Math 3 Unit 4: Rational Functions

Transcription

1 Math Unit : Rational Functions Unit Title Standards. Equivalent Rational Expressions A.APR.6. Multiplying and Dividing Rational Expressions A.APR.7. Adding and Subtracting Rational Expressions A.APR.7. Solving Rational Equations A.REI..5 Applied Problems with Rational Equations A.REI..6 Graphing Rational Functions F.IF., F.IF.5, F.IF.7d Unit Review Additional Clovis Unified Resources Clovis Unified is dedicated to helping you be successful in Math. On the website above you will find videos from Clovis Unified teachers on lessons, homework, and reviews. Digital copies of the worksheets, as well as hyperlinks to the videos listed on the back are also available at this site.

2 Math Unit : Online Resources. Equivalent Rational Expressions. Multiplying and Dividing Rational Expressions. Adding and Subtracting Rational Expressions. Solving Rational Equations.5 Applied Problems with Rational Equations.6 Graphing Rational Functions Patrick JMT: Rational Expressions - Writing in Lowest Terms Ex Patrick JMT: Rational Expressions - Writing in Lowest Terms Ex Purple Math: Rational Expressions: More Simplifying or Khan Academy: Intro to Rational Expression Simplification Khan Academy: Multiplying & Dividing Rational Expressions: Monomials Khan Academy: Multiplying Rational Expressions Khan Academy: Dividing Rational Expressions Patrick JMT: Rational Expressions - Multiplying and Dividing or or Purple Math: Dividing Rational Expressions Khan Academy: Adding & Subtracting Rational Expressions Like Denominators Khan Academy: Adding & Subtracting Rational Expressions Unlike Denominators Patrick JMT: Rational Expressions - Adding and Subtracting. Ex Patrick JMT: Rational Expressions - Adding and Subtracting. Ex Purple Math: Adding and Subtracting Rational Expressions Patrick JMT: Solving Rational Equations Purple Math: Solving Rational Expressions Khan Academy: Solving Equation with Rational Expressions Khan Academy: Solving Equation with Rational Expressions Extraneous Solutions Patrick JMT: Rational Expressions - Solving Rational Expressions No Solution Khan Academy: Rational Equations Word Problem: Combined Rates or or Purple Math: "Work" Word Problems Patrick JMT: Graphing Some Basic Rational Functions Algebra dotcom: Graphing Rational Functions

3 Unit Worksheet Name: Equivalent Rational Expressions Date: Per: [-7] Find an equivalent rational expression in lowest terms, and identify the value(s) of the variable that must be excluded.. 6nn 0nn. xx yy yy xx. 0aa5 bb cc aa bb cc 9. dddd+dddd dddd 5. xx 9 xx xx 6. nn 5nn nn 7. ff(xx) [ff(xx)] if ff(xx) = xx 8. gg (nn) if ff(nn) = nn and gg(nn) = nn 8nn ff 9. aa aa aa 6aa+9 0. yy yy. xx yy yy 6xx. ff gg (aa) if ff(aa) = aa 9 and gg(aa) = 7 + aa. ff(xx) [gg(xx)] if ff(xx) = 9 xx and gg(xx) = xx Math Unit Worksheet

4 . xx 5xx+6 8 xx xx 5. aa aa 8aa aa+ 6. yy yy+5yy 5 yy xx 5 xx 5 8. Write a rational expression with denominator 6bb that is equivalent to aa a) b) one-half of aa bb bb c) 9. Remember that algebra is just another way to perform arithmetic, but with variables replacing numbers. a) Simplify the following rational expression: xx yy (xxxx) zz (xxyy ). yyyy b) Simplify the following rational expression without using a calculator: c) How are the calculations in parts (a) and (b) similar? How are they different? Which expression was easier to simplify? Math Unit Worksheet

5 Unit Worksheet Name: Multiplying and Dividing Rational Expressions Date: Per: [-] Multiply or divide the following and then simplify the result.. 5xx yy yy 5xx 8xx. 6xx yy yy. 7xx 6yy yy xx. aa bb bbbb 9bb cc 6aadd 5. xx+ xx+ xx xx 6. xx+8 xx 5 xx 5 5xx+0 7. xx 6xx 6 xx +xx xx +9xx+ xx 8xx+5 8. xx 8xx+ xx 6 xx+6 xx xx+ Math Unit Worksheet

