Unit 4 Rational Expressions. Mrs. Valen+ne Math III

Transcription

1 Unit 4 Rational Expressions Mrs. Valen+ne Math III

2 4.1 Simplifying Rational Expressions Simplifying Rational Expressions Expression in the form Simplifying a rational expression is like simplifying any other fraction: Factor Divide out common factors Simplify Examples:

3 4.1 Simplifying Rational Expressions Simplifying a Rational Expression Containing a Trinomial Factor both the numerator and the denominator Divide out the common factor. Simplify. State the simplified form with any restrictions on the variable Excluded value a value of a variable for which a rational expression is undefined Denominator 0 Example: x 3 or 2 because the original denominator would then equal 0. where x 3 or 2

4 4.1 Simplifying Rational Expressions Recognizing Opposite Factors If the numerator and denominator are opposites, factor 1 from one of the terms. Examples:

5 4.1 Simplifying Rational Expressions Using a Rational Expression You are choosing between two wastebaskets that have the shape of the figures in the diagram. They both have the same volume. What is the height h of the rectangular wastebasket? Give your answer in terms of a. V = πr 2 h for a cylinder and V = l*w*h for a rectangular prism V = πa 2 (2a 8) based on the cylinder V = 4a 2 * h based on the prism πa 2 (2a 8) = 4a 2 * h same volume

6 4.2 Multiplying and Dividing Rational Expressions Multiplying Rational Expressions Multiply and divide the rational expressions using the same properties as numerical fractions. In both cases, the property can be used. Examples:

7 4.2 Multiplying and Dividing Rational Expressions Using Factoring Sometimes, you may have to factor the polynomial before you can multiply and simplify. Examples

8 4.2 Multiplying and Dividing Rational Expressions Multiplying a Rational Expression by a Polynomial Follows the same rules Treat the polynomial as a fraction with 1 as the denominator Examples:

9 4.2 Multiplying and Dividing Rational Expressions Dividing Rational Expressions When dividing rational expressions, take the reciprocal of the divisor, then multiply. Examples:

10 4.2 Multiplying and Dividing Rational Expressions Dividing a Rational Expression by a Polynomial Like with multiplication, treat the polynomial as a fraction over 1. Example:

11 4.2 Multiplying and Dividing Rational Expressions Simplifying a Complex Fraction Complex fraction: a fraction with one or more fractions in the numerator, denominator, or both. Example

12 4.3 Add & Subtract Rational Expressions Adding Expressions with Like Denominators With like denominators, when adding fractions, combine the numerators. Leave the denominator the same. Examples:

13 4.3 Add & Subtract Rational Expressions Subtracting Expressions with Like Denominators With like denominators, when adding fractions, combine the numerators. Leave the denominator the same. Then simplify. Examples

14 4.3 Add & Subtract Rational Expressions Adding Expressions with Different Denominators With different denominators, you will need to find the least common denominator (LCD) of the two fractions (this is the least common multiple (LCM) of the two denominators) Example

15 4.3 Add & Subtract Rational Expressions Subtracting Expressions with different denominators Find the LCD and rewrite each fraction Simplify each numerator Subtract the numerators Simplify the fraction Example:

16 4.3 Add & Subtract Rational Expressions Using Rational Expressions A certain truck gets 25% better gas mileage when it holds no cargo than when it is fully loaded. Let m be the number of miles per gallon of gasoline the truck gets when it is fully loaded. The truck drops off a full load and returns empty. What is an expression for the number of gallons of gasoline the truck uses?

17 4.4 Inverse Variation Identifying Direct and Inverse Variations Direct Variation y = kx, where k 0 As x increases, y increases proportionally The graph is linear Inverse Variation xy = k, y = k/x, or x = k/y,where k 0 As x increases, y decreases proportionally The graph is hyperbolic k is the constant of variation Graphs give rough information, but are not enough to identify an inverse or direct variation by themselves.

18 4.4 Inverse Variation Examples: Is the relationship between the variables direct variation, an inverse variation, or neither? Write the model that corresponds to the appropriate variation. As x increases, y decreases, and a plot verifies that this is likely an inverse rela+onship. x y (2)(15) = 30 (10)(3) = 30 (4)(7.5) = 30 (15)(2) = 30 x2.5 x2 x1.5 x y x2.5 x2 x1.5 The constant of varia+on is 30, so the model for this inverse varia+on is xy = 30. Inverse y/x Direct x y xy Y=8/x y=40x x2.5 x x2 x x2 x

19 4.4 Inverse Variation Determining an Inverse Variation Suppose x and y vary inversely, and x = 4 when y = 12. What function models the inverse variation? What does the graph look like? What is y when x = 10? xy = k (4)(12) = k 48 = k xy = 48 or y = 48/x y = 48/x y = 48/10 y = 4.8

20 4.4 Inverse Variation Modeling an Inverse Variation Your math class has decided to pick up litter each weekend in a local park. Each week there is approximately the same amount of litter. The table shows the number of students who worked each of the first four weeks of the project and the time needed for the pickup. What function models the data? How many students should there be to complete the project in at most 30 minutes each week? Number of students = n +me in minutes = t n t (3)(85) = 255 (12)(21) = 252 (5)(51) = 255 (17)(15) = 255 (n)(t) is almost always 255. The model is nt = 255 nt = 255 ; t = 30 n =? n(30) = 255 n = 255/30 n = 8.5 There should be at least 9 students to complete the job in at most 30min.

