Page Review of Eponent Rules Math : Unit Radical and Rational Functions Rule : Multipling Powers With the Same Base Multipl Coefficients, Add Eponents. h h h. ( )( ). (6 )(6 ). (m n )(m n ). ( 8ab)( a b ) Rule : Dividing Powers With the Same Base Divide/Simplif Coefficients, Subtract Eponents. 6. 6 7. 8. 9. 8m n m n 7 9 6 0. 6 6 Rule : Zero Power Propert A number raised to a power of zero is (eception is zero).. p. p. 0 7. b 7 0 a. 0 Rule : Negative Eponents Move the base with the negative eponent across the fraction bar. 6. 0 7. 8. 9. r q 0. a b c 7 a b c 9 8 z. 0 z 7 z. 8 0 z 6. ( )( ). ( )( ). 6m n q 6m n q Rule : Raising a Power to a Power Multipl eponents. Simplif. 6. ( 6 ) 7. (n ) 8. ( ) 9. (a - ) 7 0. T (T 7 ) - Rule 6: Raising a Product or a Quotient to a Power Multipl each inside eponent b the outside eponent. Simplif.. (d ). ( ). (g ) -. (-pq r ). (-a b) (a b 6 ) 6. 7 7. n n n 8. 6 ab a b 9. a b a b 0. 9 76 7 00 0
Page Simplifing Radicals & Basic Operations We are familiar with taking square roots ( ) or with taking cubed roots ( ), but ou ma not be as familiar with the elements of a radical. An inde in a radical tells ou how man times ou have to multipl the root times itself to get the radicand. For eample, in 8 = 9, 8 is the radicand, 9 is the root, and the inde is because ou have to multipl the root b itself twice to get the radicand (9 9 = 9 = 8). When a radical is written without an inde, there is an understood inde of. 6 =? =? Radicand: Inde: Radicand: Inde: Root is because = = 6 Root is because = = To use our calculator: An inde of : Step : Press Step : Tpe in radicand. An inde of : Step : Press Step : Choose. Step : Tpe in the radicand. An inde: Step : Tpe in the inde. Step : Press Step : Choose : Step : Tpe in the radicand. You Tr: = 96m n 8 = v 8 = BUT not ever problem will work out that nicel! Use our calculator to find an eact answer for = The calculator will give us an estimation, but we can t write down an irrational number like this eactl it can t be written as a fraction and the decimal never repeats or terminates. The best we can do for an eact answer is use simplest radical form. Here are some eamples of how to write these in simplest radical form. See if ou can come up with a method for doing this. Compare our method with our neighbor s and be prepared to share it with the class. (Hint: do ou remember how to make a factor tree?) = = 8 =
Page Simplifing Radicals: ) ) ) Eamples:. 6. 8.. 8. 80n 6. 96 7. 8 8. 86 9. 0 0. 8. 6. 6. 8 z. 9 7 z. 87 z Simplifing Radicals Homework Simplif each epression... 000. 6.. 8n 8 6. 98k 7. r 7 8. m 9. 9 0.. 0. 6a b 8. 8 7 7. 6 6. 8 7 7 6. 6
Page Rational Eponents & Radicals Simplifing Eponents: Evaluate each epression. 0 Where would Enter, and into the calculator. Enter into the calculator. fall in this list? Without using calculator predict the value of. Bases with fraction eponents can be written as radicals p n b n b p Radical epressions can be written with bases with fractions for eponents n p b b p n We have alread done this when simplifing radicals: because and because Even if a fraction cannot reduce evenl, we can still write radicals using fraction eponents. Write using rational (fraction) eponents: ab t Write each in Radical Form..).).) ( ).)..8 ( ).) 6.)
