Ready To Go On? Skills Intervention 9-1 Multiple Representations of Functions

Transcription

1 9A Read To Go On? Skills Intervention 9-1 Multiple Representations of Functions Using Multiple Representations to Solve Problems The table shows the sum of the interior angles of polgons and the number of sides of the polgons. Use a graph and an equation to find the sum of the interior angles of a -gon. Step 1 Graph the data. Degrees Number of sides Sum of the angles (degrees) Sides The data appears to be. Step Write an equation to represent the data. Let the sum of the interior angles and the number of sides of the polgon. m m( 1 ) Find the slope using an two points. Write point-slope form. ( ) Substitute values into point-slope form. Simplif. Step 3 Evaluate the function for a polgon with sides. 180( ) 360 Substitute. Simplif. The sum of the interior angles of a polgon with sides is. 16 Holt Algebra

2 9A Read To Go On? Problem Solving Intervention 9-1 Multiple Representations of Functions Often functions can be represented in a variet of was. Use the method that is easiest for ou. A hot air balloon is descending from an altitude of 1000 feet at a rate of 8 feet per second. Create a table, equation, and graph to represent the hot air balloon s altitude, a, with relation to time, t. When will it reach the ground? Understand the Problem 1. Describe the hot air balloon s descent. Make a Plan. What do ou need to determine? Solve 3. Create a table. Let t equal time and a equal altitude. t (seconds) a (feet) 1000 First differences: 8 The first differences are, so a model is appropriate.. Write an equation to model: Altitude is equal to 1000 minus feet per second. 5. Graph the equation. a-intercept: a Solve the equation for t when a 0: t t-intercept: 6. The hot air balloon will reach the ground after seconds. 0 Look Back 17 Holt Algebra Altitude (ft) h Time (s) 7. Check the intercepts b graphing the equation on our graphing calculator t

3 9A Read To Go On? Skills Intervention 9- Piecewise Functions Find these vocabular words in Lesson 9- and the Multilingual Glossar. Vocabular piecewise function step function Evaluating a Piecewise Function Evaluate each piecewise function at and 6. A. f () { if if 6 Evaluate the function at. Because 6, use the rule for 6. f ( ) ( ) Evaluate the function at 6. Because 6 6, use the rule for 6. f (6) 6 3 B. f () { 7 if 3 8 if 6 5 if 6 Evaluate the function at. Because, use the rule for. f ( ) Evaluate the function at 6. Because 6 6, use the rule for. f (6) 3(6) Graphing Piecewise Functions Graph the piecewise function. f () { 1 Because the function is divided at, evaluate both branches of the function at. if 3 if 3 Plot the point (3, 1) with a/an circle and draw a horizontal ra to the. Substitute 3 into the function f (). (3) Plot the point (3, ) with a/an circle and draw a ra to the with a slope of. 18 Holt Algebra

4 9A Read To Go On? Problem Solving Intervention 9- Piecewise Functions A piecewise function is a combination of one or more functions. If the function is constant for each interval in its domain, the function is called a step function. Maureen drove from her house in the suburbs to her office in the cit. She drove 10 minutes through town at an average speed of 30 mi/h, 30 minutes on the highwa at an average speed of 65 mi/h, and 6 minutes in cit traffic at an average speed of 15 mi/h. Write a piecewise function to represent Maureen s distance versus time. Then graph the function. Understand the Problem 1. Maureen s trip can be described b how man steps?. What do ou need to determine? Make a Plan 3. Make a table to organize the data. Use the distance formula to find Maureen s rate for each leg of her drive. Solve. Graph the function. 5. Write a linear function for each leg. Town: d 0.5 Highwa: d t 35 6 Cit: d t The function rule is: { t if 0 t d(t) t if 10 t t if 0 t 6 Through Town 30 Rate (mi/h) Time (h) Distance (mi) On the Highwa 1 In Cit Traffic 1 10 Distance (mi) Time (min) Look Back 6. Use our graph to find out how far Maureen has traveled after 5 min: 0 min: 5 min: Does our function make sense? 19 Holt Algebra

