Cluster A.2: Linear Functions, Equations, and Inequalities
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1 A.2A: Representing Domain and Range Values: Taxi Trips Focusing TEKS A.2A Linear Functions, Equations, and Inequalities. The student applies mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities. Readiness Standard Focusing Mathematical Process A.1A Apply mathematics to problems arising in everyday life, society, and the workplace. A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. A.1E Create and use representations to organize, record, and communicate mathematical ideas. Additional TEKS: A.12B Evaluate functions, expressed in function notation, given one or more elements in their domains. Supporting Standard Performance Task The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service charge of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices the passenger can expect to pay? What are the domain values that make sense in the situation? In this situation, what is the difference in the range values between the two functions representing the different types of trips? Justify your reasoning. Answer: Cost range is 2.75 y for any type of trip. The range of the regular trip with Main Taxi Company is 2.75 y and for an airport trip is 7.25 y Domain values are 0 x 75. The minimum and maximum range values are $4.50 higher for an airport trip due to an extra airport fee than for a non-airport trip. Page 1 of 9
2 Mathematically Speaking In this task, students are being asked to evaluate a real-world situation and determine reasonable domain and range values within the context of the situation. First, students must determine which part of the situation reflects the input or domain values and which part reflects the output or range values. They must think about the reasonable values for the context of the problem, and differentiate this from domain and range values that apply to a function rule without limitations. To express the range values as an inequality, the student may need to infer or write functions to represent a regular taxi trip and a trip involving the airport followed by evaluating these functions within the confines of the situation to calculate the minimum and maximum values that make sense in the context of the real-world problem. The solution process presented involves evaluating the function rule for different values. Students could also use a graph and its corresponding table of values in a graphing calculator to solve the problem, limiting the possible x-values to whole numbers from 0 to 75. Possible Solution The cost of the taxi ride is determined by the number of whole miles traveled plus the initial charge, and the airport fee if it applies to the trip. The input variable is the length of the trip in whole miles, which is the domain, or x-values, for the situation. The output variable is the total cost of the taxi trip, which is the range, or y-values, for the situation. For this situation, the domain is limited to whole numbers in the inequality 0 x 75 because in the problem it is stated that the taxi company only offers service within a 75-mile distance. The range values would be the output values that correspond to the domain values when those domain values are input into a function rule. For this situation, there are two possible function rules; one for a regular trip and one for a trip to the airport. The rule when the passenger does not wish to travel to the airport is composed of the initial service charge of $2.75 and the cost of the travel, which is $2.50 times the number of miles. Written as an equation, the regular trip fee is y = 2.5x The rule when the passenger does wish to travel to the airport is composed of the initial service charge of $2.75, the airport fee of $4.50, and the cost of the travel, which is $2.50 times the number of miles. Written as an equation, the airport trip fee is y = 2.5x + ( ) or y = 2.5x Page 2 of 9
3 The smallest amount a passenger would pay would be for an input of 0 miles for a trip that does not involve the airport. Since the regular trip carries an initial fee of $2.75, the minimum range value would be $2.75. This is because when the value of 0 is substituted for miles in the function y = 2.5x , the solution is The greatest value would be the highest possible cost of a trip, which would be going to the airport and using the maximum number of miles. This value is found by using the maximum number of miles the taxi service will travel in the area, which in this case is 75 miles. The maximum value of 75 for miles should be substituted into the function y = 2.5x + ( ) or y = 2.5x miles $2.50 per mile = $ To find the maximum cost, add the initial trip charge, $2.75 and the airport fee, $4.50: $ $7.25 = $ Since the total cost of a taxi trip can include the smallest and largest values, the cost range for a passenger is represented by the inequality $2.75 y $ for the situation. Reasonableness Realistically a passenger will use a taxi service for at least 1 mile rather than pay only the trip charge without any mileage travel. In this case, the minimum cost of a taxi would be considering 1 mile at $2.50 plus the initial charge of $2.75 making the minimum price in the range $5.25 instead of $2.75. In this line of thinking, the range of cost of the taxi service is $5.25 y Compare the range values to examine the differences. For a regular trip (no airport): 2.75 y For an airport trip: 7.25 y = = 4.50 The range values for the airport trip are 4.50 more than the regular trip. This is because of an added airport fee. Look For a correctly written inequality for the range of prices and domain of miles a solution strategy for evaluating the function rules or use of a graph and table to determine a reasonable set of range values if the student chooses to use technology, the ability to interpret domain and range using either a graph or a table that the student understands the difference in the ranges is due to an added airport fee student justification of choices of solution strategy Page 3 of 9