Page You tr: Rewrite each of the following epressions in radical form... ( 7). (6 ). 9 8. a 6. 7 7. 8.. Now, reverse the rule ou developed to change radical epressions into rational epressions... 7. 6. 9. 7 6. 7 Rational Eponents and Radicals Homework: Write each epression in radical form.. 7... 7. 6 6. 6 7. () 8. () 9. (0n) 0. a 6. (6v).. m Write each epression in eponential form.. ( 0) 6.. ( ) 6. ( ) 7. 6 8. 0 9. ( m ) 0. ( 6 ). v. 6p. ( a). ( k)
Page 6 Graphing Square Root Functions Make a table for each function. f() = f() = 0 6 7 8 9 f() 0 6 7 8 9 f() Ignore the points with decimals. What do ou notice about the other points? These functions are of each other. B definition, this means the and the. Plot the points from the tables above. As a result, the graphs have the same numbers in their points but the and the coordinates have. This causes the graphs to have the the line. but to be over The Square Root Function Reflect the function f() = over the line =. Problems? We have to define the Square Root as. This means that we will onl use the side of the graph.
Page 7 The result: f() = Characteristics of the graph Verte End Behavior Domain Range Smmetr Pattern Transforming the Graphs Now that we know the shapes we can use what we know about transformations to put that shape on the coordinate plane. Remember: Translate Reflect Vertical Change Domain: Range: ) f() = Transformations: ) f() = + Transformations: ) f() = Transformations: Domain: Range: As : Domain: Range: As : Domain: Range: As :
Page 8 ) f() = Transformations: ) f() = + Transformations: 6) f() = Transformations: Domain: Range: As : Domain: Range: As : Domain: Range: As : (HONORS) Sometimes the functions are not in graphing form. We ma have to use some of our algebra skills to transform the equations into something we can use. E: f() = This is not in graphing form. Transformations: Domain: Range: As : E: f() = 9 + 6 form. This is not in graphing Transformations: Domain: Range: As :
Page 9 Classwork: Graph each function. Then state the domain and range.. f() = +. f() = +. f() = ( ) +. f() = + + Write an equation for each graph. Then state the domain and range. Describe the transformations in the graphs of each equation. Then state the domain and range. 8. f ( ) 9. f( ) 0. ( ) f. f ( )
Page 0 Homework. Graph each. Write the Domain and Range.... ) ) ) ) D: D: D: R: R: R:.. 6. ) 6) 7) 8) D: D: D: R: R: R: 7. 8. 9. 9) 0) D: D: D: ) R: R: R: 0... ) ) ) ) D: D: D: R: R: R:... D: D: D: R: R: R:
Page Solving Radical Equations Notes Etraneous Solutions:... 9. 6. 6. 7. The number of people,, involved in reccling in a communit is modeled b the function 90 00, where is the number of months the reccling plant has been open. (a) Find the number of people involved in reccling eactl months after the plant opened. (b) After how man months will 90 people be involved in reccling? 8. The period of a pendulum (T), in seconds, is the length of time it takes for the pendulum to make L one complete swing back and forth. The formula T gives the period T for a pendulum of length L in feet. If ou want to build a grandfather clock with a pendulum that swings back and forth once ever seconds, how long, to the nearest tenth of a foot, would ou make the pendulum?
Page Solving Radical Equations Classwork Solve each equation.. 8. 9. 7. 7. 6. 7. (Calculator) 8. 6 9. 7 0. (Calculator). The velocit of a free falling object is given b V gh where V is the velocit (in meters per second), g is acceleration due to gravit (in meters per second square), and h is the distance (in meters) the object has fallen. The value g depends on which bod/planet is attracting the object. If the object hits the surface with a velocit of 0 meters per second, from what height was it dropped in each of the following situations? a. You are on Earth where g = 9.8 m/s? b. You are on the moon where g =.7 m/s? c. You are on Mars where g =.7 m/s?
