Algebra II Notes Rational Functions Unit 6. 6.6 Rational Functions Math Background Previously, you Simplified linear, quadratic, radical and polynomial functions Performed arithmetic operations with linear, quadratic, radical and polynomial functions Identified the domain, range and -intercepts of real-life functions Graphed linear, quadratic and polynomial functions Composed linear, quadratic and radical functions Created functions to represent a real life situation Solved linear, quadratic, radical and polynomial functions Transformed parent functions of linear, quadratic, radical and polynomial functions In this unit you will Simplify rational epressions Perform arithmetic operations with rational epressions Graph rational functions with and without technology Solve problems involving rational equations and inequalities Compose rational functions with other functions Create rational functions to represent real life situations You can use the skills in this unit to Use the structure of an epression to identify ways to rewrite it. Use long division to perform partial fraction decomposition. Interpret the domain and its restrictions of a real-life function. Identify etraneous solutions of a rational equation. Describe how a rational function graph is related to its parent function. Model and solve real-world problems with rational functions using graphs Interpret the composition of functions as applied to real world problems. Vocabulary Composition of Functions The act of combining two mathematical functions. Domain of a rational epression the set of all real numbers ecept the value(s) of the variable that result in division by zero when substituted into the epression. Etraneous Solutions A root of a transformed equation that is not a root of the original equation because it was ecluded from the domain of the original equation. Horizontal Asymptote A horizontal line that the graph of a function approaches as tends to plus or minus infinity. It describes the function s end behavior. Partial Fraction Decomposition the operation that consists in epressing the fraction as a sum of a polynomial and one or several fractions with a simpler denominator. Rational epression a fraction with a polynomial in the numerator and a nonzero polynomial in the denominator. Also known as an algebraic fraction. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Simplify a rational epression Write the fraction so there are no common factors other than or -. Undefined An epression in mathematics which does not have meaning and so which is not assigned an interpretation. For eample, division by zero is undefined in the field of real numbers. Vertical asymptote A vertical line that the curve approaches more and more closely but never touches as the curve goes off to positive or negative infinity. The vertical lines correspond to the zeroes of the denominator of the rational function. Essential Questions How do you find sums, differences, products, and quotients of rational epressions? What knowledge and skills are required to rewrite simple rational epressions? How do I solve a rational equation? How are etraneous solutions generated from a rational equation? How can the composition of two functions be used to represent real life applications? Overall Big Ideas The rules for addition, subtraction, multiplication and division of rational epressions are analogous to those for rational numbers. Rational epressions can be rewritten using properties of fractions and elementary numerical algorithms. We solve rational equations by transforming the equation into a simpler form to solve. However, this can produce solutions that do not eist in the original domain. Real world applications can be modeled as a composition of two functions where the input is generally a function of time. Skill To transform the graph of y. To simplify rational epressions. To add, subtract, multiply, and divide rational epressions. To solve rational equations and inequalities. To compose a rational function with other functions. To create and apply a rational function to a real life situation. (e.g. concentrations from chemistry) Related Standards F.IF.B.- Relate the domain of any function to its graph and, where applicable, to the quantitative relationship it describes. For eample, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. *(Modeling Standard) Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 F.BF.B.