Functions. Academic Vocabulary causation. Math Terms

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1 Functions Unit Overview In this unit, ou will build linear models and use them to stud functions, domain, and range. Linear models are the foundation for studing slope as a rate of change, intercepts, and direct variation. You will learn to write linear equations given varied information and epress these equations in different forms. Essential Questions How can ou show mathematical relationships? Wh are linear functions useful in real-world settings? 01 College Board. All rights reserved. Ke Terms As ou stud this unit, add these and other terms to our math notebook. Include in our notes our prior knowledge of each word, as well as our eperiences in using the word in different mathematical eamples. If needed, ask for help in pronouncing new words and add information on pronunciation to our math notebook. It is important that ou learn new terms and use them correctl in our class discussions and in our problem solutions. Academic Vocabular causation Math Terms relation function vertical line test independent variable dependent variable continuous discrete -intercept relative maimum relative minimum etrema -intercept parent function absolute value function direct variation constant of variation indirect variation inverse function one-to-one arithmetic sequence eplicit formula recursive formula slope-intercept form point-slope form standard form scatter plot trend line correlation line of best fit linear regression quadratic regression quadratic function eponential regression eponential function Embedded Assessments This unit has three embedded assessments, following Activities, 11, and 1. The will give ou an opportunit to demonstrate what ou have learned. Embedded Assessment 1: Representations of Functions p. 11 Embedded Assessment : Linear Functions and Equations p. 17 Embedded Assessment : Linear Models and Slope as Rate of Change p. 07

2 UNIT Getting Read Write our answers on notebook paper. Show our work. 1. Cop and complete the table of values List the integers that make this statement true. <. Evaluate for a = and b =. a. a 5 b. b + a. Name the point for each ordered pair. a. (, 0) b. ( 1, ) c. (, ) S R Eplain how ou would plot (, ) on a coordinate plane. T. Which of the following equations represents the data in the table? A. = 1 B. = 1 C. = + 1 D. = If + =, what is the value of? A. B. C. 0 D.. Which of the following are the coordinates of a point on this line? A. ( 1, ) B. (1, ) C. ( 1, ) D. (1, ) 1 01 College Board. All rights reserved. SpringBoard Mathematics Algebra 1, Unit Functions

3 Functions and Function Notation Vending Machines Lesson 5-1 Relations and Functions Learning Targets: Represent relations and functions using tables, diagrams, and graphs. Identif relations that are functions. SUGGESTED LEARNING STRATEGIES: Visualization, Create Representations, Think-Pair-Share, Interactive Word Wall, Paraphrasing Use this machine to answer the questions below. DVD Vending Machine ACTIVITY 5 A 1 Insert mone and push the buttons below. B C A1 A A B1 B B C1 C C Remove Purchased DVDs Here 01 College Board. All rights reserved. 1. What DVD would ou receive if ou inserted our mone and pressed: a. A1? b. C? c. B?. Assuming the machine were filled properl, describe what would happen if ou pressed the same button twice. Activit 5 Functions and Function Notation 5

4 ACTIVITY 5 Lesson 5-1 Relations and Functions Each time ou press a button, an input, ou ma receive a DVD, an output.. In the DVD vending machine situation, does ever input have an output? Eplain our response. MATH TERMS A mapping is a visualrepresentation of a relation in which an arrow associates each input with its output.. Each combination of input and output can be epressed as a mapping written input output. For eample, B The Amazing Insectman. a. Write as mappings each of the possible combinations of buttons pushed and DVDs received in the vending machine. b. Create a table to illustrate how the inputs and outputs of the vending machine are related. MATH TERMS An ordered pair shows the relationship between two elements, written in a specific order using parentheses notation and a comma separating the two values. MATH TERMS A relation is information that can be represented b a set of ordered pairs. Mappings that relate values from one set of numbers to another set of numbers can be written as ordered pairs. A relation is a set of ordered pairs. Relations can have a variet of representations. Consider the relation {(1, ), 01 College Board. All rights reserved. SpringBoard Mathematics Algebra 1, Unit Functions

