MAC 1140 Module 6 Nonlinear Functions and Equations II

Rev.S102 Learning Objectives Upon completing this module, you should be able to 1.identify a rational function and state its domain. 2.find and interpret vertical asymptotes. 3.find and interpret horizontal asymptotes. 4.solve rational equations. 5.solve applications involving rational equations. 6.solve applications involving variations. 7.solve polynomial inequalities. 8.solve rational inequalities. Click link to download other modules.

Rev.S103 Learning Objectives (Cont.) 9. learn properties of rational exponents. 10.learn radical notation. 11.use power functions to model data. 12.solve equations involving rational exponents. 13.solve equations involving radical expressions. Click link to download other modules.

Rev.S104 Nonlinear Functions and Equations II Click link to download other modules. 4.6Rational Functions and Models 4.7More Equations and Inequalities 4.8Radical Equations and Power Functions There are three sections in this module:

Rev.S105 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). Click link to download other modules.

Rev.S106 What is a Vertical Asymptote? Click link to download other modules. In this graph, the line x = 2 is a vertical asymptote.

Rev.S107 What is a Horizontal Asymptote? Click link to download other modules. In this graph, the line y = 25 is a horizontal asymptote.

Rev.S108 Let’s Look at an Example Click link to download other modules. 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

Rev.S109 How to Find Vertical and Horizontal Asymptotes? Click link to download other modules.

Rev.S1010 More Examples Click link to download other modules. 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. a)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

Rev.S1011 Examples (Cont.) Click link to download other modules. 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: x 2  9 = 0 x =  3 are the vertical asymptotes.

Rev.S1012 Examples (Cont.) Click link to download other modules. 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.

Rev.S1013 What is a Slant/Oblique Asymptote? Click link to download other modules. 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.

Rev.S1014 Example Click link to download other modules. 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?

Rev.S1015 Example (Cont.) Click link to download other modules. Solution (Cont.) b)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?

Rev.S1016 How to Solve Rational Equation? Click link to download other modules. Solve Solution SymbolicGraphicalNumerical

Rev.S1017 One More Example Click link to download other modules. 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.

Rev.S1018 Direct Variation Click link to download other modules. The nonzero number k is called the constant of variation or the constant of proportionality. In the area formula for a circle, A = π r 2, the π is the constant of variation; so, k = π in this case.

Rev.S1019 Inverse Variation Click link to download other modules. This inverse variation occurs when we have two quantities that vary inversely; the increase of one quantity will decrease the other quantity.

Rev.S1020 Example of Application Click link to download other modules. 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 proportionality k. 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.

Rev.S1021 How to Solve Polynomial Inequalities? Click link to download other modules. 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 of test values can be used to determine the intervals where the inequality holds.

Rev.S1022 Solving Polynomial Inequalities in Four Steps Click link to download other modules.

Rev.S1023 Let’s Look at This Example Click link to download other modules. 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.

Rev.S1024 Let’s Look at This Example (Cont.) Click link to download other modules. Step 3: The boundary numbers separate the number line into four disjoint intervals:

Rev.S1025 Let’s Look at This Example (Cont.) Click link to download other modules. Step 4: Complete a table of test values. The solution set is IntervalTest Value xx 3 + 7x xPositive/Negative –6–24Negative –48Positive –1–4Negative 118Positive

Rev.S1026 Let’s Look at This Example (Cont.) Click link to download other modules. Graphically:

Rev.S1027 How to Solve Rational Inequalities? Click link to download other modules. Inequalities involving rational expressions are called rational inequalities.

Rev.S1028 Example Click link to download other modules. Solve Solution Step 1: Rewrite the inequality in the form

Rev.S1029 Example (Cont.) Click link to download other modules. 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: NumeratorDenominator 1 – x = 0 x = 1 x + 4 = 0 x = –4

Rev.S1030 Example (Cont.) Click link to download other modules. Step 4: Use a table to solve the inequality. The interval notation is (–4, 1]. IntervalTest Value x(1–x)/(x + 4)Positive/Negative –5–6Negative –23/2Positive 2–1/6Negative 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.

Rev.S1031 Let’s Review Some Properties of Rational Exponents Click link to download other modules.

Rev.S1032 Let’s Practice Some Simplification Click link to download other modules. Simplify each expression by hand. a) 8 2/3 b) (–32) –4/5 Solutions

Rev.S1033 Let’s Practice Some Simplification (Cont.) Click link to download other modules. Use positive rational exponents to write each expression. a) b) Solutions

Rev.S1034 What are Power Functions? Click link to download other modules. Power functions typically have rational exponents. A special type of power function is a root function. Examples of power functions include: f 1 (x) = x 2, f 2 (x) = x 3/4, f 3 (x) = x 0.4, and f 4 (x) =

Rev.S1035 What are Power Functions? (Cont.) Click link to download other modules. 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) = x p/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.

Rev.S1036 Example Click link to download other modules. 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) = 0.2w 2/3, 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.) a) Approximate S(0.75) and interpret the result. b) What weight corresponds to a surface area of 0.45 square meter?

Rev.S1037 Example (Cont.) Click link to download other modules. Solution a)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 square meter. b)To answer this, we must solve the equation 0.2w 2/3 = 0.45.

Rev.S1038 Example (Cont.) Click link to download other modules. 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.

Rev.S1039 How to Solve Equations Involving Rational Exponents? Click link to download other modules. Example Solve 4x 3/2 – 6 = 6. Approximate the answer to the nearest hundredth, and give graphical support. Solutions Symbolic SolutionGraphical Solution 4x 3/2 – 6 = 6 4x 3/2 = 12 (x 3/2 ) 2 = 3 2 x 3 = 9 x = 9 1/3 x = 2.08

Rev.S1040 How to Solve Equations Involving Rational Exponents? (Cont.) Click link to download other modules. 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.

Rev.S1041 How to Solve Equations Involving Radicals? Click link to download other modules. Some equations may contain a cube root. Solve Solution Both solutions check, so the solution set is

Rev.S1042 How to Solve Equations Involving Negative Exponents? Click link to download other modules. Example Solve Solution Since

Rev.S1043 What have we learned? We have learned to 1.identify a rational function and state its domain. 2.find and interpret vertical asymptotes. 3.find and interpret horizontal asymptotes. 4.solve rational equations. 5.solve applications involving rational equations. 6.solve applications involving variations. 7.solve polynomial inequalities. 8.solve rational inequalities. Click link to download other modules.

Rev.S1044 What have we learned? (Cont.) 9.learn properties of rational exponents. 10.learn radical notation. 11.use power functions to model data. 12.solve equations involving rational exponents. 13.solve equations involving radical expressions Click link to download other modules.

Rev.S1045 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 Click link to download other modules.