- Identify the effect on the graph of replacing f() by f() + k, k f(), f(k), and f( + k) for specific values of k (both positive and negative); find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic epressions for them. Include simple radical, rational, and eponential functions, note the effect of multiple transformations on a single graph, and emphasize common effects of transformations across function types. A.SSE.A.- Use the structure of an epression, including polynomial and rational epressions, to identify ways to rewrite it. For eample, see y y factored as y y. as, thus recognizing it as a difference of squares that can be A.APR.D.6 a ( ) Rewrite simple rational epressions in different forms; write b ( ) in the form r ( ) q ( ), where b ( ) a ( ), b ( ), q ( ), and rare ( ) polynomials with the degree of rless ( ) than the degree of b ( ), using inspection, long division, or, for the more complicated eamples, a computer algebra system. A.APR.D.7 Understand that rational epressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational epression; add, subtract, multiply, and divide rational epressions. A.REI.A. Solve simple rational and radical equations in one variable, and give eamples showing how etraneous solutions may arise. A.CED.A.- Create equations and inequalities in one variable and use them to solve problems. Include equations arising from all types of functions, including simple rational and radical functions. *(Modeling Standard) F.BF.A.c Compose functions. For eample, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. *(Modeling Standard) A.CED.A.- Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate aes with labels and scales. Use all types of equations. *(Modeling Standard) A.CED.A.- Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling contet. Use all types of equations. For eample, represent inequalities describing nutritional and cost constraints on combinations of different foods. *(Modeling Standard) Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Unit 6.: Graphing Rational Functions Notes, Eamples, and Eam Questions Rational Function: a function of the form f p, where q p and q are polynomial functions. E : Use a table of values to graph the function y. y - y - Note: The graph of y has two branches. The -ais 0 is a vertical asymptote. Domain and Range: All real numbers not equal to zero. y is a horizontal asymptote, and the y-ais 0 A horizontal asymptote is a horizontal line that the graph of a function approaches as tends to plus or minus infinity. As grows very large or very small, the function approaches this value. A vertical asymptote is a vertical line near which the function grows to infinity or negative infinity. It is a place where the function is undefined. 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. y. y. y 6. y 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 on the general a equation of a hyperbola y k. Horizontal Asymptote: y k Vertical Asymptote: h 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. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 E : Sketch the graph of y Horizontal Asymptote: y Vertical Asymptote: The asymptotes are shown as red dashed lines. Plot points to the left and right of the vertical asymptote:,, 0,,,,, Sketch the branches in the second and fourth quadrants. Note the transformations: Reflected (a < 0), shifted up and right. 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 The asymptotes are shown as red dashed lines. Plot points to the left and right of the vertical asymptote:,,,, 0,,,0-0 - 0 - -0 Using long division: The quotient is:. Rewriting this, we get: ( ). Using transformations in this form, it tells us to reflect the graph, go up by and go left one unit. By dividing the rational function so that the numerator is one less degree than the denominator, we can use transformations to draw the graph of our rational function. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 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 ) p is less than the degree of q, then y 0 is a horizontal asymptote. o If the degree of o If the degree of p is equal to the degree of is a horizontal asymptote. p is greater than the degree of asymptote. It will have a slant asymptote. o If the degree of q, then y = (the ratio of the leading coefficients) q, then the graph has no horizontal 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 0-0 - 0 - -0 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 E : Graph the function y. Find any local etrema. 0 -intercepts: 0 0 Vertical Asymptotes:, Horizontal Asymptote: y Plot points between and outside the vertical asymptotes 0 y 0 0,0. Local maimum is 0. It occurs at the point y 0-0 - 0 - -0 Alg II Notes Unit 6. 