Algebra II Notes Unit Nine: Rational Equations and Functions

Transcription

1 Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems. Recall Direct Variation: y varies directly with if y k, where k is the constant of variation and k 0. Inverse Variation: and y show inverse variation if k 0. k y where k is the constant of variation and E: Do the following show direct variation, inverse variation, or neither? a. y 4.8 Solve for y. 4.8 y Inverse Variation b. y 8 Solve for y. y 8 Neither y c. Solve for y. y Direct Variation E: and y vary inversely, and y 6 when.5. Write an equation that relates and y. Then find y when. 4 Use the equation k y. Substitute the values for and y. 6 k.5 Solve for k. k Write the equation that relates and y. Use the equation to find y when. 4 9 y 9 4 y 9 4 Application Problem Involving Inverse Variation E: The volume of gas in a container varies inversely with the amount of pressure. A gas has volume 75 in. at a pressure of 5 lb/in.. Write a model relating volume and pressure. Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

2 Use the equation V k. Substitute the values of V and P. 75 P k 5 Solve for k. k Write the equation that relates V and P. Checking Data for Inverse Variation V 875 P E: Do these data show inverse variation? If so, find a model. W If H k, then there eists a constant, k, such that k HW. W H Check the products HW from the table. k The product k is constant, so H and W vary inversely. H 8 W Joint Variation: when a quantity varies directly as the product of two or more other quantities E: The variable y varies jointly with and z. Use the given values to write an equation relating, y, and z., y, z 5 Use the equation y kz. Substitute the values in for, y, and z. k 5 Solve for k. Write the equation that relates, y, and z. k 5 y 5 z E: Write an equation for the following.. z varies jointly with and y. z k y. y varies inversely with and z. y k z Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

3 . y varies directly with and inversely with z. k y z You Try: The ideal gas law states that the volume V (in liters) varies directly with the number of molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in kilopascals). The constant of variation is denoted by R and is called the universal gas constant. Write an equation for the ideal gas law. Then estimate the universal gas constant if V = 5.6 liters; n = mole; t = 88 K; P = 9.5 kilopascals. QOD: Suppose varies inversely with y and y varies inversely with z. How does vary with z? Justify your answer algebraically. Sample CCSD Common Eam Practice Question(s): The value of s varies jointly with m and p. If s = 0 when m = and p = 4, what is the value of s when m = 7 and p =? A B. 8 5 C D. 5 Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

4 Syllabus Objectives: 9. The student will graph rational functions with and without technology. 9. The student will identify domain, range, and asymptotes of rational functions. 9.6 The student will develop mathematical models involving rational epressions to solve real-world problems. Rational Function: a function of the form functions f p, where q p and q are polynomial E: Use a table of values to graph the function y. y y Note: The graph of y has two branches. The -ais y 0 is a horizontal asymptote, and the y- ais 0 is a vertical asymptote. Domain and Range: All real numbers not equal to zero. Eploration: Graph each of the functions on the graphing calculator. Describe how the graph compares to the graph of y. Include horizontal and vertical asymptotes and domain and range in your answer... y y. 4. y y 5 5. y 4 6. y Page 4 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

5 Hyperbola: the graph of the function y a k h E: From the eploration above, describe the asymptotes, domain and range, and the effects of a a on the general equation of a hyperbola y k. h Horizontal Asymptote: y k Vertical Asymptote: h Domain: All real numbers not equal to h. Range: All real numbers not equal to k. As a gets bigger, the branches move farther away from the origin. If a 0, the branches are in the first and third quadrants. If a 0, the branches are in the second and fourth quadrants. E: Sketch the graph of y Horizontal Asymptote: y Vertical Asymptote: Plot points to the left and right of the vertical asymptote:,, 0,5,,, 4, Sketch the branches in the second and fourth quadrants. More Hyperbolas: graphing in the form a b y c d Horizontal Asymptote: a y Vertical Asymptote: c c d 0 d c E: Sketch the graph of y. y 0 Horizontal Asymptote: y Vertical Asymptote: 0 Plot points to the left and right of the vertical asymptote: 4 4,,,, 0,,, Page 5 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

6 Application Problems with Rational Functions E: The senior class is sponsoring a dinner. The cost of catering the dinner is $9.95 per person plus an $8 delivery charge. Write a model that gives the average cost per person. Graph the model and use it to estimate the number of people needed to lower the cost to $ per person. What happens to the average cost per person as the number increases? Model: Average cost = (Total Cost) / (Number of People) A They need at least 7 people to lower the cost to $ per person. The average cost approaches $9.95 as the number of people increases. You Try: Write a rational function whose graph is a hyperbola that has a vertical asymptote at and a horizontal asymptote at y. Can you write more than one function with the same asymptotes? QOD: In what line(s) is the graph of inverse of this function? y symmetric? What does this symmetry tell you about the Page 6 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

