6.2 Describing Functions

Transcription

1 ? LESSN. Describing Functions ESSENTIAL QUESTIN Proportionalit.5.H Identif eamples of proportional and nonproportional functions that arise from mathematical and real-world problems. Also.5.G What are some characteristics that ou can use to describe functions? EXPLRE ACTIVITY.5.G Investigating a Constant Rate of Change The U.S. Department of Agriculture defines heav rain as rain that falls at a rate of 1.5 centimeters per hour. A The table shows the total amount of rain that falls in various amounts of time during a heav rain. Complete the table. Time (h) Total Amount of Rain (cm) B C Plot the ordered pairs from the table on the coordinate plane at the right. How much rain falls in 3.5 hours? Heav Rainfall Houghton Mifflin Harcourt Publishing Compan D E F Plot the point corresponding to 3.5 hours of heav rain. What do ou notice about all of the points ou plotted? Is the total amount of rain that falls a function of the number of hours that rain has been falling? Wh or wh not? Reflect 1. Suppose ou continued to plot points for times between those in the table, such as 1. hours or.5 hours. What can ou sa about the locations of these points? Total Amount of Rain (cm) Time (h) Lesson. 13

2 Math n the Spot m.hrw.com Graphing Linear Functions The relationship ou investigated in the previous activit can be represented b the equation = 1.5, where is the time and is the total amount of rain. The graph of the relationship is a line, so the equation is a linear equation. Since there is eactl one value of for each value of, the relationship is a function. It is a linear function because its graph is a nonvertical line. EXAMPLE 1.5.H Math Talk Mathematical Processes Carrie said that for a function to be a linear function, the relationship it represents must be proportional. Do ou agree or disagree? Eplain. The temperature at dawn was F and increased steadil F ever hour. The equation = + gives the temperature after hours. State whether the relationship between the time and the temperature is proportional or nonproportional. Then graph the function. STEP 1 STEP Compare the equation with the general linear equation = m + b. = + is in the form = m + b, with m = and b =. Therefore, the equation is a linear equation. Since b 0, the relationship is nonproportional. Choose several values for the input. Substitute these values for in the equation to find the output. + (, ) 0 (0) + (0, ) () + 1 (, 1) () + 1 (, 1) () + 0 (, 0) STEP 3 YUR TURN Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.. State whether the relationship between and in = 0.5 is proportional or nonproportional. Then graph the function. - Temperature ( F) Temperatures Time (h) Houghton Mifflin Harcourt Publishing Compan Personal Math Trainer nline Assessment and Intervention m.hrw.com - 1 Unit

3 Determining Whether a Function is Linear The linear equation in Eample 1 has the form = m + b, where m and b are real numbers. Ever equation in the form = m + b is a linear equation. The linear equations represent linear functions. Equations that cannot be written in this form are not linear equations, and therefore are not linear functions. Math n the Spot m.hrw.com EXAMPLE.5.H Houghton Mifflin Harcourt Publishing Compan A square tile has a side length of inches. The equation = gives the area of the tile in square inches. Determine whether the relationship between and is linear and, if so, if it is proportional. STEP 1 STEP STEP 3 STEP YUR TURN Choose several values for the input. Substitute these values for in the equation to find the output. Graph the ordered pairs. Identif the shape of the graph. The points suggest a curve, not a line. Draw a curve through the points to represent the solutions of the function. Describe the relationship between and. The graph is not a line so the relationship is not linear. nl a linear relationship can be proportional, so the relationship is not proportional. 3. A soda machine makes _ 3 gallon of soda ever minute. The total amount that the machine makes in minutes is given b the equation = _ 3. Use the table and graph to determine whether the relationship between and is linear and, if so, if it is proportional. Time (min), Amount (gal), (, ) (1, 1) (, ) (3, 9) 1 (, 1) Amount (gal) Making Soda Time (min) Animated Math m.hrw.com Math Talk Mathematical Processes How can ou use the numbers in the table to decide whether or not the relationship between and is linear? Personal Math Trainer nline Assessment and Intervention m.hrw.com Lesson. 15

4 Guided Practice Plot the ordered pairs from the table. Then graph the function represented b the ordered pairs and tell whether the function is linear or nonlinear. Tell whether the function is proportional. (Eamples 1 and ) 1. = 5 -. = - Input, utput, Input, utput, Eplain whether each equation is a linear equation. (Eample ) 3. = 1. = 1? ESSENTIAL QUESTIN CHECK-IN 5. Eplain how ou can use a table of values, an equation, and a graph to determine whether a function represents a proportional relationship. Houghton Mifflin Harcourt Publishing Compan 1 Unit

5 Name Class Date. Independent Practice.5.G,.5.H m.hrw.com Personal Math Trainer nline Assessment and Intervention. State whether the relationship between and in = 5 is proportional or nonproportional. Then graph the function. 7. The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approimated with the equation = Is the relationship between and linear? Is it proportional? Eplain Kile spent $0 on rides and snacks at the state fair. If is the amount she spent on rides, and is the amount she spent on snacks, the total amount she spent can be represented b the equation + = 0. Is the relationship between and linear? Is it proportional? Eplain. Houghton Mifflin Harcourt Publishing Compan 9. Represent Real-World Problems The drill team is buing new uniforms. The table shows, the total cost in dollars, and, the number of uniforms purchased. Number of uniforms, Total cost ($), a. Use the data to draw a graph. Is the relationship between and linear? Eplain. b. Use our graph to predict the cost of purchasing 1 uniforms.. Marta, a whale calf in an aquarium, is fed a special milk formula. Her handler uses a graph to track the number of gallons of formula the calf drinks in hours. Is the relationship between and linear? Is it proportional? Eplain. Total cost ($) Total amount of formula (gal) Drill Team Uniforms 10 Number of uniforms Marta s Feedings Time (h) Lesson. 17

6 11. Critique Reasoning A student claims that the equation = 7 is not a linear equation because it does not have the form = m + b. Do ou agree or disagree? Wh? 1. Make a Prediction Let represent the number of hours ou read a book and represent the total number of pages ou have read. You have alread read 70 pages and can read 30 pages per hour. Write an equation relating hours and pages ou read. Then predict the total number of pages ou will have read after another 3 hours. FCUS N HIGHER RDER THINKING Work Area 13. Draw Conclusions Rebecca draws a graph of a real-world relationship that turns out to be a set of unconnected points. Can the relationship be linear? Can it be proportional? Eplain our reasoning. 1. Communicate Mathematical Ideas Write a real-world problem involving a proportional relationship. Eplain how ou know the relationship is proportional. 15. Justif Reasoning Show that the equation + 3 = 3( + 1) is linear and that it represents a proportional relationship between and. Houghton Mifflin Harcourt Publishing Compan Image Credits: Photodisc/ Gett Images 1 Unit