Page Solving Radical Equations Homework Solve each equation.... 6. 7 6. 6. 7. 8 8. 9. 9 0. 6. The speed that a tsunami (tidal wave) can travel is modeled b the equation S 6 d where S is the speed in kilometers per hour and d is the average depth of the water in kilometers. a.) What is the speed of the tsunami when the average water depth is 0. kilometers? (round to nearest tenth) b.) Solve the equation for d. c.) A tsunami is found to be traveling at 0 kilometers per hour. What is the average depth of the water? (round to three decimal places)
Page Radical Applications Warm Up Solve the following equations.. 9.. 0 Solving Radical Equations Application Worksheet. A pendulum can be measured with the equation T = π L, where T is the time in seconds, G is the G force in gravit (0m/s ) and L is the length of the pendulum. a) find the period (to the nearest hundredth of a second) if a pendulum is 0.9m long b) find the period if the pendulum is 0.09 m long. c) solve the equation for length L. d) how long would the pendulum be if the period were eactl s? Solve the following applications.. The difference between an integer and its square root is. What is the integer?. The sum of an integer and twice its square root is. What is the integer?. The sum of an integer and three times its square root is 0. Find the integer.
Page Use d( h) h, where d is the distance to the horizon in miles from a given height h in feet for # & 6.. If a plane flies at a height 0,000 ft, how far awa is the horizon? 6. Janine was looking out across the ocean from her hotel room on the beach. Her ees were 0 ft above the ground. She saw a ship on the horizon. Approimatel how far was the ship from her? When a car comes to a sudden stop, ou can determine the skidding distance (in feet) for a given speed (in miles per hour) using the formula s() = where s is skidding distance and is speed. Calculate the skidding distance for the following speeds (round to the nearest tenth). 7. mi/h 8. 6 mi/h 9. 7 mi/h 0. 0 mi/h Radical Applications Homework. Did ou ever stand on a beach and wonder how far out into the ocean ou could see? Or have ou wondered how close a ship has to be to spot land? In either case, the function dh h can be used to estimate the distance to the horizon (in miles) from a given height (in feet). a. Cordelia stood on a cliff gazing out at the ocean. Her ees were 00 ft above the ocean. She saw a ship on the horizon. Approimatel how far was she from that ship? b. From a plane fling at,000 ft, how far awa is the horizon? c. Given a distance, d, to the horizon, what altitude would allow ou to see that far?. A weight suspended on the end of a string is a pendulum. The time, in seconds, that it takes for one period is given b the radical equation and l is the length of the pendulum. a. Find the period (to the nearest hundredth of a second) if the pendulum is. m long. t l in which g is the force of gravit (0 m/s g ) b. Find the period if the pendulum is 0.0 m long. c. Solve the equation for length l. d. How long would the pendulum be if the period were eactl seconds?
Page 6. When a car comes to a sudden stop, ou can determine the skidding distance (in feet) for a given speed (in miles per hour) using the formula s, in which s is skidding distance and is speed. Calculate the skidding distance for the following speeds (round to the nearest tenth). a. mph b. 70 mph c. 90 mph d. mph e. Given the skidding distance s, what formula would allow ou to calculate the speed in miles per hour? f. Use the formula obtained in (e) to determine the speed of a car in miles per hour if the skid marks were ft long.. Solve each of the following applications. a. The sum of an integer and its square root is 6. Find the integer. b. The difference between an integer and its square root is 6. What is the integer? c. The sum of an integer and twice its square root is. What is the integer? d. The sum of an integer and times its square root is 0. Find the integer. Notes: Direct and Inverse Variation I. Direct Variation: = k Stated: varies directl as or is directl proportional to where k is the constant of proportionalit or constant of variation. Ke Idea: As gets larger, gets larger or as gets smaller, gets smaller. A line with a -intercept of 0 is a direct variation. Real world eamples of direct variation include: fruit sold b the pound, distance traveled b a car over time, characters printed from a computer per second, circumference of a circle varies directl as the diameter, and wages varing directl to hours worked. Can ou think of others? Eample : If ou bu three pounds of grapes at $.99 per pound, how much would ou pa for the grapes? What are the two variables and what is the constant of variation? How can ou write this as a linear equation?