5 Lesson 5-1 Relations and Functions ACTIVITY 5 Relations can have a variet of representations. Consider the relation {(1, ), (, ), (, 5)}, shown here as a set of ordered pairs. This relation can also be represented in these was. Table Mapping Graph Write the following numerical mappings as ordered pairs. Input Output Ordered Pairs 1 (1, ) 1 7 Check Your Understanding 01 College Board. All rights reserved.. A vending machine at the Ocean, Road, and Air show creates souvenir coins. You select a letter and a number and the machine creates a souvenir coin with a particular vehicle imprinted on it. The graph shows the vending machine letter/number combinations for the different coins. a. Make a table showing each coin s letter/number combination. b. Write the letter/number combinations as a set of ordered pairs. c. Write the letter/number combinations in a mapping diagram. Activit 5 Functions and Function Notation 7

6 ACTIVITY 5 Lesson 5-1 Relations and Functions A function is a relation in which each input is paired with eactl one output. 7. Compare and contrast the DVD Vending Machine with a function.. Suppose when pressing button C1 on the vending machine both The Dependables and The Light Knight come out. Describe how this vending machine resembles or does not resemble a function. 9. Imagine a machine where ou input an age and the machine gives ou the name of anone who is that age. Compare and contrast this machine with a function. Eplain b using eamples and create a representation of the situation. 10. Create an eample of a situation (math or real-life) that behaves like a function and another that does not behave like a function. Eplain wh ou chose each eample to fit the categor. a. Behaves like a function: 01 College Board. All rights reserved. b. Does not behave like a function: SpringBoard Mathematics Algebra 1, Unit Functions

7 Lesson 5-1 Relations and Functions ACTIVITY Determine whether the ordered pairs and equations represent functions. Eplain our answers. a. {(5, ), (, ), (7, )} b. {(, 5), (, ), (5, )} c. {(5, ), (, ), (7, )} d. = 5, where represents input values and represents output values e. = +, where represents input values and represents output values 1. Attend to precision. Using positive integers, write two relations as lists of ordered pairs below, one that is a function and one that is not a function. Function: Not a function: Check Your Understanding 01 College Board. All rights reserved. 1. Does the mapping shown represent a function? Eplain Does the graph shown represent a function? Eplain Activit 5 Functions and Function Notation 9

8 ACTIVITY 5 Lesson 5-1 Relations and Functions LESSON 5-1 PRACTICE For the Bingo card below, suppose that a combination of a column letter and a row number, such as B1, represents an input and the number at that location, such as 7, represents an output. Use this information for Items B I N G O FREE What output corresponds to I? 1. What input corresponds to 5? 17. Does ever input have a numerical output? Eplain. 1. Construct viable arguments. Eplain wh each of the following is not a function. a. b c. =, where represents input values and represents output values. 01 College Board. All rights reserved. 70 SpringBoard Mathematics Algebra 1, Unit Functions

9 Lesson 5- Domain and Range ACTIVITY 5 Learning Targets: Describe the domain and range of a function. Find input-output pairs for a function. SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Discussion Groups, Marking the Tet, Sharing and Responding The set of all inputs for a function is known as the domain of the function. The set of all outputs for a function is known as the range of the function. 1. Consider a vending machine where inserting 5 cents dispenses one pencil, inserting 50 cents dispenses pencils, and so forth up to and including all 10 pencils in the vending machine. a. Identif the domain in this situation. WRITING MATH The domain and range of a function can be written using set notation. For eample, for the function {(1, ), (, ), (5, )}, the domain is {1,, 5} and the range is {,, }. b. Identif the range in this situation.. For each function below, identif the domain and range. a. b. input output Domain: Domain: 01 College Board. All rights reserved. c. Range: Range: d. {( 7, 0), (9, ), (,.5)} Domain: Range: Domain: Range: Activit 5 Functions and Function Notation 71

10 ACTIVITY 5 Lesson 5- Domain and Range. Consider a machine that echanges quarters for dollar bills. Inserting one dollar bill returns four quarters and ou ma insert up to five one-dollar bills at a time. a. Is 7 a possible input for the relation this change machine represents? Justif our response. b. Could.5 be included in the domain of this relation? Eplain wh or wh not. c. Reason abstractl. What values are not in the domain? Justif our reasoning. d. Is a possible output for the relation this change machine represents? Justif our response. e. Could be included in the range of this relation? Eplain wh or wh not. f. What values are not in the range? Justif our reasoning. 01 College Board. All rights reserved. 7 SpringBoard Mathematics Algebra 1, Unit Functions

