UNIT 2: RATIONAL EXPRESSIONS

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1 INTRODUCTION UNIT 2: RATIONAL EXPRESSIONS In this unit you will learn how to do arithmetic operations with rational expressions. You will also learn how to graph rational functions, as well as solve rational equations. RATIONAL FUNCTION A rational function has the form polynomials and 0. where and are There are five things you have learned how to do with fractions: simplify, add, subtract, multiply, and divide. You will do these same things with rational expressions. You will learn that rational functions look different from other functions you have worked with. Here are a couple of graphs of rational functions: You will notice that these functions seem to avoid some vertical (and possibly horizontal) walls. These are asymptotes. Asymptotes are points of infinite discontinuity. A hole is also a point of discontinuity. A hole exists on the graph of a rational function at any input value that causes both the numerator and denominator of the function to be equal to zero.

2 Section A.APR.7 Simplify rational expressions. Common Core Standard(s) A.APR.7 Multiply and divide rational expressions. A.APR.7 Add and subtract rational expressions. F.IF.7.d Graph rational functions, identifying zeros and asymptotes, when suitable factorizations are available, and showing end behavior. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

3 2.1 SIMPLIFY RATIONAL EXPRESSIONS Objectives 1) Students will be able to define a rational expression. 2) Students will be able to simplify rational expressions. Note: We are working with rational expressions, namely, fractions. Remember that we cannot divide by 0. Therefore, there will be restrictions to our answers there are certain values where the expression does NOT exist. For example, has a restriction at -5. That is, 5. Can you explain why? In this section we will be adding, subtracting, multiplying, and dividing rational expressions. The good news is, you have already learned the processes! Now we are just applying it to good old algebra. Compare the two problems below. Arithmetic Algebraic Simplify 6 8 Simplify Rewrite as factors: Cancel out common factors Rewrite as factors: Cancel out common factors The restrictions are 4, 2

4 Your turn: Simplify. Identify any restrictions Practice: Set A Simplify each rational expression. Write the restrictions for each answer #% 7. $ & 10. % #% 13. # ! " 5. $ $" & ## 17. " & 14. #!!$ $ & #!! 3. #$" ##% $ # #%& 12. # # #!!

5 2.2 MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS Compare the two problems below. Objectives 1) Students will learn how to multiply rational expressions. 2) Students will learn how to divide rational expressions. 3) Students will be able to determine whether rational expressions are closed under multiplication and division. Multiplying Rational Expressions Arithmetic Algebraic Multiply Multiply Multiply straight across Multiply straight across Factor when possible Cancel out any common factors. ; 3, 6 $ Your turn: Multiply. Identify any restrictions. 2 There are restrictions! Since the denominator cannot equal 0, x cannot equal -3 nor

6 Compare the two division problems below. Dividing Rational Expressions Arithmetic Algebraic Divide Divide Rewrite as a multiplication problem, flip the second fraction. 2 3 * Rewrite as a multiplication problem, flip the second fraction ,+*, There are restrictions! Since the denominator cannot equal 0, x cannot equal -3 nor -4. This can be written as 3, Your turn: Divide

7 Practice: Set B Find the product. If possible, simplify. Write the restrictions, if there are any. 1. &- $- / #! 2. # $!! #%# 4.! #$#! Practice: Set C Find the quotient. If possible, simplify. Write the restrictions, if there are any. 1.!- $ #! $! # 2.!! #! #! 4.

8 Simplifying Rational Expressions Multiple Operations Note: When working with several operations at once, the easiest approach is to get rid of any division by rewriting it as a multiplication problem. Example Rewrite as multiplication. (Flip the second fraction.) 1 4,+2 6+8, 3 +3 Factor any trinomials. Multiply straight across Cancel out any common factors. Write your final answer Are there any restrictions? Practice: Set D Perform the indicated operations. If possible, simplify. Determine the restrictions, if there are any. 1. # #%! #! 2. $ #! & #& # ### 3. Are rational expressions closed under multiplication and division? Explain.

9 2.3 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Objectives 1) Students will learn how to add rational expressions. 2) Students will learn how to subtract rational expressions. Again, the process for addition and subtraction will be the same as the arithmetic method. Compare the two problems given. Compare the two problems below. Adding Rational Expressions Add Arithmetic Add Algebraic Multiply the top and bottom of each fraction by the denominator of the other fraction. Multiply Multiply the top and bottom of each fraction by the denominator of the other fraction. Multiply ,++, ,+4, Add Add Simplify There are restrictions! 4 and 5.

