Date Period Unit 9: Rational Functions DAY TOPIC Direct, Inverse and Combined Variation Graphs of Inverse Variation Page 484 In Class 3 Rational Epressions Multipling and Dividing 4 Adding and Subtracting Rational Epressions 5 Simplifing Comple Fractions with Addition and Subtraction 6 7 Partial Fractions 8 Solving Rational Equations 9 Rational Equations Word Problems 0 Review Page of 8
Unit 9 (Rational Functions), Da : Direct Variation Direct variation: A linear function defined b an equation of the form = k., where k 0. Constant of variation: When and are variables, ou can write equates the constant k. k, so the ratio : In general, an equation shows a direct variation if is equal to the product of a nonzero constant and a function of. The following eamples illustrate direct variation 5 5 3 Eamples: For each function, determine whether varies directl with. If so, find the constant of variation... 8 3 5 0 4 7 5 6 3. 4. -6-3 4 - - 3 4 6 7 For each function, determine whether varies directl with. If so, find the constant of variation. 5. 3 = 6. = + 3 7. 8. = Page of 8
Inverse Variation: A function of the form k or k where k 0 Eamples 9. Suppose that and var inversel, and = 3 when = -5. Write the function that models the inverse variation. 0. Suppose that and var inversel, and = 0.3 when =.4. Write the function that models the inverse variation.. Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write a function that models the direct and inverse variations. a. 0.5 6.5 6 8 b. 0. 0.6. 4 c. If varies directl as the square of, and = 50 when =, find when = 6. (using proportions) d. If varies directl as and = 3 when =, find when =. e. The illumination, I, from a light source varies inversel as the square of the distance, d. If the illumination is 0 foot-candles when the distance is m, find the illumination when the distance is 4m. f. If varies inversel as, and = 3 when = 8, what is the value of when = 5? (using proportions) Page 3 of 8
g. The amount of commission is directl proportional to the amount of sales. A realtor received a commission of $8,000 on the sale of a $5,000 house. How much would the commission be on a $30,000 house? h. A train travels at a constant rate. If it travels 00 miles in hours, how far will it travel in 7 hours? Combined variation: combines direct and inverse variations in more complicated relationships Eamples: Combined variation varies directl with the square of varies inversel with the cube of z varies jointl with and z varies jointl with and and inversel with w z varies directl with and inversel with the product of w and Equation form Gmm. Newton s Law of Universal Gravitation is modeled b the formula F. F is the gravitational d force between two objects with masses m and m, and d is the distance between the objects. G is the gravitational constant. Describe Newton s Law as a combined variation. 3. The formula for the area of a trapezoid is A hb ( ) b. Describe this relationship. Closure: Write an equation that shows what is meant for a quantit w to var jointl as and and inversel as z? Page 4 of 8
Unit 9 (Rational Functions), Da : Graphing inverse variation Unlike the graph of direct variation, the graph of inverse variation is not linear. Rather, it is a hperbola- there are alwas two parts to the graph k b c Note that the lines never cross the aes (unless it is a translation)-- the get closer and closer to = 0 and = 0, but and never equal zero. To graph an inverse variation, make a data table and plot points. Then connect the points with a smooth (not straight) curve. There should be two curves -- one in the first quadrant (where both and are positive) and one in the third quadrant (where both and are negative). Horizontal Translations In our own words, describe the shift: Vertical Translations In our own words, describe the shift: Page 5 of 8
Sketch the graph of the following: 3 Remember k b c To do this:. Draw asmptotes. Find our vertices: if k is positive, our graph is in the st and 3 rd quadrant. If k is negative, then the graph is in the nd and 4 th quadrant. 3. The graph intersects = at k, k and k, k, where k is the constant of variation. So, for this problem we have (,) and (-,-) as points. Plot these points starting at the NEW ORIGIN. Sketch the graph of To do this: 3. draw the asmptotes. find our vertices, but remember, this graph is in the nd and 4 th quadrant. So ignore the - and take, but our ordered pair is (-,) and (, -) because this graph MUST BE IN THE nd and 4 th QUADRANT. 3. graph Page 6 of 8
