of a rational function has a degree one more than the degree of the denominator. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  14. Example Let a) Use a calculator to graph f. b) Identify any asymptotes. c) Sketch a graph of f that includes the asymptotes. Solution a) Reminder: Slant/Oblique Asymptote occurs when the numerator of a rational function has a degree one more than the degree of the denominator. Is that true in this rational function? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  15. Example (Cont.) Solution (Cont.) • Asymptotes: The function is undefined when x 2 = 0 or when x = 2. * Vertical asymptote at x = 2 * Oblique asymptote at y = x + 2 c) How about horizontal asymptote? Why don’t we have it in this rational function? How can you tell from the function itself? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  16. How to Solve Rational Equation? • Solve • Solution • SymbolicGraphicalNumerical http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  17. One More Example Solve Solution Multiply by the LCD to clear the fractions. When 1 is substituted for x, two expressions in the given equation are undefined. There are no solutions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  18. Direct Variation The nonzero number k is called the constant of variation or the constant of proportionality. In the area formula for a circle, A = π r2, the π is the constant of variation; so, k = π in this case. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  19. Inverse Variation This inverse variation occurs when we have two quantities that vary inversely; the increase of one quantity will decrease the other quantity. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  20. Example of Application At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. a) Find the constant of proportionalityk. b) Determine the intensity at a distance of 2.5 meters. Solution a) Substitute d = 3 and I = 0.88 into the equation and solve for k. b) Let and d = 2.5. The intensity at 2.5 meters is 1.27 watts per square meter. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  21. How to Solve Polynomial Inequalities? • An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, another expression. • To solve Polynomial Inequalities, we need the following: • Boundary numbers (x-values) are found where the inequality holds. • A graph or a table oftest values can be used to determine the intervals where the inequality holds. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  22. Solving Polynomial Inequalities in Four Steps http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  23. Let’s Look at This Example Solve symbolically and graphically. Solution Symbolically Step 1: Write the inequality as Step 2: Replace the inequality symbol with an equal sign and solve. The boundary numbers are –5, –2, and 0. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  24. -8 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 5 Let’s Look at This Example (Cont.) Step 3: The boundary numbers separate the number line into four disjoint intervals: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  25. Let’s Look at This Example (Cont.) Step 4: Complete a table of test values. The solution set is http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  26. Let’s Look at This Example (Cont.) Graphically: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  27. How to Solve Rational Inequalities? • Inequalities involving rational expressions are called rational inequalities. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  28. Example • Solve • Solution • Step 1: Rewrite the inequality in the form http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  29. Example (Cont.) • Step 2: Find the zeros of the numerator and the denominator. • Step 3: The boundary numbers are – 4 and 1, which separate the number line into three disjoint intervals: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  30. Example (Cont.) Step 4: Use a table to solve the inequality. The interval notation is (–4, 1]. • Caution: When solving a rational inequality, it is essential not to multiply or divide each side of the inequality by the LCD if the LCD contains a variable. This techniques often leads to an incorrect solution set. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  31. Let’s Review Some Properties of Rational Exponents http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  32. Let’s Practice Some Simplification • Simplify each expression by hand. • 82/3 b) (–32)–4/5 • Solutions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  33. Let’s Practice Some Simplification (Cont.) • Use positive rational exponents to write each expression. • b) • Solutions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  34. What are Power Functions? • Power functions typically have rational exponents. • A special type of power function is a root function. • Examples of power functions include: • f1(x) = x2, f2(x) = x3/4 , f3(x) = x0.4, and f4(x) = http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  35. What are Power Functions? (Cont.) • Often, the domain of a power function f is restricted to nonnegative numbers. • Suppose the rational number p/q is written in lowest terms. The the domain of f(x) = xp/q is all real numbers whenever q is odd and all nonnegative numbers whenever q is even. • The following graphs show 3 common power functions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  36. Example • Modeling Wing Size of a Bird: Heavier birds have larger wings with more surface areas than do lighter birds. For some species the relationship can be modeled by S(w) = 0.2w2/3, where w is the weight of the bird in kilograms and S is surface area of the wings in square meters. (Source: C. Pennycuick, Newton Rules Biology.) • Approximate S(0.75) and interpret the result. • What weight corresponds to a surface area of 0.45 square meter? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  37. Example (Cont.) Solution • S(0.75) = 0.2(0.75)2/3 The wings of a bird that weighs about 0.75 kilogram have the surface area of about 0.165 square meter. • To answer this, we must solve the equation 0.2w2/3 = 0.45. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  38. Example (Cont.) Solution (cont.) Since w must be positive, the wings of a 3.4 kilogram bird must have a surface area of about 0.45 square meter. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  39. How to Solve Equations Involving Rational Exponents? Example Solve 4x3/2– 6 = 6. Approximate the answer to the nearest hundredth, and give graphical support. Solutions Symbolic Solution Graphical Solution 4x3/2 – 6 = 6 4x3/2 = 12 (x3/2)2 = 32 x3 = 9 x = 91/3 x = 2.08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  40. How to Solve Equations Involving Rational Exponents? (Cont.) Check Substituting these values in the original equation shows that the value of 1 is an extraneous solution because it does not satisfy the given equation. Therefore, the only solution is 6. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  41. How to Solve Equations Involving Radicals? Some equations may contain a cube root. Solve Solution Both solutions check, so the solution set is http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  42. How to Solve Equations Involving Negative Exponents? ExampleSolve Solution Since http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  43. What have we learned? We have learned to • identify a rational function and state its domain. • find and interpret vertical asymptotes. • find and interpret horizontal asymptotes. • solve rational equations. • solve applications involving rational equations. • solve applications involving variations. • solve polynomial inequalities. • solve rational inequalities. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  44. What have we learned? (Cont.) 9. learn properties of rational exponents. • learn radical notation. • use power functions to model data. • solve equations involving rational exponents. • solve equations involving radical expressions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

  45. Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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