6 9. xx 9xx 0 xx +xx 6 xx xx 0. xx +6xx 6xx +5xx xx 6 xx +7xx+5. xx +xx 6 xx xx xx xx 8 xx xx. 9 xx xx +6xx+9 xx 9 xx+9. xx xx xx +5xx xx +xx 7 xx +xx. xx +xx+8 xx 50 xx +6xx+5 xx +xx Math Unit Worksheet

7 5. Suppose that x = tt +tt tt and y = tt +tt 8 tt, for tt, tt, tt, and tt. Show that the tt value of xx yy does not depend on the value of tt. 6. Determine which of the following numbers is larger without using a calculator, or 0. (Hint: We can compare two positive quantities aa and bb by computing the quotient aa. If aa >, then aa > bb. bb bb Likewise, if 0 < aa <, then aa < bb.) bb 7. One of two numbers can be represented by the rational expression xx, where xx 0 and xx. xx a. Find a representation of the second number if the product of the two numbers is. b. Find a representation of the second number if the product of the two numbers is 0. Math Unit Worksheet

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9 Unit Worksheet Name: Adding and Subtracting Rational Expressions Date: Per: [-5] Perform the indicated operation. Write your final answer in simplest form xx xx xx yy 7xx yy. xx + xx+5 xx+ xx+. xx +5xx+6 xx+ xx xx+ 5. xx + 9 xx 6 6 xx 6. xx xx+ xx xx 7. xx xx 9 + xx+5 9 xx 8. 5 xx + xx 9. 6xxxx 5xx xx+ xx. 8 xx+ xx. 5 xx + xx Math Unit Worksheet

10 . + xx xx xx +xx+. + xx 9 xx xx+ 5. xx +xx 5 xx +xx 0 [6-] Simplify each rational expression completely. + yy xx + yy 6. xx+ xx 7. yy xx yy 8. 7xx xx 7 9. mm+ 7 mm 9 0. xx yy xx + yy xx. 9 xx 6 xx. Place one or more expressions from the EXPRESSIONS OPTIONS in each box to create an equation that is true for all values of x. (Assume no denominator equals zero) = x+ x x EXPRESSIONS OPTIONS x x x (x + ) Math Unit Worksheet

11 . Place one or more expressions from the EXPRESSIONS OPTIONS in each box to create an equation that is true for all values of x. (Assume no denominator equals zero) = x x x EXPRESSIONS OPTIONS x x (x - ) [-9] Perform the indicated operation. Write your final answer in simplest form.. 0 xx +5xx xx xx 8 xx 6. 8xx 5 xx xx 9 xx 5xx 8. xx 8xx 0 5xx xx 7xx+ 9xx 9. 5xx 5xx 8 5xx +7xx+ Math Unit Worksheet

12 EXTENSION: 0. Suppose that xx 0 and yy 0. We know from our work in this section that is equivalent to. Is it xx yy xxxx also true that + xx yy is equivalent to? Provide evidence to support your answer. xx+yy. Suppose that x = tt tt +tt and y = +tt. Show that the value of xx + yy does not depend on the value of tt.. Show that for any real numbers aa and bb, and any integers xx and yy so that xx 0, yy 0, xx yy, and xx yy, yy xx xx + bbbb aaaa yy xx + yy aaaa bbbb = (aa bb). xx yy. Suppose that nn is a positive integer. a. Simplify the expression + +. nn nn+ b. Simplify the expression nn nn+ nn+ c. Simplify the expression nn nn+ nn+ nn+ d. If this pattern continues, what is the product of nn of these factors? Math Unit Worksheet

13 Unit Worksheet Name: Solving Rational Equations Date: Per: [-8] Solve the following equations and then state whether each equation has no real solution, one real solution, two real solutions, or infinitely many solutions.. = 6 xx+ xx. xx = xx 5 xx+7 xx. ff(xx) = [ff(xx)] 8 ff(xx) if ff(xx) = xx. = 6 xx+ 9xx+ 5. xx = xx xx+ 6. xx xx+ = xx 7. = xx+ xx = 8 xx xx xx xx [9-7] Solve the following equations and check for extraneous solutions. 9. xx+ + xx+ = xx 0. xx xx+ + xx +5xx+6 = 5 xx+ Math Unit Worksheet