Unit 9 Da 3: Multipling and Dividing Rational Epressions Factor the following:. 3. 4 9 3. 5 6 4. 0 0 A rational epression is in simplest form when its numerator and denominator are polnomials that have no common divisors. Simplest form 3 Not in simplest form ( 3) 3( 3 Eamples:. Simplif and state an restrictions on the variable 0 5 a. 9 0 b. 3 7 4 9 c. 6 3 6 8 d. 6 6 5 6. Multipl and state an restrictions on the variables 7 3 6 a. 4 8 5 b. a a 4 a a a Page 7 of 8
c. 3 5 5 5 3 7 3. Divide and state an restrictions on variables 4 5( 4) a. 3 ( )(7 5) b. 3 6 3 ( )( 5) ( )( 7) Simplif each rational epression. Be sure to state restrictions. 5. 6. 7. 8. Page 8 of 8
Unit 9 Da 4: Adding and Subtracting Rational Epressions Add the following. Simplif where possible.. 3. 4 3. 3 3 3c 5c c c. Eplain the steps ou used to simplif in question. 3. How is adding rational epressions (the eamples in #) similar to adding fractions? 4. Finding the LCM a. Find the least common multiple of 4 36 and 6 36 54 b. Find the LCM of 3 9 30 and 6 30 5. Adding/Subtracting Rational epressions. a. 5 54 33 3 b. 4 48 7 4 c. 5 5 35 Page 9 of 8
Unit 9 Da 5: Simplifing Comple Fractions with + and - Simplif the following. Be sure to state restrictions on the variables... Multipl or divide. Write our answer in simplest form. Be sure to state restrictions on the variables. 3. 4. Comple Fractions: fractions that have a fraction in the numerator, denominator, or BOTH.. Simplif the following: 3 5 4. 3. 3 Page 0 of 8
Unit 9 Da 7: Partial Fractions Resolve: 4 4 ( )( 5) into partial fractions. 4 4 A B ( )( 5) ( ) ( 5) 4 4 A ( 5) B ( ) ( )( 5) ( ) ( 5) ( 5) ( ) 44 A( 5) B( ) A5ABB AB5AB A ( B) 5AB 4 4 A( 5) B( ) ( )( 5) ( )( 5) ( 5)( ) A B 4 solve using sstems 5A B 4 A=7, B=-3 SO, 4 4 7 3 ( )( 5) ( ) ( 5) Tr: (ou should get ( )( ) ( ) ( ) ) Page of 8
Date Period Unit 9, Da 7: Partial Fractions Resolve each of the following rational epressions into partial fractions. 5 5 0.. 6 3 8 3. 54 8 4. 5 6 9 5. 6 6. Page of 8
Unit 9, Da 8: Solving Rational Equations Just as etraneous solutions, solutions that are introduced into a problem that are NOT solutions to the original problem, can be introduced when raising both sides of an equation to a power (unit 7), the can be introduced when ou multipl both sides of an equation b the same algebraic epression. So what does this mean we have to do once we solve a rational equation? Solve and remember to check for etraneous solution.. 5 5. 3 6 9 Since we have a sum/difference in these eamples, we have to find the LCD first. 3. 4 3 4. 5 5. 5 5 3 6 Page 3 of 8
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Unit 9, Da 9: Word Problems. The aerodnamic covering on a biccle increases a cclist s average speed b 0mi/h. The time for a 75-mile trip is reduced b hours. What is the average speed for the trip using the aerodnamic covering? Epress the equation in words: Remember: d=rt Define Without covering Distance (miles) Rate (mile/hour) Time (hours) With covering. Mr. and Mrs. Smith have to paint 6000 square feet of walls in their house (ep, that s right the live in a mansion!). Mr. Smith works twice as fast as Mrs. Smith (not reall but for the sake of the problem we will go with it). Working together, the can complete the job in 5 hours. How long would it take each of them to work alone? Epress the equation in words: Define: Mr. Smith Mrs. Smith Together Time (hours) Rate (square ft/hour) Page 5 of 8
Unit 9, Da 9: Word Problems Page 6 of 8
Unit 9 (Rational Functions), Da 0: Review. Write the function and find the value of z. z varies directl with the square of and inversel with. If z = 9 when = 3 and = 3, find z when = and = 4.. Write an equation to model the data in the table: - - 4-0.5-0.5 Perform each operation and simplif. State an restrictions. 3. 3 3 4 7 4. 4 64 8 83 4 3 5. 77 3 4 6. 7. 8. 4 Resolve into partial fractions. 9 3 9. 0. 5 Solve each equation. Check for etraneous solutions.. 6 6. 6 Word Problems Write and equation and solve each. 3. It takes Patt one hour more to clean the house than Bill if the each work alone. If the work 5 together, it takes them hours to clean the house. How long does it take each of them working 7 alone? 4. The denominator of a fraction is more than the numerator. If the numerator is decreased b 5 and the denominator is decreased b 6, the result is. Find the fraction. 3 Graph each of the following and write the equation of the asmptotes. Show all work. Page 7 of 8
5. 4 3 6. 5 7. 3 5 Page 8 of 8