7 Sample CCSD Common Eam Practice Question(s): What is the graph of f? Page 7 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

8 Syllabus Objectives: 9. The student will graph rational functions with and without technology. 9. The student will identify domain, range, and asymptotes of rational functions. 9.6 The student will develop mathematical models involving rational epressions to solve real-world problems. Graphs of Rational Functions f p q -intercepts: the zeros of p Vertical Asymptotes: occur at the zeros of q Horizontal Asymptote: describes the END BEHAVIOR of the graph (as and ) o If the degree of p is less than the degree of q, then y 0 is a horizontal asymptote. p is equal to the degree of o If the degree of coefficients) is a horizontal asymptote. o If the degree of p is greater than the degree of horizontal asymptote. q, then y = (the ratio of the leading q, then the graph has no E: Graph the function y. -intercepts: 0 0 Vertical Asymptotes:, Horizontal Asymptote: y 0 Plot points between and outside the vertical asymptotes 0 y 0 y Note: The point 0,0 lies on a horizontal asymptote. This is valid since horizontal asymptotes describe end behavior. A graph can NEVER have a point eist on a vertical asymptote since vertical asymptotes represent where the function does not eist. Local (Relative) Etrema: the local (relative) maimum is the largest value of the function in a local area, and the local (relative) minimum is the smallest value of the function in a local area Page 8 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

9 E: Graph the function 0 y. Find any local etrema. 4 -intercepts: 0 40 Vertical Asymptotes:, Horizontal Asymptote: y Plot points between and outside the vertical asymptotes y ,0. Local maimum is 0. It occurs at the point y Slant Asymptote: If the degree of the numerator is one greater than the degree of the denominator, then the slant asymptote is the quotient of the two polynomial functions (without the remainder). E: Sketch the graph of f and to find the local etrema.. Use the graphing calculator to check your answer 4 -intercepts: 0 0, 40 Vertical Asymptotes: 4 Horizontal Asymptote: none Slant Asymptote: y 6 Plot points to the left and right of the vertical asymptote y Local Minimum: Local Maimum: 9.65 y Page 9 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

10 Application Problems with Local Etrema E: A closed silo is to be built in the shape of a cylinder with a volume of 00,000 cubic feet. Find the dimensions of the silo that use the least amount of material. Volume of a Cylinder: V r h 00,000 00,000 h r rh Using the least amount of material is finding the minimum surface area, S, of the cylinder. Surface Area of a Cylinder: Substitute h from above: S r rh 00,000 00,000 S r r r r r Graph the function for surface area and find the minimum value. The minimum surface area occurs when the radius is 5.5 ft. The height is 00,000 00,000 h 50. ft. r 5.5 You Try: Sketch the graph of 90 y. QOD: Describe how to find the horizontal, vertical, and slant asymptotes of a rational function. Sample CCSD Common Eam Practice Question(s): What are the asymptotes of the function f? 5 A. y = 0, = 5 B. y = 0, = 5, = 5 C. y =, = 5 D. y =, = 5, = 5 Page 0 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

11 Syllabus Objective: 9.4 The student will simplify, add, subtract, multiply, and divide rational epressions. Recall: When simplifying fractions, we divide out any common factors in the numerator and denominator E: Simplify The numerator and denominator have a common factor of 4. They can be rewritten. 45 Now we can divide out the common factor of 4. The remaining numerator and denominator have no common factors (other than ), so the fraction is now simplified Simplified Form of a Rational Epression: a rational epression in which the numerator and denominator have no common factors other than Simplifying a Rational Epression. Factor the numerator and denominator. Divide out any common factors E: Simplify the epression Factor. Divide out common factors Recall: When multiplying fractions, simplify any common factors in the numerators and denominators, then multiply the numerators and multiply the denominators. E: Multiply 5 4. Divide out common factors Multiplying Rational Epressions. Factor numerators and denominators (if necessary).. Divide out common factors.. Multiply numerators and denominators. Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

12 E: Multiply Factor Divide out common factors. Multiply. E: Find the product Factor Divide out common factors Multiplying Rational Epressions with Monomials Use the properties of eponents to multiply numerators and denominators, then divide. E: Multiply 4 6 y 0 y. y 8y Use the properties of eponents and simplify. 6 y 0 y 60 y 5 y y 8y 6y Recall: When dividing fractions, multiply by the reciprocal. E: Find the quotient. Dividing Rational Epressions Multiply the first epression by the reciprocal of the second epression and simplify. Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