5 9A Read To Go On? Skills Intervention 9-3 Transforming Functions Transforming Piecewise Functions Given f () { 6 8 if, write the rule for h(), a horizontal if shift of f () 3 units left. Replace ever in the function with. { 6( ) 8 if ( 3) h() f ( 3) ( 3 ) if ( 3) h() { 10 9 if if Simplif. Identifing Intercepts Identif the - and -intercepts of f (). Without graphing g (), identif its - and -intercepts. A. f () 1 5 and g () f () f (0) 1 ( ) 5 Find the -intercept of the original function. 1 5 Find the -intercept of the original function. 5 1 The -intercept of f () is and the -intercept is. g () is a of f () b a factor of. The -intercept of g () is and the -intercept is. Check our answer: Graph f () and g () to verif the intercepts. B. f () 8 and g () f (5) f (0) 8 Find the -intercept of the original function. 8 Find the -intercept of the original function. 8 8 The -intercept of f () is and the -intercept(s) is/are. g () is a of f () b a factor of, and a reflection of f () across the ais. The -intercept of g() is and the -intercept(s) is/are. Check our answer: Graph f () and g () to verif their intercepts. 150 Holt Algebra

6 9A 9-1 Multiple Representations of Functions 1. A plane is descending from an altitude of 3,000 feet at a rate of 5 feet per second. Create a table and an equation to represent the plane s altitude, a, with relation to time, t. t (s) a (ft) Read To Go On? Quiz. The population of a certain bacteria at different times is shown in the table. Time (hours) Number of bacteria ,00 a. Find an appropriate model for the population of bacteria. b. Assuming the growth in population continues, when will the bacteria s population equal 1,31,00? 9- Piecewise Functions 3. Graph the function f () { if 1 if 1.. The cost to park in a parking garage in Boston is $5 for the first hours and $ for each additional hour. Sketch a graph of the cost of parking in the garage for 0 to 6 hours, and write a piecewise function for the graph Holt Algebra

7 9A Read To Go On? Quiz continued Write a piecewise function for each graph Transforming Functions Identif the - and -intercepts of f (). Without graphing g (), identif its - and -intercepts. 7. f () and g () f () 8. f () 1 and g () f ( 1 ) -intercept(s) of f () -intercepts(s) of f () -intercept(s) of g () -intercept(s) of f () -intercepts(s) of f () -intercept(s) of g () -intercepts(s) of g () 9. Given f () and g () f () 1, graph g (). -intercepts(s) of g () 15 Holt Algebra

8 9A Read To Go On? Enrichment Eploring Step Functions A step function is a special tpe of piecewise function whose graph resembles steps, not lines or curves. One common step function is the greatest integer function, or rounding-down function. Given an number, the greatest integer function is defined as: f () the greatest integer less than or equal to. Find f () for the following values of _ The domain of the greatest integer function is the set of Real numbers. What is the range of the greatest integer function? 6. Graph the greatest integer function for all values of such that The cost of a telephone call between Boston and Paris is $0.90 for the first minute and $0. for each additional minute or portion of a minute. a. Use the greatest integer function to model the cost of a call, C. b. How much does an 8 minute and 5 second phone call cost? 153 Holt Algebra

9 9B Read To Go On? Skills Intervention 9- Operations with Functions Find this vocabular word in Lesson 9- and the Multilingual Vocabular Glossar. composition of functions Evaluating Functions Given f () 6 5, g () 1 6 7, and h (), find each 9 function or value. A. (g f )( ) (g f )( ) 6 7 ( 5) Substitute function rules. Combine like terms. B. (fh)( 1) (fh)( ) ( 5) 9 (fh)( 1) (6( 33 ) 5) ( ) 9 Substitute function rules. Substitute 1 for. Simplif. C. h g ( ) h h( ) g ( ) ( ) ( )( ) ( )( ) Substitute function rules. Factor completel. Note that 9 or 3. Simplif the fraction. where 9 or. Divide out common factors ( ) ( ) and simplif. D. g (h ( 3)) h ( 3) 1 9 Find h ( 3). g () 6( ) 7 Substitute our result for h ( 3) into g. 15 Holt Algebra

10 9B Read To Go On? Problem Solving Intervention 9- Operations with Functions You can multipl functions b following this rule: (fg)( ) f ( ) g ( ). During a sale, a shoe store is selling boots for 5% off. Preferred customers also receive an additional 10% off. Write a composite function to represent the final cost of a pair of boots that originall cost c dollars. Then find the cost of a pair of boots originall priced at $9 that a preferred customer wants to bu. Understand the Problem 1. Describe the shoe store s sale. Make a Plan. What do ou need to determine? 3. Write a function P for the final price of boots after the sale. P (c ) c. Write a function D for the price after appling the preferred customer discount. D (c ) c Solve 5. Find the composition P (D(c )). P (D (c)) P ( c) Substitute the rule of D into P. (0.90c) Appl the rule for P. c Simplif. 6. Find the cost of the boots that originall cost $9. Look Back 7. To check our answer, find 5% of $9: Sale savings 5%(9) Then subtract the savings from $9 and take 10% of that value. Preferred customer discount 10%(9 savings) Cost 9 sale savings preferred customer discount 8. Does our solution in Eercise 6 make sense? 155 Holt Algebra