Page 7 Data that represents direct variation: - 0 0 What is the constant of proportionalit? - 6 - Eample : If varies directl as and = 0 when =9, then what is when =? Eample : When a biccle is pedaled in a certain gear, the distance traveled varies directl to the number of pedal revolutions. The biccle travels 6 meters for ever pedal revolutions. How man revolutions would be needed to travel 600 meters? Eample : A refund r ou get varies directl as the number of cans ou reccle. If ou get a $.7 refund for 7 cans, how much should ou receive for 00 cans? II. Inverse Variation: = k or = k Stated: varies inversel as or is inversel proportional to where k is the constant of proportionalit or constant of variation Ke Idea: As gets larger, gets smaller or as gets smaller, gets larger. Real world eamples of indirect variation include: Boles Law of Gases is a real world eample of inverse variation. Likewise, for a trip to Mrtle Beach, the greater our car speed, the less time it would take ou to get there. If a rectangle has an area of square units, then as the length increases the width decreases. Data that represents inverse variation: 6 What is the constant of variation? 9 6 0 Eample: If varies inversel as and = when = 9, then what is when = 7? Eample : If varies inversel as the square of and = 0 when =, find when =.
Page 8 Eample : Find when =, if varies inversel as and = when = 6. Eample : The amount of resistance in an electrical circuit required to produce a given amount of power varies inversel with the square of the current. If a current of.8amps requires a resistance of 0 ohms, what resistance will be required b a current of. amps? (HONORS) Combined Variation: describes a situation where a variable depends on two (or more) other variables, and varies directl with some of them and varies inversel with others (when the rest of the variables are held constant). k z Eample: The number of hours needed to assemble computers varies directl as the number of computers and inversel as the number of workers. If workers can assemble computers in 9 hours, how man workers are needed to assemble 8 computers in 8 hours? For each problem: a) write a function of variation to represent the situation and b) solve for the indicated information.. The number of gallons g of fuel used on a trip varies directl with the number of miles m traveled. If a trip of 70 miles required gallons of fuel, how man gallons are required for a trip of 00 miles?. Karen earns $8.0 for working si hours. If the amount m she earns varies directl with h the number of hours she works, how much will she earn for working 0 hours?. A bottle of 0 vitamins costs $.. If the cost varies directl with the number of vitamins in the bottle, what should a bottle with 0 vitamins cost?. Wei received $. in interest on the $0 in her credit union account. If the interest varies directl with the amount deposited, how much would Wei receive for the same amount of time if she had $000 in the account?. The volume V of a gas kept at a constant temperature varies inversel as the pressure p. If the pressure is pounds per square inch, the volume is cubic feet. What will be the volume when the pressure is 0 pounds per square inch? 6. The time to complete a project varies inversel with the number of emploees. If people can complete the project in 7 das, how long will it take people?
Page 9 7. The time needed to travel a certain distance varies inversel with the rate of speed. If it takes 8 hours to travel a certain distance at 6 miles per hour, how long will it take to travel the same distance at 60 miles per hour? 8. The number of revolutions made b a tire traveling over a fied distance varies inversel to the radius of the tire. A -inch radius tire makes 00 revolutions to travel a certain distance. How man revolutions would a 6-inch radius tire require to travel the same distance? 9. (Challenge) An egg is dropped from the roof of a building. The distance it falls varies directl with the square of the time it falls. If it takes seconds for the egg to fall eight feet, how long will it take the egg to fall 00 feet? 0. (Challenge) The time needed to paint a fence varies directl with the length of the fence and inversel with the number of painters. If it takes five hours to paint 00 feet of fence with three painters, how long will it take five painters to paint 00 feet of fence? Direct and Inverse Variation Worksheet Find the Missing Variable: ) varies directl with. If = - when =, find when = -6. ) varies inversel with. If = 0 when = 6, find when = -. ) varies inversel with. If = 7 when = -, find when =. ) varies directl with. If = when = -8, find when =.6. ) varies directl with. If = 7 when =, find when =. Classif the following as: a) Direct b) Inverse c) Neither 6) m = -p 9 8) d = t 7) r = 9) c = t e 0) n = ½ f ) z =. t ) c = v ) u = 8 i What is the constant of variation for the following? ) = ) = 6) = ½ 7) = 9