4 Differentiation: Simplified Task Main Taxi Company offers taxi services within a 75-mile distance of any point in the city area. The taxi charge is based on the number of trip miles. The rate per whole mile for the taxi is $2.50. The taxi service also charges a service charge of $3.00 for any regular trip or $8.00 for an airport trip, both of which are charged as soon as a passenger gets into the car. The cost of taxi service for a regular trip is represented using the function y = 2.5x + 3. The cost of taxi service for an airport trip is represented using the function y = 2.5x + 8. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices a passenger can expect to pay for either type of trip? Justify your reasoning. Answer: Regular trip: 3 y ; Airport trip: 8 y Differentiation: Enriching Task Main Taxi Company offers taxi services within a 75-mile distance of any point in the city area. The taxi charge is based on the number of trip miles. The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service charge of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices the passenger can expect to pay? What are the domain values that make sense in the situation? In this situation, what is the difference in the range values between the two functions representing the different types of trips? Justify your reasoning. City Car Company offers taxi services within the same 75-mile city area. Their airport fee is $4.75, their initial trip fee is $3.50, and their mileage rate is $2.25 per whole mile. How does the cost for City Car Company compare to Main Taxi Company? Justify your reasoning. Answer: Domain values are 0 x 75 for both. The range of the regular trip with Main Taxi Company is 2.75 y and for an airport trip is 7.25 y The minimum and maximum values differ by $4.50 due to an extra airport fee. Main Taxi Company s range is 2.75 y City Car Company s range is 3.50 y 177 At 75 miles and an airport trip, the maximum cost for City Car Company is $17.75 less than for Main Taxi Company, but if there is no airport trip and the trip is less than 3 miles, Main Taxi Company is less expensive. Page 4 of 9
5 Scaffolded Task with Answers The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service chart of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. 1. What values make up the domain in this situation? Write these values using an inequality statement. The number of miles is the domain and is given as within a 75-mile distance. The minimum number of miles then is 0 and the maximum is 75. The inequality is 0 x If a passenger uses Main Taxi Company for a regular trip, what function rule describes the situation? y = 2.5x If a passenger uses Main Taxi Company for a regular trip, what is the minimum charge? What is the maximum charge for a regular trip? Minimum is $2.75 (the service charge only). Maximum is 2.5(75) = $ Write the minimum and maximum charges for a regular trip as a range using an inequality y If a passenger uses Main Taxi Company for an airport trip, what function rule describes the situation? y = 2.5x + ( ) or y = 2.5x If a passenger uses Main Taxi Company for an airport trip, what is the minimum charge? What is the maximum charge for an airport trip? Minimum is $7.25 (the service and airport charges only). Maximum is 2.5(75) = $ Write the minimum and maximum charges for an airport trip as a range using an inequality y What is the least amount a passenger would pay based on either trip type? What is the most a passenger would pay based on either trip type? Write these values as a range using an inequality. Least amount is $2.75. Most is $ The inequality is 2.75 y What is the difference in the minimum cost between the two trip types? What is the difference in the maximum cost between the two trip types? What does this amount represent? Difference in minimum costs is $4.50. Difference in maximum costs is $4.50. This amount is the airport charge. Page 5 of 9
6 Name Date Performance Task: A.2A Representing Domain and Range Values: Taxi Trips The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service charge of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices the passenger can expect to pay? What are the domain values that make sense in the situation? In this situation, what is the difference in the range values between the two functions representing the different types of trips? Justify your reasoning. Procedural Conceptual Communication Total points: Page 6 of 9
7 Name Date Performance Task: A.2A Representing Domain and Range Values: Taxi Trips The rate per whole mile for the taxi is $2.50. The taxi service also charges a service charge of $3.00 for any regular trip or $8.00 for an airport trip, both of which are charged as soon as a passenger gets into the car. The cost of taxi service for a regular trip is represented using the function y = 2.5x + 3. The cost of taxi service for an airport trip is represented using the function y = 2.5x + 8. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices a passenger can expect to pay for either type of trip? Justify your reasoning. Procedural Conceptual Communication Total points: Page 7 of 9
8 Name Date Performance Task: A.2A Representing Domain and Range Values: Taxi Trips The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service charge of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. If a passenger wants to use Main Taxi Company, what inequality can be used to represent the possible prices the passenger can expect to pay? What are the domain values that make sense in the situation? In this situation, what is the difference in the range values between the two functions representing the different types of trips? City Car Company offers taxi services within the same 75-mile city area. Their airport fee is $4.75, their initial trip fee is $3.50, and their mileage rate is $2.25 per whole mile. How does the cost for City Car Company compare to Main Taxi Company? Justify your reasoning. Procedural Conceptual Communication Total points: Page 8 of 9
9 Name Date Performance Task: A.2A Representing Domain and Range Values: Taxi Trips The rate per whole mile for the taxi is $2.50. The taxi service also charges an initial service chart of $2.75. When a passenger wants to go to the airport, there is also a $4.50 airport fee. 1. What values make up the domain in this situation? Write these values using an inequality statement. 2. If a passenger uses Main Taxi Company for a regular trip, what function rule describes the situation? 3. If a passenger uses Main Taxi Company for a regular trip, what is the minimum charge? What is the maximum charge for a regular trip? 4. Write the minimum and maximum charges for a regular trip as a range using an inequality. 5. If a passenger uses Main Taxi Company for an airport trip, what function rule describes the situation? 6. If a passenger uses Main Taxi Company for an airport trip, what is the minimum charge? What is the maximum charge for an airport trip? 6. Write the minimum and maximum charges for an airport trip as a range using an inequality. 7. What is the least amount a passenger would pay based on either trip type? What is the most a passenger would pay based on either trip type? Write these values as a range using an inequality. 8. What is the difference in the minimum cost between the two trip types? What is the difference in the maximum cost between the two trip types? What does this amount represent? Page 9 of 9
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