14 . = ff(xx)+ if ff(xx) = xx. x+ = + ff(xx) ff(xx) xx+ xx. ff(xx)+ ff(xx) = xx ff(xx) ff(xx)+ ff(xx) if ff(xx) = xx. 5xx = xx+ xx xx 5. x + = xx xx+ xx 6. = xx xx xx 7. x = xx xx x xx 6xx+8 Math Unit Worksheet

15 8. Given ff(xx) = 5 xx+ and gg(xx) =. Solve the equation ff(xx) = gg(xx) algebraically. xx 9. Given ff(xx) = 7 xx 5 and gg(xx) =. Solve the equation ff(xx) gg(xx) = 0 algebraically. xx 0. Given ff(xx) = xx xx+ 8 and gg(xx) =. Solve the equation (ff gg)(xx) = 0 algebraically. xx+6. Given ff(xx) = xx+ xx 6 and gg(xx) = xx. Solve the equation ff(xx) = gg(xx) algebraically.. Create and solve a rational equation that has 0 as an extraneous solution.. Create and solve a rational equation that has as an extraneous solution. Math Unit Worksheet

16 EXTENSION:. Two lengths aa and bb, where aa > bb, are in golden ratio if the ratio of aa + bb is to aa is the same as aa is to bb. Symbolically, this is expressed as aa bb = aa+bb aa. We denote this common ratio by the Greek letter phi (pronounced fee ) with symbol φφ, so that if aa and bb are in common ratio, then φφ = aa bb = aa+bb φφ is the positive number that satisfies the equation φφ = φφ+ value for φφ. φφ aa. By setting bb =, we find that φφ = aa and. Solve this equation to find the numerical. Remember that if we use xx to represent an integer, then the next integer can be represented by xx +. a. Does there exist a pair of consecutive integers whose reciprocals sum to 5? Explain how you know. 6 b. Does there exist a pair of consecutive integers whose reciprocals sum to? Explain how you know. c. Does there exist a pair of consecutive even integers whose reciprocals sum to? Explain how you know. d. Does there exist a pair of consecutive even integers whose reciprocals sum to 5? Explain how you 6 know. Math Unit Worksheet

17 Unit Worksheet 5 Name: Applied Problems with Rational Equations Date: Per:. Aubrey can wash all the windows of a retail store in 6 hours. Maxwell can wash all the windows of the same retail store in 9 hours. a) Write an equation that can be used to find the time t, in hours, it would take Aubrey and Maxwell to wash all the windows of the retail store together. b) Solve the equation for t that you wrote in part a). Der can wash all the dishes in the house in 0 minutes. Her brother Yashua can wash all the dishes in the house in 0 minutes. a) Write an equation that can be used to find the time t, in minutes, it would take Der and Yashua to wash all dishes in the house together. b) Solve the equation for t that you wrote in part a). A group of friends decide to evenly divide the $7 cost of watching a premiere boxing match on pay-per-view TV. Initially, there are x friends, but then friends decide not to watch the boxing match and spend their money to see, Star Wars: Episode : The Neverending Force, at the movie theatre instead. This causes each remaining friend to have to each pay $ more. a) Create an equation that represents the situation and can be used to solve for x. b) Use the equation you created in part a) to solve for x, the initial number of friends.. A group of college students rent a large house to live in and agree to evenly divide the $00 monthly rent. Initially, there are n college students, but then 5 additional college students decide to join the group, causing each college student to pay $0 less per month. a) Create an equation that represents the situation and can be used to solve for n. b) Use the equation you created in part a) to solve for n the initial number of college students. Math Unit Worksheet 5