6.6 Rational Functions Page 6 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 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 6: Sketch the graph of local etrema. f -intercepts: 0 0, 0 Vertical Asymptotes: Horizontal Asymptote: none since the degree of the numerator is greater than the degree of the denominator. Slant Asymptote: y 6. The asymptote is in red. To find the slant asymptote, use long division. 6 The quotient is: 6. 6 6. Use the graphing calculator to check your answer and to find the -0 - -0 - -0-0 0 - y 0 0-0 - -0 - -0 Plot points to the left and right of the vertical asymptote 9 6 0 y 9... 0.7 0. 0.6 Local Minimum: 0.8 Local Maimum: 9.6 Alg II Notes Unit 6. 6.6 Rational Functions Page 7 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. What is the graph of f SAMPLE EXAM QUESTIONS? Ans: A. What are the asymptotes of the function: f? A. y = 0, = B. y =, =, = C. y =, = D. y = 0, =, =. Which set contains all the real numbers that are not part of the domain of A. {8} C. {-} B. {-, 8} D. {-8, } Ans: B f( )? Ans: D Alg II Notes Unit 6. 6.6 Rational Functions Page 8 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. Which is the graph of f? y A. y B. y C. y D. Ans: D Alg II Notes Unit 6. 6.6 Rational Functions Page 9 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. What are the asymptotes of the function f? 0 A. 6,, y B. 6,, y C., 6, y D., 6, y Ans: A 6. Which intervals correctly define the domain of f( ) A. (,) and (, ) B. (, ) and (, ) C. (, ) and (, ) D. (, ) and (, ) Ans: B 7. Which statement is true for the function f( )? A. is not in the range of the function. B. is not in the domain of the function. C. - is not in the range of the function. D. - is not in the domain of the function. Ans: D Unit 6.: Simplify Rational Epressions Recall: When simplifying fractions, we divide out any common factors in the numerator and denominator E: Simplify 6 0. The numerator and denominator have a common factor of. They can be rewritten. Now we can divide out the common factor of. The remaining numerator and denominator have no common factors (other than ), so the fraction is now simplified. Alg II Notes Unit 6. 6.6 Rational Functions Page 0 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 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 if you can. Divide out any common factors E 7: Simplify the epression Factor. 6. 6 Divide out common factors. 6 6 E 8: Simplify the epression Factor. 6. 6 Divide out common factors. 6 6 E 9: Simplify the epression Factor. 9 8 8 9. 9 9 9 Divide out common factors. 9 9 9 9 Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Unit 6.: Multiply and Divide Rational Epressions Recall: When multiplying fractions, simplify any common factors in the numerators and denominators, then multiply the numerators and multiply the denominators. E: Multiply. Divide out common factors. 6 0 7 7 0 7 Multiplying Rational Epressions. Factor numerators and denominators (if necessary).. Divide out common factors.. Multiply numerators and denominators. E 0: Multiply Factor. 7. 9 Divide out common factors. Multiply. E : Find the product. 9 6 Factor. 7 8 9 6 9 6 Divide out common factors. 9 6 9 6 Multiplying Rational Epressions with Monomials Use the properties of eponents to multiply numerators and denominators, then divide. E : Multiply 6 y 0 y. y 8y Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Use the properties of eponents and simplify. 7 6y 0y 60y 6 y y 8y y Recall: When dividing fractions, multiply by the reciprocal. E: Find the quotient. Dividing Rational Epressions 0 6 0 0 6 0 6 8 Multiply the first epression by the reciprocal of the second epression and simplify. E : Divide. 8 6 Multiply by the reciprocal. 6 8 Factor and simplify. E : Find the quotient of 8 0 and. Multiply by the reciprocal. 8 0 Factor and simplify. E : Simplify. 9 ( ) Multiply by the reciprocal. 9 Factor and simplify. ( ) ( ) ( ) QOD: What is the factoring pattern for a sum/difference of two cubes? Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 SAMPLE EXAM QUESTIONS. Simplify the epression. 7 A. B. C. D. 9 Ans: A. Simplify the epression: 6 A. C. B. D. Ans: D. Multiply. Simplify your answer. 6 8 y 9y z z y A. 8 9 6 y z C. B. 6 yz D. 6 z yz Ans: C. Which epression represents the quotient? 6 8 z z z A. z z C. z z B. z z D. z z Ans: A Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. Which epression is equivalent to y 8 y for all y, 0? 