10 Compare the two problems given. Subtracting Rational Expressions Arithmetic Algebraic Subtract. Subtract Multiply the top and bottom of each fraction by the denominator of the opposite fraction. Combine. 3 2 * * Factor the denominator of the first fraction Notice that the first and second fraction both have +2 in the numerator. To get the second fraction to have the same denominator as the first, multiply the top and bottom by , 6, There are restrictions: 2, and 2.

Find the sum. 2+4 +1 +3 2 3 Your turn Find the difference. 2+4 +1 3 2 3 Practice: Set E Find the sum. If possible, simplify. Determine the restrictions. 1. $ + &!&!& 3. $ # + & 5. $ +! 2. + # #% 4.

11 Find the sum Your turn Find the difference Practice: Set E Find the sum. If possible, simplify. Determine the restrictions. 1. $ + &!&!& 3. $ # + & 5. $ +! 2. + # #% 4.! $ + 6. $ + # " Find the difference. If possible, simplify. Determine the restrictions. 7. " ## ## " & & $ $ $ Simplify and determine the restrictions. 8. # $ # $ $ $! #%

12 2.4 GRAPHING RATIONAL FUNCTIONS Objectives 1) Students will learn how to identify the intercepts, asymptotes, and end behavior of a rational function. 2) Students will learn how to sketch the graph of a rational function. 3) Students will be able to interpret rational function graphs from real world situations. The following key features will help us graph rational functions: x-intercepts: These can be found by setting the numerator equal to 0 and solve for x. y-intercept: This can be found by setting all of the values equal to 0 and solve for y. Vertical asymptotes: The graph has a vertical asymptote at each real zero of the denominator (restrictions) unless the numerator and the denominator share a factor in which it would have a hole. Horizontal asymptotes: Horizontal asymptotes can be crossed: their purpose is to show end behavior. There are three rules to help you determine the horizontal asymptote: Rule 1: If the degree of the numerator is less than the degree of the denominator, then there is a horizontal asymptote at 70 (the x-axis). Rule 2: If the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote. There is an oblique (slanted) asymptote. Rule 3: If the degree of the numerator is the same as the degree of the denominator, then the horizontal asymptote is the fraction of the leading coefficients of the polynomials. For example, $! has a horizontal asymptote at $.

13 Example Sketch a graph of! without using technology. Factor the denominator and rewrite the expression Your turn Sketch the graph of the function without using technology Find the x-intercept by setting the numerator equal to 0: Find the y-intercept by substituting 0 for and solve for y: Find the vertical asymptotes by figuring out the restrictions: 3, and 2. The vertical asymptotes are 3, 2 Find the horizontal asymptote by using the rules given. The degree of the numerator is smaller than the denominator: rule 1 applies. There is a horizontal asymptote at 7 0. Sketch the graph.

14 Example Sketch a graph of without using technology. First, see if we can simplify the problem by factoring. We find the vertical asymptotes by setting +1 0 and 2 0 so 1,2 After factoring the rational equation, we find that the numerator and denominator have (x+1) in common, so we can factor it out. Remember that we cannot have a zero in the denominator, so +1/+1 will leave a hole in our graph. That is, 1 The denominator is greater than the numerator, so as the numbers get very large, our graph will approach the x-axis. Notice the hole at 1 Example Sketch a graph of without using technology Notice we cannot factor the numerator. (The x-intercepts are imaginary!) Using the denominator, we know that there is a vertical asymptote at 2. Use long division to divide. You will get 4 (ignore the remainder). This line is the oblique asymptote. Create a sign array to see where the function is positive or negative. (Use the intervals of the x-intercepts and vertical asymptotes) Sketch.

15 Practice: Set F Sketch the graph of each function without using technology. Then use technology to check your work. 1. #! 3. $ 2. 4.! 5. # 7. $% $ $ 6. #! 8. " $# 9. & "#%

16 2.5 SOLVING RATIONAL EQUATIONS Objectives 1) Students will be able to determine the restrictions of a rational equation. 2) Students will learn what extraneous solutions are, and how to determine if an equation has one. 3) Students will be able to solve rational equations. In this section, you will be combining all of your skills you have been working on in sections You may need to go back and review these sections if you are struggling. Up to this point, we have been simplifying rational expressions. We have not solved anything yet! It s time to solve rational equations. When it comes to solving rational equations, you will have to check for extraneous solutions. These are extra answers that do not work in the original equation. Checking answers is not an option but a necessity!