18 5. The density, D, of an object is defined as DD = MM VV volume of the object. Solve this equation for V. where M represents the mass of the object, and V represents the 6. Suppose a basketball player has a shooting percentage, P, that is inversely proportional to the square root of her distance from the basket, d according to the equation PP = kk Solve this equation for d. dd where k is the proportionality constant. 7. Joule s Law states that PP = EE where P is the power in watts, E is the voltage in volts, and R is the resistance in ohms. Solve this equation for E. RR 8. Newton s law of universal gravitation, FF = GGmm mm, measures the force of gravity between two masses mm rr and mm, where rr is the distance between the centers of the masses, and GG is universal gravitational constant. Solve this equation for GG. 9. Consider the rational equation = +. Solve this equation for RR and simplify. RR xx yy 0. Consider an ecosystem of rabbits in a park that starts with 0 rabbits and can sustain up to 60 rabbits. An equation that roughly models this scenario is PP(tt) = 60, where PP(tt) represents the rabbit population in year tt of the study. + 5 tt+ a. What is the rabbit population in year 0? Round your answer to the nearest whole rabbit. b. Solve this equation for tt. Describe what this equation represents in the context of this problem. c. At what time does the population reach 50 rabbits? Math Unit Worksheet 5

19 Unit Worksheet 6 Name: Graphing Rational Functions Date: Per:. ff(xx) = xx+ a) Sketch the graph of ff on the xxxx-plane given. 5 y b) Find the yy-intercept of ff. c) Find the xx-intercept of ff. x d) State the domain of ff. e) State the range of ff. 5 5 f) State the equation(s) of any vertical asymptotes of ff. g) State the equation(s) of any horizontal asymptotes of ff. 5 h) State the interval(s) on which the graph of ff is increasing? i) State the interval(s) on which the graph of ff is greater than zero?. gg(xx) = xx a) Sketch the graph of gg on the xxxx-plane given. 5 y b) Find the yy-intercept of gg. c) Find the xx-intercept of gg. 5 5 x d) State the domain of gg. e) State the range of gg. f) State the equation(s) of any vertical asymptotes of gg. 5 g) State the equation(s) of any horizontal asymptotes of gg. h) State the interval(s) on which the graph of gg is decreasing? i) State the interval(s) on which the graph of gg is greater than zero? Math Unit Worksheet 6

20 . Let h(xx) = gg ff(xx) with gg(xx) = + and ff(xx) = xx a) Find h(xx) xx b) Sketch the graph of h(xx) on the xxxx-plane given. y c) Find the yy-intercept of h. 0 8 d) Find the xx-intercept of h. 6 e) State the domain of h. f) State the range of h. x g) State the equation(s) of any vertical asymptotes of h. 6 h) State the equation(s) of any horizontal asymptotes of h. i) State the interval(s) on which the graph of h is increasing? j) State the interval(s) on which the graph of h is less than zero?. Let jj(xx) = ff gg(xx) with ff(xx) = and gg(xx) = xx + xx a) Find jj(xx) b) Sketch the graph of j on the xy-plane given. c) Find the y- intercept of jj. 5 y d) Find the x- intercept of jj. x e) State the domain of jj. f) State the range of jj. 5 5 g) State the equation(s) of any vertical asymptotes of jj. h) State the equation(s) of any horizontal asymptotes of jj. 5 i) State the interval(s) on which the graph of jj is decreasing? j) State the interval(s) on which the graph of jj is less than zero? Math Unit Worksheet 6

21 For problems 5-8, state the equations of any horizontal and vertical asymptotes of each function. 5. ff(xx) = xx 6. gg(xx) = xx h(xx) = + 8. jj(xx) = xx+ xx 7 9. Given ff(xx) = and gg(xx) = 6 xx a) Solve the equation ff(xx) = gg(xx) by b) Solve the equation ff(xx) = gg(xx) algebraically. Graphing ff(xx) and gg(xx) on the xxxx-plane below. 5 y 5 5 x c) How are the intersection points on the graph related to the algebraic solutions? 5 0. Given ff(xx) = xx and gg(xx) = a) Solve the equation (ff gg)(xx) = 0 by b) Solve the equation (ff gg)(xx) = 0 algebraically. graphing ff(xx) and gg(xx) on the xxxx-plane below. 5 y 5 5 x c) Explain the relationship between solving an equation graphically and solving algebraically. 5 Math Unit Worksheet 6