11 Lesson 5- Domain and Range ACTIVITY 5. Make sense of problems. Each of the functions that ou have seen has a finite number of ordered pairs. There are functions that have an infinite number of ordered pairs. Describe an difficulties that ma eist tring to represent a function with an infinite number of ordered pairs using the four representations of functions that have been described thus far. MATH TERMS A finite set has a fied countable number of elements. An infinite set has an unlimited number of elements. 5. Sometimes, machine diagrams are used to represent functions. In the function machine below, the inputs are labeled and the outputs are labeled. The function is represented b the epression a. What is the output if the input is = 7? =? = 1? b. Epress regularit in repeated reasoning. Is there an limit to the number of input values that can be used with this epression? Eplain. 01 College Board. All rights reserved. Consider the function machine below Use the diagram to find the (input, output) ordered pairs for the following values. a. = 5 b. = c. = 10 5 Activit 5 Functions and Function Notation 7

12 ACTIVITY 5 Lesson 5- Domain and Range 7. Make a function machine for the epression Use it to find ordered pairs for =, =, = 0.5, and =. Creating a function machine can be time consuming and awkward. The function represented b the diagram in Item 5 can also be written algebraicall as the equation = For each function, find ordered pairs for =, = 5, =, and = Create tables of values. a. = 9 b. = 1 Check Your Understanding 9. The set {(, 5), ( 1, ), (, ), (0, 1)} represents a function. Identif the domain and range of the function. 10. Identif the domain and range for each function. a. b College Board. All rights reserved. 7 SpringBoard Mathematics Algebra 1, Unit Functions

13 Lesson 5- Domain and Range ACTIVITY 5 LESSON 5- PRACTICE Identif the domain and range Model with mathematics. At an arcade, there is a machine that accepts game tokens and returns tickets that can be redeemed for prizes. Inserting 5 tokens returns tickets and inserting 10 tokens returns tickets. You must insert tokens in multiples of 5 or 10, and ou have a total of 0 tokens. a. Identif the domain in this situation. b. Identif the range in this situation. 1. For the function machine shown, cop and complete the table of values College Board. All rights reserved For each function below, find ordered pairs for = 1, =, = 1, and = 0.. Write our results as a set of ordered pairs. a. = b. = Activit 5 Functions and Function Notation 75

14 ACTIVITY 5 Lesson 5- Function Notation MATH TIP It is important to recognize that f() does not mean f multiplied b. Learning Targets: Use and interpret function notation. Evaluate a function for specific values of the domain. SUGGESTED LEARNING STRATEGIES: Create Representations, Discussion Groups When referring to the functions in Item in Lesson 5-, it can be confusing to distinguish among them since each begins with =. Function notation can be used to help distinguish among different functions. For instance, the function = 9 in Item a can be written: This is read as f of and f () is equivalent to. MATH TIP Notice that f() =. For a domain value, the associated range value is f(). f () = 9 } f is the name of the function. is the input variable. 1. To distinguish among different functions, it is possible to use different names. Use the name h to write the function from Item b using function notation. Function notation is useful for evaluating functions for multiple input values. To evaluate f() = 9 for =, ou substitute for the variable and write f() = 9 (). Simplifing the epression ields f() = 1.. Use function notation to evaluate f() = 9 for = 5, =, and = College Board. All rights reserved. 7 SpringBoard Mathematics Algebra 1, Unit Functions

15 Lesson 5- Function Notation ACTIVITY 5. Use the values for and f() from Item. Displa the values using each representation. a. list of ordered pairs b. table of values c. mapping d. graph. Given the function f() = 9 as shown above, what value of results in f() = 1? 5. Evaluate each function for = 5 and =. a. f() = 7 b. g() = 01 College Board. All rights reserved. c. h( )=. Reason quantitativel. Recall the mone-changing machine from Item in Lesson 5-, in which customers can insert up to five onedollar bills at a time and receive an equivalent amount of quarters. The function f() = represents this situation. What does represent? What does f() represent? Activit 5 Functions and Function Notation 77