13 E: Divide Multiply by the reciprocal Factor and simplify. 4 4 E: Find the quotient of 8 0 and 4 4. Multiply by the reciprocal Factor and simplify You Try: Simplify. 4 9 QOD: What is the factoring pattern for a sum/difference of two cubes? Sample CCSD Common Eam Practice Question(s): Simplify the epression: A. B. C. D. 4 4 Page of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

14 Syllabus Objective: 9.4 The student will simplify, add, subtract, multiply, and divide rational epressions. Recall: To add or subtract fractions with like denominators, add or subtract the numerators and keep the common denominator. E: Find the difference Adding and Subtracting Rational Epressions with Like Denominators Add or subtract the numerators. Keep the common denominator. Simplify the sum or difference. E: Subtract E: Add. 6 6 Recall: To add or subtract fractions with unlike denominators, find the least common denominator (LCD) and rewrite each fraction with the common denominator. Then add or subtract the numerators. E: Add. 5 0 Note: To find the LCD, it is helpful to write the denominators in factored form. 5 5, 0 5 LCD = 5 60 Adding and Subtracting Rational Epressions with Unlike Denominators. Find the least common denominator (in factored form).. Rewrite each fraction with the common denominator.. Add or subtract the numerators, keep the common denominator, and simplify. 4 E: Find the sum. 6 Factor and find the LCD. 4 LCD = Rewrite each fraction with the LCD Page 4 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

15 Add the fractions. 84 Note: Our answer cannot be simplified because the numerator cannot be factored. We will leave the denominator in factored form. E: Subtract: 69 9 Factor and find the LCD. LCD = Rewrite each fraction with the LCD. Subtract the fractions. 6 Comple Fraction: a fraction that contains a fraction in its numerator and/or denominator Simplifying a Comple Fraction Method : Multiply every fraction by the lowest common denominator. Method : Add or subtract fractions in the numerator/denominator, then multiply by the reciprocal of the fraction in the denominator. E: Use Method to simplify the comple fraction. 4 4 Multiply every fraction by the LCD = and divide out the common factors. 4 4 Simplify the remaining fraction. 4 Page 5 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

16 E: Use Method to simplify the comple fraction. 4 Add the fractions in the denominator Multiply by the reciprocal of the denominator and simplify. 5 5 You Try: Perform the indicated operations and simplify. 4 5 QOD: Eplain if the following is a true statement. The LCD of two rational epressions is the product of the denominators. Page 6 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

17 Syllabus Objective: 9.5 The student will solve equations involving rational epressions. Recall: When solving equations involving fractions, we can eliminate the fractions by multiplying every term in the equation by the LCD. E: Solve the equation. 4 9 Multiply every term by the LCD = 9. Solve the equation , 0 Rational Equation: an equation that involves rational epression To solve a rational equation, multiply every term by the LCD. Then check your solution(s) in the original equation. E: Solve the equation Multiply every term by the LCD = Solve the equation Check. true 8 8 6, so 8 E: Solve the equation Multiply every term by the LCD = Solve the equation. Page 7 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

18 Check This solution leads to division by zero in the original equation. Therefore, it is an etraneous solution. This equation has no solution. 6 E: Solve the equation 4 Multiply every term by the LCD = Solve the equation , Check true true Solutions:, Recall: When solving a proportion, cross multiply the two ratios. E: Solve the proportion Solving a Rational Equation by Cross Multiplying: this can be used when each side of the equation is a single rational epression Cross multiply. E: Solve the equation Solve , Page 8 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

19 Check division by zero! true 7 Solution: (Note: 4 is an etraneous solution.) 50 0 E: use the graph of the rational model y 8 the graphing calculator. to find the value of when y. on 0.67 You Try: Solve the rational equation QOD: When is cross-multiplying an appropriate method for solving a rational equation? Sample CCSD Common Eam Practice Question(s):. Solve 7 n 4n. A. B. C. D. n n 49 n 4 n 84 Page 9 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions

20 . Solve the rational equation for y: A. y = B. y = 0 C. y = D. y = 4 5 5y 0 5 y y y y The equivalent resistance R of two resistors in parallel, with resistances r and r, is given by the formula: R r r If r has twice the resistance of r, what is the value of r if the equivalent resistance is 0 ohms? A. 5 ohms B. 0 ohms C. 45 ohms D. 90 ohms Sample SAT Question(s): Taken from College Board online practice problems. The projected sales volume of a video game cartridge is given by the function sp 000, where s is p a the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 00,000 cartridges are sold at $0 per cartridge, how many cartridges will be sold at $0 per cartridge? (A) 0,000 (B) 50,000 (C) 60,000 (D) 50,000 (E) 00,000 Page 0 of 0 McDougal Littell: Alg II Unit 9 Notes: Rational Eq and Functions