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23 Math Unit Review Worksheet Name: Rational Expressions and Functions Date: Per: Simplify and state any excluded values.. xx 8 xx. xx 5xx xx 7xx+0. xx xx +6xx +8xx. xx +xx xx xx +5xx xx Multiply or divide and simplify completely. 5. xx yy 9yy xx 6. 7xx yyzz 9xx 8yy zz 7. xx xx 0 xx xx+ xx 8 xx 0 8. xx 8 xx xx 0xx 0 xx +xx 6 9. xx xx+ xx xx 6 0. xx +0xx+75 xx 5 xx xx 5. xx+ 0xx +xx 5xx xx 6xx +xx. xx 6xx +8xx xx 8xx+6 xx 0xx 0xx Math Unit Review Worksheet

24 Add or subtract and simplify completely.. 5xx 7 + xx +5xx xx +5xx. 5ff(xx) 5 ff(xx) ff(xx) 5 ff(xx) if ff(xx) = xx aa 6aa nn 6nn 7. 5xx + 0 xx xx 8. xx xx xx xx 9. 5 xx+ + 7 xx 0. 5 ff(xx) ff(xx)+ if ff(xx) = xx +. xx + xx+6 xx+. + xx xx xx +xx+. xx xx xx 9. Place one or more expressions from the EXPRESSIONS OPTIONS in each box to create an equation that is true for all values of x. (Assume no denominator equals zero). EXPRESSIONS OPTIONS = x x x x x x ( x ) Math Unit Review Worksheet

25 Simplify each complex fraction. 5. xx xx + xx 6. xx xx 7. + xx 9 xx Solve for x. Then state whether the equation has no real solutions, one real solution, two real solutions, or infinitely many real solutions = 5 xx xx 9. xx = xx xx+ 0. = xx xx. xx+ 6 = xx+ Solve for x. Check for extraneous solutions.. 9 = ff(xx) 6 ff(xx) ff(xx) ff(xx) if ff(xx) = xx = 0 xx xx. + = xx xx+ xx +xx 5. xx + xx xx = xx Math Unit Review Worksheet

26 6. xx + xx 5 = xx xx 7. = xx 8xx+ xx + xx xx 6 8. Given ff(xx) = 6 xx 5 and gg(xx) =. Solve the equation (ff gg)(xx) = 0 algebraically. xx 9. A group of students decides to purchase a billboard for their school that costs $0 and to divide the cost evenly among the students. Initially there are x students, but then nine more students decide to join the group, causing each student to pay $ less. a) Create an equation that represents the situation and can be used to solve for xx, the initial number of students. b) Use the equation you created in part a) to solve for xx, the initial number of students. 0. Chet can change all the tires on a truck in 0 minutes. Destiny can change all the tires on the same truck in 5 minutes. a) Write an equation that can be used to find the time tt, in minutes, it would take Chet and Destiny to change all the tire on the truck if they work together. b) Solve the equation for tt that you wrote in part a) Math Unit Review Worksheet

27 For problems, do all of the following: a) Sketch the graph of ff. g) State the equation(s) of any horizontal asymptotes of ff. b) Find the y- intercept of ff. h) State the interval(s) on which the graph of ff is increasing? c) Find the x- intercept of ff. i) State the interval(s) on which the graph of ff is decreasing? d) State the domain of ff. j) State the interval(s) on which the graph of f is greater than zero? e) State the range of ff. k) State the interval(s) on which the graph of ff is less than zero f) State the equation(s) of any vertical asymptotes of ff.. ff(xx) = xx+ 5 y. ff(xx) = +. ff(xx) = xx xx+ 5 y 5 y x x x Math Unit Review Worksheet

28 For problems and 5: If ff(xx) = (gg h)(xx), then a) find ff(xx) and b) state the equations for any horizontal and vertical asymptotes of ff(xx).. gg(xx) = 5 & h(xx) = xx 5. gg(xx) = + & h(xx) = xx + xx xx 6. Given ff(xx) = 8 xx and gg(xx) =, a) Solve the equation ff(xx) = gg(xx) by graphing ff(xx) and gg(xx) on the same xxxx-plane. 5 y x b) Solve the equation ff(xx) = gg(xx) algebraically c) How are the intersection point(s) on the graph related to the algebraic solution(s)? Math Unit Review Worksheet