11 9B Read To Go On? Skills Intervention 9-5 Functions and Their Inverses Find these vocabular words in Lesson 9-5 and the Multilingual Glossar. Vocabular one-to-one function horizontal-line test Using the Horizontal-Line Test and Writing Rules for Inverses Find the inverse of each function. Then state the domain and range of the inverse. A. f () 7 Step 1 Graph the function on our graphing calculator. Does it pass the horizontal-line test? Step Find the inverse. B. f ( ) a function. 7 7 Therefore, the inverse Rewrite the function using instead of f (). Interchange and. Note the range restriction. Isolate the radical. Square both sides of the equation. 1 Multipl. 1 Solve for. The domain of the inverse is the of f ( ):. The range of the inverse is the of f ( ): Step 1 Graph the function on our graphing calculator. Does it pass the horizontal-line test? a function. Step Find the inverse Therefore, the inverse Rewrite the function using instead of f ( ). Interchange and. Solve for. The domain of the inverse is: { } and its range is: { }. 156 Holt Algebra

12 9B Read To Go On? Skills Intervention 9-6 Modeling Real-World Data Identifing Models b Using Constant Ratios or Differences Use finite differences or ratios to determine which parent function best models each set of data. A Are the -values evenl spaced? Check the first differences between -values:, 5,, Check the second differences between -values:,, 8 Check the ratios between -values: Because the differences are constant, a/an model best models the data set. B Are the -values evenl spaced? Check the first differences between -values:, 6.15,, Check the second differences between -values:, 5.359, Check the ratios between -values:, 0.15,, Because the are constant, a/an model best models the data set. C Are the -values evenl spaced? Check the first differences between -values: 1.8,,, Check the second differences between -values: Because the are constant, a/an model best models the data set. 157 Holt Algebra

13 9B Read To Go On? Problem Solving Intervention 9-6 Modeling Real-World Data To model data in a real-world application, look for a pattern. The table shows the estimated population of a Southwestern town. Using time as the independent variable and 1980 as a reference ear, find a model that best fits the data. Use our model to predict the town s population in the ear 030. Year Population Understand the Problem 1. Describe the town s population. Make a Plan. What do ou need to determine? Solve 3. Are the ears evenl spaced? Check the first differences: Check the second differences: Check the ratios between population values: Because the are constant, a/an model is best.. Use our graphing calculator to perform a/an regression. 5. A function that models the data is: f (t) ( ) t 6. Solve for the town s population in the ear 030. Let t. f (t) ( ) 50 Look Back 7. To check our model, let t 0, 5, 10, 15, 0, and 5 and see if our function values using the function in Eercise 5 match the population figures in the table. 8. Does our model seem reasonable? 158 Holt Algebra

14 9B Read To Go On? Quiz 9- Operations with Functions Given f ( ) 1, g () 11 18, and h ( ) 10, find each function or value. 1. (g f )(). h g (5) 3. (fg )( ). h (f ( )) 5. a. Find (g f )( ). b. What is the domain of the composite function? 6. Mike imports lemon juice from Sicil. The cost of a case of lemon juice includes a 15% import ta and 50 euros for shipping. Given 1 dollar 0.83 euros, write a composite function for the total cost of a case of lemon juice in dollars, if the cost of a case is c euros,. 9-5 Functions and Their Inverses State whether the inverse of each relation is a function Holt Algebra

15 9B Read To Go On? Quiz continued Write the rule for the inverse of each function. Then state the domain and range of the inverse. 9. f () (8 ) g ( ) 7 domain: range: domain: range: 9-6 Modeling Real-World Data 11. Use finite differences or ratios to determine which parent function would best model this set of data. t a The table shows the ears that a park ranger has been monitoring the giraffe population in a game park in Namibia and the size of the giraffe population. Using time as the independent variable, find a model for the size of the giraffe population. Time (ear) Giraffe Holt Algebra

16 LESSON 9B Read To Go On? Enrichment Composition of Functions Recall that the composition of functions f and g is notated: (f g )( ) f (g ( )) The domain of (f g )( ) is all values of in the domain of g such that g ( ) is in the domain of f. In Eercises 1, find two functions f and g such that (f g )() equals the rule shown. (There is more than one correct answer. Find multiple answers if ou can.) 1 1. (f g )(). (f g )( ) 3 ( 1) 3. (f g )( ) ( ). (f g)( ) Holt Algebra