6 A. 6 y C. y B. y 9 D. Unit 6.: Add and Subtract Rational Epressions y 8 Ans: C Recall: To add or subtract fractions with like denominators, add or subtract the numerators and keep the common denominator. E: Find the difference. 8 8 6 Adding and Subtracting Rational Epressions with Like Denominators Add or subtract the numerators. Keep the common denominator. Simplify the sum or difference. E 6: Subtract. 7 7 E 7: 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. 0 Note: To find the LCD, it is helpful to write the denominators in factored form., 0 LCD = 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. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 E 8: Find the sum. 6 Factor and find the LCD. LCD = Rewrite each fraction with the LCD. 8 Add the fractions. 8 Note: Our answer cannot be simplified because the numerator cannot be factored. We can leave the denominator in factored form. E 9: Subtract: 69 9 Factor and find the LCD. LCD = Rewrite each fraction with the LCD. Subtract the fractions. 6 E 0: Perform the indicated operations and simplify. Find the LCD. LCD = ( )( ) Rewrite each fraction with the LCD. ( ) ( ) ( ) ( ) Add and subtract the fractions. ( )( ) ( )( ) ( )( ) 6 ( )( ) ( )( ) Note: Denominator can be left in factored form. Alg II Notes Unit 6. 6.6 Rational Functions Page 6 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 QOD: Eplain if the following is a true statement. The LCD of two rational epressions is the product of the denominators. SAMPLE EXAM QUESTIONS. A board of length cm was cut into two pieces. If one piece is 7 cm, epress the length of the other board as a rational epression. A. B. C. D. ( )( ) ( ) 6 ( ) 6 ( )( ) Ans: C Unit 6. Solve rational equations 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. 9 Multiply every term by the LCD = 9. Solve the equation. 9 9 9 9 66 0 0 0 0 0, 0 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. Alg II Notes Unit 6. 6.6 Rational Functions Page 7 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 E : Solve the equation Multiply every term by the LCD = Solve the equation. 6 8 8 Check. E : Solve the equation true 8 8 6, so 8 Multiply every term by the LCD = + Solve the equation. 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 Multiply every term by the LCD = 6 6 Solve the equation. 60 0 0, Check. 6 6 true 6 6 true Solutions:, Alg II Notes Unit 6. 6.6 Rational Functions Page 8 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Recall: When solving a proportion, cross multiply the two ratios. E: Solve the proportion 0. 0 0 Solving a Rational Equation by Cross Multiplying: this can be used when each side of the equation is a single rational epression E : Solve the equation. Cross multiply. Solve. Check. 0 0 division by zero! 0 0, true 7 Solution: (Note: is an etraneous solution.) E 6: Solve the rational equation. 7 0 This is not a proportion. We cannot cross multiply. We must find the LCD and multiply. Multiply every term by the LCD = ( ) ( ) 7 0 ( ) ( ) (7) () ()(0 ) Solve the equation. 6 0 7 6 0 (7 )( ) 0, 7 0 is true. is a solution. 6 9 Check the solution: 0 7 is true. is also a solution. 0 7 7 7 Alg II Notes Unit 6. 6.6 Rational Functions Page 9 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 0 0 E 7: use the graph of the rational model y 8 graphing calculator. to find the value of when y. on the 0 0 Put 8 into Y and put. in Y. Go to CALC to find the point of intersection. 0.67 QOD: When is cross-multiplying an appropriate method for solving a rational equation? Unit 6. Solve rational inequalities To solve rational inequalities, which are inequalities that contain one or more rational epressions follow these steps:. State the ecluded values. These are the values for which the denominator is 0.. Solve the related equation.. Use the values determined from the previous steps to divide a number line into intervals.. Test a value in each interval to determine which intervals contain values that satisfy the inequality. E 8: Solve 9 Step : The ecluded value for this inequality is. Step : Solve 9 Cross multiply 98 9 88 Step : Separate the number line into intervals at the solutions and the ecluded value. Alg II Notes Unit 6. 6.6 Rational Functions Page 0 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Step : Test a sample in each interval. Use = 0, = 7/8 and =. 0 0 9 0 0 9 FALSE 7 8 7 8 9 7 8 7 9 TRUE 9 9 FALSE The statement is true for 7/8, so the solution is: 9. E 9: Solve 7 9 Step : The ecluded value for this inequality is 0. Step : Solve 7 LCD = 9 9 7 9 9 9 9 6 7 Step : Separate the number line into intervals at the solutions and the ecluded value. Step : Test a sample in each interval. Use = -, = and = 6. 