16 ACTIVITY 5 Lesson 5- Function Notation A function whose domain is the set of positive consecutive integers forms a sequence. The terms of the sequence are the range values of the function. For the sequence, 7, 10, 1,, f(1) =, f() = 7, f() = 10, and f() = Consider the sequence,, 0,,,,,. a. What is f()? b. What is f(7)? Check Your Understanding. Evaluate the functions for the domain values indicated. a. p() = + 1 for = 5, 0, b. h(t) = t 5t for t =, 0, 5, 7 9. Consider the sequence 7,, 1, 5, 9,. a. What is f()? b. What is f(5)? LESSON 5- PRACTICE Use the function = for Items Write the function in function notation. 11. Evaluate the function for =. Epress our answer in function notation. 1. Make use of structure. For what value of does f() =? 1. Consider the sequence 1, 1,,, 5,,. What is f()? 01 College Board. All rights reserved. 7 SpringBoard Mathematics Algebra 1, Unit Functions

17 Functions and Function Notation Vending Machines ACTIVITY 5 ACTIVITY 5 PRACTICE Write our answers on notebook paper. Show our work. Lesson 5-1 Use the Beverage Vending Machine to answer Items 1. 1 For Items 7 9, two relations are given. One relation is a function and one is not. Identif each and eplain. 7. {(5, ), (, 5), (, 5), ( 5, )} {(5, ), (, 5), (5, ), ( 5, )} College Board. All rights reserved. 1. List all of the possible inputs.. List all of the possible outputs.. Which output results from an input of C? A. Juice B. Iced tea C. Latte D. Cocoa. Which number/letter combination would ou input if ou wanted the machine to output juice? A. A B. 1B C. B D. 1D 5. In a mapping of the relation shown b the vending machine, what drink would 1D map to?. In a table of the relation shown b the vending machine, what number/letter combination would correspond to cocoa? What value(s) of in the relation below would create a set of ordered pairs that is not a function? Justif our answer. {(0, 5) (1, 5) (, ) (, 7)} 7 Activit 5 Functions and Function Notation 79

18 ACTIVITY 5 Functions and Function Notation Vending Machines 11. Does the graph shown represent a function? Eplain. Lesson 5- Use the graph for Items Identif the domain of the relation represented in the graph. 1. Identif the range of the relation represented in the graph. 1. Does the relation shown in the graph represent a function? Eplain. Lesson 5- Use the function machine for Items How would ou write the function shown in the function machine in function notation? 1. What is the value of f( )? 17. What value(s) of results in f() =? 1. Given the function f() = 5, determine the value of f( ). The first seven numbers in the Fibonacci sequence are: 0, 1, 1,,, 5,. Use this information for Items 19 and What is f()? 0. What is f()? MATHEMATICAL PRACTICES Construct Viable Arguments and Critique the Reasoning of Others 1. Dora said that the mapping diagram below does not represent a function because each value in the domain is paired with the same value in the range. Eplain the error in Dora s reasoning College Board. All rights reserved. 0 SpringBoard Mathematics Algebra 1, Unit Functions

19 Interpreting Graphs of Functions Shake, Rattle, and Roll Lesson -1 Ke Features of Graphs Learning Targets: Relate the domain and range of a function to its graph. Identif and interpret ke features of graphs. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Visualization, Interactive Word Wall, Discussion Groups ACTIVITY Roller coasters can be scar but fun to ride. Below is the graph of the heights reached b the cars of the Thunderball Roller Coaster over its first 150 feet of track. The graph displas a function because each input value has one and onl one output value. You can see this visuall using the vertical line test. Stud this graph to determine the domain and range. Height Above Ground (feet) Thunderball Roller Coaster Graph Distance Along the Track (feet) MATH TERMS The vertical line test is a visual check to see if a graph represents a function. For a function, ever vertical line drawn in the coordinate plane will intersect the graph in at most one point. This is equivalent to having each domain element associated with one and onl one range element. 01 College Board. All rights reserved. The domain gives all values of the independent variable: in this case, the distance along the track in feet. The domain values are graphed along the horizontal or -ais. The domain of the function above can be written in set notation as: {all real values of : 0 150} Read this notation as: the set of all real values of, between 0 and 150, inclusive. The range gives the values of the dependent variable: in this case, the height of the roller coaster above the ground in feet. The range values are graphed on the vertical or -ais. The range of the function above can be written in set notation as: {all real values of : } Read this notation as: the set of all real values of, between 10 and 110, inclusive. MATH TERMS An independent variable is the variable for which input values are substituted in a function. A dependent variable is the variable whose value is determined b the input or value of the independent variable. Activit Interpreting Graphs of Functions 1