21 4.4 Inverse Variation Using Combined Variation Combined variation: when one quantity varies with respect to at least two others. Joint variation: when one quantities varies directly with at least two others. Equations: z varies jointly with x and y z varies jointly with x and y and inversely with w z varies directly with x and inversely with the product wy

22 4.4 Inverse Variation Example: The number of bags of grass seed n needed to reseed a yard varies directly with the area a to be seeded and inversely with the weight of w of a bag of seed. If it takes two 3-lb bags to seed an area of 3600ft 2, how many 3-lb bags will seed 9000ft 2?

23 4.4 Inverse Variation Applying Combined Variation Gravitational potential energy PE is a measure of energy. PE varies directly with an object s mass and its height in meters above the ground. Physicists use g to represent the constant of variation, which is gravity. A skateboarder has a mass of 58kg and a potential energy of J at the top of a 4m halfpipe. What is the gravitational potential energy of a 65kg skateboarder on the same halfpipe? PE = gmh = g (58)(4) = 232g PE = 9.8mh PE = (9.8)(65)(4) PE = 2548J 9.8 = g PE = 9.8mh

24 4.4 Inverse Variation The formula for the Ideal Gas Law is PV=nRT where P is the pressure in kilopascals (kpa), V is the volume in liters (L), T is the temperature in Kelvin (K), n is the number of moles of gas, and R=8.314 in the universal gas constant. Write an equa+on to find the volume in terms of P, n, r, and T. What volume is needed to store 5 moles of helium gas at 350K under a pressure of 190kPA? A 10L cylinder is filled with hydrogen gas to a pressure of 5000kPA. The temperature of gas is 300K. How many moles of hydrogen gas are in the cylinder?

25 4.5 The Reciprocal Function Family Graphing an Inverse Variation Function Reciprocal functions: Parent function: f(x) = 1/x, where x 0 General form: Asymptotes are lines the graph approaches but does not touch. The vertical asymptote for a reciprocal function is x = h. The horizontal asymptote for a reciprocal function is y = k. Inverse variation function: f(x) = a/x, where x 0. a determines stretches, compressions, and reflections on x-axis.

26 4.5 The Reciprocal Function Family Example: What is the graph of y = 8/x, x 0? Identify the x- and y- intercepts and the asymptotes of the graph. Also state domain and range. There are no x- or y-intercepts as the asymptotes of the graph are y =0 and x = 0. The domain is all real numbers except x = 0 and the range is all real numbers except y = 0.

27 4.5 The Reciprocal Function Family Identifying Reciprocal Function Transformations Branches Each part of the graph of a reciprocal function In Quadrants I and III when a is positive. In Quadrants II and IV when a is negative. All stretches ( a >1)/shrinks( a <1) remain in the same Quadrants. Example: For each given value of a, how do the graphs of y = 1/x and y = a/x compare? What is the effect of a on the graph? a = 6 a = 0.25 a = 6

28 4.5 The Reciprocal Function Family Graphing a Translation Start by graphing the asymptotes. Then translate the graph. Draw the branches through these points.

29 4.5 The Reciprocal Function Family Example: What is the graph of? Identify the domain and range. h = 4 k = 6 Ver+cal asymptote: x = 4 Horizontal asymptote: y = 6 The graph is moved 4 units right and 6 units up. Use the points (1,1) and ( 1, 1) from the parent func+on to select points on this graph. (1,1) + (4,6) (5,7) ( 1, 1) + ( 4, 6) ( 3, 5) Draw the branches through these points.

30 4.5 The Reciprocal Function Family Writing the Equation of a Transformation If you know the asymptotes and the value of a of a reciprocal function, you can write the equation of the function. Use the intersection of the new asymptotes to determine how the graph was translated (h and k). Example: The graph below is a translation of y = 2/x. What is an equation for the function? The asymptotes cross at ( 3, 4). This means that h = 3 and k = 4 are the asymptotes. Therefore, the equa+on is

31 4.5 The Reciprocal Function Family Using a Reciprocal Function The rowing club is renting a 57-passenerg bus for a day trip. The cost of the bus is $750. Five passengers will be chaperones. If the students who attend share the bus cost equally, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How many students must ride the bus to make the cost per student no more than 20? To share the cost equally, divide $750 by n. With a capacity for 57 people and 5 chaperones, there are a maximum of 52 students. Domain is 1 x 52. Graph by plokng the func+on and y = 20. Determine where they intersect. At least 38 students must ride the bus.