7 ( ) 9 7 9 TRUE 9 7 () 9 7 9 FALSE 9 6 7 (6) 6 9 7 8 6 9 TRUE 8 9 The statement is true for = - and = 6. Therefore, the solution is 0 or. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 E 0: Solve Step : The ecluded value for this inequality is. Step : Solve LCD = ( ) ( ) ( ) ( ) (8) (6)( ) 6 60 ( 6)( ) 0 6 or Step : Separate the number line into intervals at the solutions and the ecluded value. Step : Test a sample in each interval. Use = -, = 0, = and = 8. TRUE 0 0 0 0 FALSE TRUE 6 8 8 8 8 9 0 7 FALSE The statement is true for = - and =. Therefore, the solution is or 6. SAMPLE EXAM QUESTIONS. Solve 7 n n. A. B. n C. n D. n 9 n 8 Ans: A Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. Solve the rational equation for y: y 0 y y y y 8 A. y = C. y = 0 B. y = D. y = Ans: C 7. Solve the equation. 9 7 7 A. -8 C. 7 B. 8 D. No solution Ans: A. Solve: A. B. C., D., Ans: A. Use the following epressions to answer the questions. A () Eplain how to add two rational epressions. Simplify A B as an eample. Be sure to give reasons for each step. Simplify completely. AB 6 B 76 C 7 D Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 () Eplain how subtracting (such as simplifying A B ) would be different from adding. AB 6 6 () Eplain how to multiply rational epressions, simplifying B C () Show how to divide B by C. BC BC as an eample. () Eplain how you would solve a rational equation likec D. Then describe, in detail, the strategy for solving an equation like B C D. Do not completely solve the equations. Instead, concentrate on the first two or three steps in solving; show what to do and eplain why. First, I need to check the denominators: they tell me that cannot equal zero or (since these values would cause division by zero). Re-check at the end, to make sure any solutions are "valid". C 7 D 7 7 C D X Cross product 7 7 so 07 0 In the end, using the quadratic formula the solutions are: valid solutions. Now, B C D When you were adding and subtracting rational epressions, you had to find a common denominator. Now that you have equations (with an "equals" sign in the middle), you are allowed to multiply through by the LCD (because you have two sides to multiply on) and get rid of the denominators entirely. In other words, you still need to find the common denominator, but you don't necessarily need to use it in the same way. Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 7 Now let s multiply through by LCD = So we get 7 Which is equivalent to: 7, 7 This is a quadratic equation with the following real solutions: Unit 6. Compose a rational function with other functions Recall that the composition of functions refers to the combining of functions in a manner where the output from one function becomes the input for the net function. In math terms, the range (the y-value answers) of one function becomes the domain (the -values) of the net function. E : Let f( ) and g ( ). Find ( f g )( ) and ( g f )( ). ( f g)( ) ( f( g( )) ( ) ( g f)( ) ( g( f( )) ( ) E : Let f ( ) and g ( ). Find ( f g)( ) and ( g f)( ). ( f g)( ) ( f( g( )) ( ) 6 9 8 6 9 ( ) ( g f)( ) ( g( f( )) 7 ( ) Alg II Notes Unit 6. 6.6 Rational Functions Page of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Unit 6.6 Create and Apply Rational Functions Application Problems with Rational Functions E : The senior class is sponsoring a dinner. The cost of catering the dinner is $9.9 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) 9.9 8 A They need at least 7 people to lower the cost to $ per person. The average cost approaches $9.9 as the number of people increases. 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. Alg II Notes Unit 6. 6.6 Rational Functions Page 6 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 The minimum surface area occurs when the radius is. ft. The height is 00, 000 00, 000 h 0. ft. r. E : Every year, the junior and senior classes at Hillcrest High School build a house for the community. If it takes the senior class days to complete a house and 7 days for the junior class to complete a house, how long would it take if they worked together? Find the rate for each of the classes: The senior class can complete house in days, so their rate is of a house per day. The rate for the junior class is of a house per day. We will be combining the two rates. 