20 ACTIVITY Lesson -1 Ke Features of Graphs CONNECT TO The absolute maimum of a function f() is the greatest value of f() for all values in the domain. The absolute minimum of a function f() is the least value of f() for all values in the domain. Unlike relative maimums and relative minimums, absolute maimums and absolute minimums ma correspond to the endpoints of graphs. MATH TIP AP An open interval is an interval whose endpoints are not included. For eample, 0 < < 5 is an open interval, but 0 5 is not. The graph above shows data that are continuous. The points in the graph are connected, indicating that domain and range are sets of real numbers with no breaks in between. A graph of discrete data consists of individual points that are not connected b a line or curve. Man other useful pieces of information about a function can be determined b looking at its graph. The -intercept of a function is the point at which the graph of the function intersects the -ais. The -intercept is the point at which = 0. A relative maimum of a function f() is the greatest value of f() for values in a limited open domain interval. A relative minimum of a function f() is the least value of f() for values in a limited open domain interval. Because the must occur within open intervals of the domain, relative maimums and relative minimums cannot correspond to the endpoints of graphs. Use the Thunderball Roller Coaster Graph on the previous page for Items Reason abstractl. What is the -intercept of the function shown in the graph, and what does it represent?. Identif a relative maimum of the function represented b the graph.. Identif the absolute maimum of the function represented b the graph. Interpret its meaning in the contet of the situation.. Identif a relative minimum of the function represented b the graph. 01 College Board. All rights reserved. 5. Identif the absolute minimum of the function represented b the graph. Interpret its meaning in the contet of the situation. SpringBoard Mathematics Algebra 1, Unit Functions

21 Lesson -1 Ke Features of Graphs ACTIVITY Suppose ou got on a roller coaster called Cougar Mountain that immediatel started climbing the track in a linear fashion, as shown in the graph. Height Above Ground (feet) Distance Along the Track (feet). Identif the domain and range of the function. 7. Identif the -intercept of the function.. Identif the absolute maimum and minimum of the function. 01 College Board. All rights reserved. 9. Does the function have an relative maimum or minimum values? Eplain. 10. How are the etrema different on this linear graph versus the nonlinear graph for the Thunderball Roller Coaster? MATH TERMS Etrema refers to all maimum and minimum values. Activit Interpreting Graphs of Functions

22 ACTIVITY Lesson -1 Ke Features of Graphs Check Your Understanding 11. The graph below shows five points that make up the function h. Is the function h continuous? Eplain A function has three relative maimums:, 10., and. One of the relative maimums is also the absolute maimum. What is the absolute maimum? Tell whether each statement is sometimes, alwas, or never true. Eplain our answers. 1. A relative minimum is also an absolute minimum. 1. An absolute minimum is also a relative minimum. Tom hiked along a circular trail known as the Juniper Loop. The graph shows his distance d from the starting point after t minutes. Distance from Start (km) d Time (minutes) 15. Identif the domain and range of the function shown in the graph. t 01 College Board. All rights reserved. 1. Identif the absolute minimum of the function. What does it represent? SpringBoard Mathematics Algebra 1, Unit Functions

23 Lesson -1 Ke Features of Graphs ACTIVITY 17. In this function, the absolute minimum corresponds to two points on the graph. What are the two points? What do the represent in this contet? 1. Identif the absolute maimum of the function. What does it represent? 19. What points on the graph correspond to the absolute maimum? What does this mean in the contet of Tom s hike? 0. Identif an relative minimums for the function shown in the graph. 1. Identif an relative maimums for the function shown in the graph. 01 College Board. All rights reserved. Check Your Understanding. What are the independent and dependent variables for the function representing Tom s hike?. Eplain how to determine the maimum and minimum values of a function b eamining its graph.. Is it possible for a function to have more than one absolute maimum or absolute minimum value? Eplain. Activit Interpreting Graphs of Functions 5