17 Example: Example: First off, write down any restrictions. 0. In order to solve this example, we need the same denominator. Then solve for x ,, This is a true statement. 8 is a solution ,+3, , 3, 3,+3, , 1 Check for extraneous solutions: In the original equation, 1, and 1. In our answers, 1 is an extraneous solution. Therefore, our final answer is 3.

18 Practice: Set G Solve each equation. Make sure to check for extraneous solutions. 1. ## $ # $ 2. # $ " 3. & " " 4. + # # 5. $ & $! 6. 1! $ $ #! " $ "# $ 7+ #% # "

19 Work Problems Now, I m sure you are sitting there after having fun with all these fractions asking why? How is this applicable to math? Well, one of many applications is work. Well, work rates. Example Pat can put up a sheet of plywood over the hole in the bunker in 2 minutes. Chris can do that same job in 3 minutes. If they work together, how long would it take? If you automatically thought, well, 2.5 minutes, of course! Think again. Does that make sense? It would take Pat even longer when someone is helping her? Your turn Jack doesn t eat pizza very often because he hates feeling all greasy. When he does eat pizza, he can finish ten 16 pizzas in 2 hours. Phoenix can finish six 16 pizzas in 4 hours. How long would it take them to eat one 16 pizza if they worked together? Write it as a rate: Pat # MNOOP QRSTPOM Chris # MNOOP $ QRSTPOM We want to know how long it would take them to do 1 sheet if they worked together. That is, 1 sheet 2 minutes + 1 sheet 3 minutes 1 sheet x minutes At this point, this shouldn t make you break a sweat. You are solving: You can cross-multiply to solve: So if Pat and Chris work together, it will take them! minutes. Which is 1.2 minutes. Or 1 minute and 12 seconds.

20 Practice: Set H Solve each equation. Make sure to check for extraneous solutions. 1. Moe and Curly have been hired to paint the interior of a school during summer break. Moe can paint the entire school in 80 hours by himself. Working together, it takes Moe and Curly 50 hours to paint the entire school. Write and solve a rational equation to determine how long it would take Curly to paint the interior of the school by himself. Show your work. 2. It takes Jane 4 hours to mow a field. June can mow the same field in 2 hours. How long will it take them to mow the field if they work together? 3. So far in your volleyball match, you have put into play 37 of the 44 serves you have attempted. Solve the equation &% $" to find the number of consecutive serves you need #%% to put into play in order to raise your service percentage to 90%. 4. A speed skater travels 9 kilometers in the same amount of time that it takes a second skater to travel 8 kilometers. The first skater travels 4.38 kilometers per hour faster than the second skater. a) Use the verbal model below to write an equation that relates the skating times of the skaters. `abcdefg hij bkdcgj # bkdcgj # bgg` labcdefg hij bkdcgj mkdcgj bgg` b) Solve the equation in part (a) to find the speeds of both skaters.

21 2.6 SOLVING RADICAL EQUATIONS Objectives 1) Students will learn how to solve radical equations. 2) Students will learn how to determine if the solution of a radical equation is extraneous. Note: There will be extraneous solutions extra answers that are not true for the original equation. Again, checking answers is not optional, but necessary! You need to figure out which answers are not true. Solving radical equations 1) Isolate the radical on one side of the equation. 2) Get rid of the radical (you get rid of a square root by squaring it, a cubed root by cubing it, etc.) 3) Check each answer in the original equation. Example Solve Your turn Solve o Check. Is it extraneous? Practice: Set I Solve each radical equation. Check for extraneous solutions / /

22 Solving Radical Equations: Rational Exponents Note: If you have a rational exponent, such as, do you know how to eliminate the exponent? Hint: a fraction multiplied by its reciprocal is 1. Example: +2 $ $ $ Your turn +3 $ 35 Check! 62+2 $ 160 This is a true statement. 62 is a solution. Practice: Set J Solve each radical equation. Check for extraneous solutions o o3 $

Domain: The domain of f is all real numbers except those values for which Q(x) =0.

Domain: The domain of f is all real numbers except those values for which Q(x) =0. Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such