7 Solve: Multiply by the LCD: 7c 7 c 7c 7c 7c 7 c cc7 c7 c 8 Check: 7 8 7 7 7 so c 8 days E 6: Sandra is rowing a canoe on Stanhope Lake. Her rate in still water is 6 miles per hour. It takes Sandra hours to travel 0 miles round trip. Assuming that Sandra rowed at a constant rate of speed, determine the rate of the current. Time with Time against Total time d the current the current Use the formula d rt, or t. r 6 r 6 r hours Solve: LCD = (6 r)(6 r) 6r 6r (6 r) (6 r) (6 r) (6 r) 6 r 6 r (6 r) (6 r) 6 r 8 r r 0 r0 r 08 r 60 08 r (6 r)(6 r) Since speed cannot be negative, the speed of the current is miles per hour. E 7: Mia adds a 70% acid solution to milliliters of a solution that is % acid. How much of the 70% acid solution should be added to create a solution that is 60% acid? Each solution has a certain percentage that is acid. The percentage of acid in the final solution must equal the amount of acid divided by the total solution. Original Added New Pe rcen = a mount of acid Amount of Acid 0.() 0.7() 0.()+0.7() tage of acid in solution total solution Total Solution + Alg II Notes Unit 6. 6.6 Rational Functions Page 7 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 Solve: percent amt. of acid 00 total solution 60 (0.)() 0.7 Cross Multiply 00 60( ) 00(.8 0.7 ) 70 6080 70 0 0 Check: 60 (0.)() 0.7( ) 00 7. 8 0.6 66 0.6 0.6 Mia needs to add milliliters of the 70% acid solution. QOD: Describe how to find the horizontal, vertical, and slant asymptotes of a rational function. QOD: 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 y symmetric? What does this symmetry tell you about the inverse of this function? SAMPLE EXAM QUESTIONS. 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. ohms C. 0 ohms B. ohms D. 90 ohms Ans: C. Last week, Wendy jogged for a total of 0 miles and biked for a total of 0 miles. She biked at a rate that was twice as fast as her jogging rate. () Suppose Wendy jogs at a rate of r miles per hour. Write an epression that represents the amount of time she jogged last week and an epression that represents the amount to time she biked last week. (hint: distance rate time ) t t jogged biked d 0 r r d 0 r r r Alg II Notes Unit 6. 6.6 Rational Functions Page 8 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6 () Write and simplify an epression for the total amount of time Wendy jogged and biked last week. 0 t total r r r () Wendy jogged at a rate of miles per hour. What was the total amount of time Wendy jogged and biked last week? ttotal ( hours ) r. The rate of heat loss from a metal object is proportional to the ratio of its surface area to its volume. () What is the ratio of a steel sphere s surface area to volume? Here are the formulas for sphere s surface area and volume, where r is radius: So the ratio of sphere s surface area to volume is: r r r () Compare the rate of heat loss for two steel spheres of radius meters and meters, respectively. The smaller sphere (the one with the radius of ) will lose heat at a rate of /=. and the bigger one at a rate of. Since time is inverse proportion to heat rate we could also say that if it takes the big sphere an hour to cool down, is going to take the smaller one / of that time which is 0 minutes.. A sight-seeing boat travels at an average speed of 0 miles per hour in the clam water of a large lake. The same boat is also used for sight-seeing in a nearby river. In the river, the boat travels.9 miles downstream (with the current) in the same amount of time it takes to travel.8 miles upstream (against the current). Find the current of the river. t downstream t upstream.9.8 0 r 0 r.9 0.8 0 r r.8r.9r.9 0.80.7r 8 6.7r r.68 mi/ h Alg II Notes Unit 6. 6.6 Rational Functions Page 9 of 0 //0
Algebra II Notes Rational Functions Unit 6. 6.6. A baseball player s batting average is found by dividing the number of hits the player has by the number of at-bats the player has. Suppose a baseball player has hits and 0 at-bats. Write and solve an equation to model the number of consecutive hits the player needs in order to raise his batting average to 0.00. Eplain now you found your answer. Let be the number of consecutive hits the player needs. In that case the number of hits will become (+) and the number of at-bats will be (0+), so the equation representing his batting average will be: 0.00 0 0. 0.6 7.67 cross product, followed by solving the linear equation gives us: 0. 0.6 7.67 and since is a whole number we need to round it up to. So the number of consecutive hits the player needs in order to raise his batting average to 0.00 is hits. 6. 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 p a, where s is 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) 0,000 (C) 60,000 (D) 0,000 (E) 00,000 Ans: C Alg II Notes Unit 6. 6.6 Rational Functions Page 0 of 0 //0