24 ACTIVITY Lesson -1 Ke Features of Graphs LESSON -1 PRACTICE Model with mathematics. Use the graph below for Items 5 0. d Bath Water Depth 1 5 Minutes Since Bath Began t 01 College Board. All rights reserved. Depth of Bath Water (in.) What are the independent and dependent variables? Eplain.. Use set notation to write the domain and range of the function. 7. Is the function discrete or continuous? Eplain.. What is the -intercept? Interpret the meaning of the -intercept in this contet. 9. Identif an relative maimums or minimums of the function. 0. Identif the absolute maimum and absolute minimum values. Interpret their meanings in this contet. SpringBoard Mathematics Algebra 1, Unit Functions

25 Lesson - More Comple Graphs ACTIVITY Learning Targets: Relate the domain and range of a function to its graph and to its function rule. Identif and interpret ke features of graphs. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Levels of Questions, Think Aloud, Create Representations, Summarizing Eamine the graph of the function f( ) =, graphed below. ( 1 ) Describe how this graph is different from the graphs in Lesson College Board. All rights reserved. Eample A Give the domain and range of the function f( ) =. ( 1 ) Then find the -intercept, the absolute maimum, and the absolute minimum. To find the domain and range: Step 1: Stud the graph. The sketch of this graph is a portion of the function represented b the equation f( ) = ( 1 ). Step : Look for values for which the domain causes the function to be undefined. Look how the graph behaves near =. Solution: The domain and range of f( ) = can be written: ( 1 ) Domain: {all real values of : } Range: {all real values of : > 0} MATH TIP Notice the result when = is substituted into f(). f() = 1 = 1 ( ) 0 Division b zero is undefined in mathematics. Activit Interpreting Graphs of Functions 7

26 ACTIVITY Lesson - More Comple Graphs To determine the -intercept and identif an maimums or minimums: Stud the graph. We can see that the function intersects the -ais at (0, 0.5). The value of f() keeps getting larger as approaches from both sides. The value of f() approaches, but never reaches, 0 as gets further from on both sides. Solution: The -intercept is (0, 0.5). The function does not have an absolute maimum or minimum. Tr These A The function f() = + is graphed below a. Identif the domain and range of the function. Domain: Range: b. Identif the -intercept. c. Identif an relative or absolute minimums of the function. 01 College Board. All rights reserved. d. Identif an relative or absolute maimums of the function. SpringBoard Mathematics Algebra 1, Unit Functions

27 Lesson - More Comple Graphs ACTIVITY. The equation = 1 is graphed below a. Identif the domain and range. Domain: Range: b. What is the -intercept of = 1? c. Identif an relative or absolute minimums of = 1. d. Identif an relative or absolute maimums of = 1. e. Construct viable arguments. Eplain whether this equation represents a function and how ou determined this. 01 College Board. All rights reserved.. The function = is graphed below. a. Identif the domain and range. Domain: Range: b. What is the -intercept of the function =? Activit Interpreting Graphs of Functions 9

28 ACTIVITY Lesson - More Comple Graphs c. Identif an relative or absolute minimums of =. d. Identif an relative or absolute maimums of =. MATH TIP The domain is restricted to avoid situations where division b zero or taking the square root of a negative number would occur.. If ou have access to a graphing calculator, work with a partner to graph the equations listed in the table below. Each equation is a function. a. Using the graphs ou create, determine the domain and range for each function from the possibilities listed below the chart. b. Select the appropriate domain from choices 1 and record our answer in the Domain column. Then select the appropriate range from choices a f and record the appropriate range in the Range column. c. When the chart is complete, compare our answers with those from another group. Function Domain Range = + = + 5 = 9 = + 1 = + = Possible Domains Possible Ranges 1) all real numbers a) all real numbers ) all real, such that b) all real, such that 0 ) all real, such that 0 c) all real, such that ) all real, such that d) all real, such that 0 5) all real, such that 0 e) all real, such that 0.5 ) all real, such that 0 f) all real, such that 01 College Board. All rights reserved. 90 SpringBoard Mathematics Algebra 1, Unit Functions