32 4.6 Graphing Rational Expressions Finding Points of Discontinuity Rational function: a function containing a rational expression. If a function has a polynomial in its denominator, its graph has a gap at each zero of the polynomial. One-point hole Vertical asymptote The domain is all real numbers except the zeros of the denominator. These graphs are discontinuous graphs (graphs with a break in the domain) If there are no values that make the denominator zero, the graph is a continuous graph.

33 4.6 Graphing Rational Expressions Point of Discontinuity If a is a real number of which the denominator of a rational function f(x) is zero, then a is not the domain of f(x). The graph of f(x) is not continuous at x = a and the function has a point of discontinuity at x = a. The graph of has a removable discontinuity at x = -2. The hole in the graph is called a removable discontinuity because you could make the function continuous by redefining it at x = -2 so that f(-2) = 1. The graph of has a non-removable discontinuity at x = 2. There is no way to redefine the function at 2 to make the function continuous.

34 4.6 Graphing Rational Expressions When you are looking for discontinuities, remember to factor the denominator. The discontinuity caused by (x a) n is removable if the numerator also has (x a) n as a factor. Example: What are the domain and points of discontinuity of each rational function? Are the points of discontinuity removable or non-removable? What are the x- and y- intercepts? Domain: all real numbers except x = 1,3 The points of discon+nuity are non-removable x-intercept: (-3,0) y-intercept: (0,1) Domain: all real numbers except x = 4 The point of discon+nuity is removable x-intercept: (-1,0) y-intercept: (0,1)

35 4.6 Graphing Rational Expressions Finding Vertical Asymptotes If a rational function has a non-removable discontinuity at x = a, the graph of the rational function will have a vertical asymptote at x = a. The graph of the rational function f(x) = P(x) / Q(x) has a vertical asymptote at each real zero of Q(x) if P(x) and Q(x) have no common zeros. If P(x) and Q(x) have (x a) m and (x a) n as factors, respectively and m < n, then f(x) also has a vertical asymptote at x = a. Example: What are the vertical asymptotes for the graph of Since 2 and 3 are the zeros of the denominator and neither is a zero of the numerator, the lines x = 2 and x = 3 are ver+cal asymptotes. Since 3 is zeros of the denominator with no match in the numerator, x = 3 is a ver+cal asymptote. 2 is a zero of both numerator and denominator, so it is a hole.

36 4.6 Graphing Rational Expressions Finding Horizontal Asymptotes To find the horizontal asymptotes of the graph of a rational function, compare the degree of the numerator m to the degree of the denominator n. If m < n, the graph has a horizontal asymptote y = 0 (x-axis) If m > n, the graph has no horizontal asymptote If m = n, the graph has horizontal asymptote y = a/b where a is the coefficient of the greatest degree in the numerator and b is the coefficient of the greatest degree in the denominator. Examples: What is the horizontal asymptote for the rational function? m = n; y = 2/1 à y = 2 m < n; y = 0 m > n; none

37 4.6 Graphing Rational Expressions Graphing a Rational Function By finding all intercepts and asymptotes, you can get a reasonable graph. Sometimes, a few extra points will be necessary. Example: What is the graph of the rational function Horizontal asymptote: m = n; y = 1/1 à y = 1 Ver+cal asymptotes: x = 2, x = 2 x-intercepts (roots of numerator): (3,0) & ( 4, 0) y-intercept (x = 0): (0,3) More points: x y 3 6/ /3 4 2/3

38 4.6 Graphing Rational Expressions Using a Rational Function You work in a pharmacy that mixes different concentrations of saline solutions for its customers. The pharmacy has a supply of two concentrations, 0.5% and 2%. The function below gives the concentration of the saline solution after adding x ml of the 0.5% solution to 100mL of the 2% solution. How many ml of the 0.5% solution must you add for the combined solution to have a concentration of 0.9%? Graph the func+on under Y1 and under Y2 on the calculator. Find the point of intersec+on. It will take 275mL of the 0.5% solu+on.

39 4.7 Solving Rational Equations Solving a Rational Equation A rational equation contains at least one rational expression Find the Least Common Denominator (LCD) Multiply both sides by the LCD Simplify and Solve Check for extraneous solutions Write the solutions Example:

40 4.7 Solving Rational Equations Using Rational Equations A flight across the US takes longer east to west than it does west to east. Assume that winds are constant in the eastward direction. When flying westward, the headwind decreases the airplane s speed. When flying eastward, the tailwind increases its speed. The time for a round trip between Chicago and San Francisco (1850mi one way) is 7.75hr. If the airplane cruises at 480 mi/h, what is the speed of the wind? Let x = wind speed Rate * +me = distance, so +me = distance/rate West to East 1850 Dist Rate Time x Speed is posi+ve. East to west x West-to-east wind speed is about 35mi/h

41 4.7 Solving Rational Equations Using a Graphing Calculator to Solve a Rational Equation To solve using the graphing calculator, plot each side of the equation and see where they intersect. Example: The solu+on is x = 0