Page 0 Answer the following questions. 8) If and var directl, as decreases, what happens to the value of? 9) If and var inversel, as increases, what happens to the value of? 0) If and var directl, as increases, what happens to the value of? ) If and var inversel, as decreases, what happens to the value of? Classif the following graphs as a) Direct b) Inverse c) Neither ) ) ) ) Answer the following questions: 6) The electric current I, is amperes, in a circuit varies directl as the voltage V. When volts are applied, the current is amperes. What is the current when 8 volts are applied? 7) The volume V of gas varies inversel to the pressure P. The volume of a gas is 00 cm under pressure of kg/cm. What will be its volume under pressure of 0 kg/cm? 8) The number of kilograms of water in a person s bod varies directl as the person s mass. A person with a mass of 90 kg contains 60 kg of water. How man kilograms of water are in a person with a mass of 0 kg? 9) On a map, distance in km and distance in cm varies directl, and km are represented b cm. If two cities are 7cm apart on the map, what is the actual distance between them? 0) The time it takes to fl from Los Angeles to New York varies inversel as the speed of the plane. If the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h? Solving Rational Equations Rational Equations Restricted/Ecluded Values
Page Steps for solving rational equations. Proportions. Set the cross products equal.. Solve the equation. Rational Equations with unlike denominators. Find the Least Common Denominator (LCD). Multipl each term b the LCD.. Simplif each term.. Solve the equation. Solve each equation. State an restricted/ecluded values... 7. a a 6. b b b 7 7 k. 6. k k m 7. m m 8. p 9. p p 0. a a a. b b b. k k 8k k k 6
Page Solving Rational Equations Practice Solve each equation. State an restricted/ecluded values... 7.. 6. 6. 6 7. 8. b b 9. 0. 9 n n 8.. ( )
Page Graphing Inverse Variation Sketch a graph of 0. Make a table of values that include positive and negative values of. -0-8 - - - - 0 8 0 Graph the points and connect them with a smooth curve. The graph has two parts- each part is called a branch. The -ais is called the asmptote. The -ais is called the Asmptote. The Parent Graph of : Transforming Inverse Variation Functions: The parent graph of a rational function is. Like all parent graphs, it passes starts at the point (, ). It has a horizontal asmptote at and a vertical asmptote at. The graph approaches but does not cross these lines. The parent graph of a rational function has a domain of and a range of. The parent graph of a rational function is alwas decreasing.
Page Steps to Graph Inverse (Rational) Functions: Center of Asmptotes at: (, ) or (, ) (Starting Point) If a is positive, graph If a is negative, graph Domain: Range: and a k h To graph:. Find C.O.A.,draw a vertical and horizontal line thru point. Use pattern Over Up a a -a - - -a. Connect points with a smooth curve approaching asmptotes Eamples: For each rational function, state the Center of Asmptotes, Domain and Range.. Graph each function. State the Domain and Range.. Function:. Function: D: R: D: R:. Function: D: R: 6. Function: D: R: 7. Function: D: R: 8. Function: D: R:
Page Homework: Graph each function. State the domain and range. ) ) ) ) ) 0) 9) 8) 7) 6) ) ) ) ) )
Page 6 Solving Sstems involving Radical and Inverse Functions Algebraicall. Use subsitution to combine the equations.. Solve the equation.. Plug variable back into one of the original equations to find nd variable.. Write solutions as ordered pairs. Graphicall. Tpe st equation (or left side) into.. Tpe nd equation (or right side) into.. Find the intersection(s) of the two functions. ( nd Trace, : Intersection, Enter, Enter, Enter) Solving Radical Equations/Sstems with Radical Equations Notes Solve each equation algebraicall, then check graphicall... 6... 7 6 6.
Page 7 Solving Sstems Homework Solve each sstem smbolicall. Then represent the solution on a graph. Label an ke points. 9. 6 (Calculator).. 6. 9 9 (Calculator). (Calculator) 6. Applications. 7. Find the lengths of the legs of a right triangle whose hpotenuse is feet and whose area is square feet. 8. A small television is advertised to have a picture with a diagonal measure of inches and a viewing area of square inches. What are the length and width of the screen?