29 Lesson - More Comple Graphs ACTIVITY Check Your Understanding 5. How can ou determine from a function s graph whether the function has an maimum or minimum values?. How can ou determine the domain of a function b eamining its graph? B eamining its function rule? 7. Give an eample of a function that has a restricted domain. Justif our answer. LESSON - PRACTICE The function f() = is graphed below College Board. All rights reserved.. Give the domain, range, and -intercept. 9. Identif an relative or absolute minimums. 10. Identif an relative or absolute maimums. 11. Attend to precision. Eamine the graphs below. Eplain wh one function has an absolute minimum and an absolute maimum and the other function does not. Identif the absolute minimum and maimum values of the function for which the eist. Activit Interpreting Graphs of Functions 91

30 ACTIVITY Lesson - Graphs of Real-World Situations Learning Targets: Identif and interpret ke features of graphs. Determine the reasonable domain and range for a real-world situation. SUGGESTED LEARNING STRATEGIES: Visualization, Discussion Groups, Look for a Pattern The function f() = + is graphed below. MATH TIP Graph a function b substituting several values for and generating ordered pairs. You can organize the ordered pairs in a table. There are infinitel man other solutions because the graph has infinitel man points. 1. What are the domain and range of the function? Domain: Range: In man real-world situations, not all values make sense for the domain and/ or range. For eample, distance cannot be negative; number of people cannot be a decimal or a fraction. In such situations, the values that make sense for the domain and range are called the reasonable domain and range. Eample A A tai ride costs an initial rate of $.00, which is charged as soon as ou get in the cab, plus $ for each mile traveled. The cost of traveling miles is given b the function f() = +. What are the reasonable domain and range? Step 1: Sketch a graph of the function. 01 College Board. All rights reserved. f() = + (, ) 0 (0, ) 1 5 (1, 5) 7 (, 7) 9 SpringBoard Mathematics Algebra 1, Unit Functions

31 Lesson - Graphs of Real-World Situations ACTIVITY Step : Step : Determine the reasonable domain. Think about what the variable represents. What values make sense? The variable represents the number of miles, so it does not make sense for to be negative. The reasonable domain is {: 0}. Use the reasonable domain and the graph to determine the reasonable range. From the graph, all -values corresponding to the reasonable domain values are greater than or equal to. The reasonable range is {: }. Solution: The reasonable domain is {: 0}. The reasonable range is {: }. Tr These A a. A banquet hall charges $15 per person plus a $100 setup fee. The cost for people is given b the function f() = What are the reasonable domain and range? b. Eight Ball Billiards charges $5 to rent a table plus $10 per hour of game pla, rounded to the nearest whole hour. The cost of plaing billiards for hours is given b the function f() = What are the reasonable domain and range? 01 College Board. All rights reserved.. Reason quantitativel. Are the domain and range of f() = + that ou found in Item 1 the same as the reasonable domain and range of f() = + found in Eample A? Eplain. Activit Interpreting Graphs of Functions 9

32 ACTIVITY Lesson - Graphs of Real-World Situations. The graph below represents a real-world situation a. Identif the domain and range. b. Describe a real-world situation that matches the graph. Your answers to Part (a) should be the reasonable domain and range for our situation. c. Identif the independent and dependent variables in our real-world situation. Check Your Understanding. For a function that models a real-world situation, the dependent variable represents a person s height. What is a reasonable range? Eplain. 5. A tour compan charges $5 to hire a tour director plus $75 per tour member. The total cost for a group of people is given b f() = What is the reasonable domain? Eplain. LESSON - PRACTICE Talk the Talk Cellular charges a base rate of $0 per month for unlimited tets plus $0.15/minute of talk time. The monthl cost for minutes is given b f() = College Board. All rights reserved.. Make sense of problems. What is the independent variable and what is the dependent variable? Eplain how ou know. 7. What are the reasonable domain and range? Eplain. 9 SpringBoard Mathematics Algebra 1, Unit Functions

33 Interpreting Graphs of Functions Shake, Rattle, and Roll ACTIVITY 01 College Board. All rights reserved. ACTIVITY PRACTICE Write our answers on notebook paper. Show our work. Lesson -1 Use the graph below for Items A B C D E F Which point corresponds to the absolute maimum of the function? A. B B. D C. G D. H. Which represents the range of the function shown in the graph? A. {0 10} B. {1 10} C. {0 10} D. {1 10}. Which point does not correspond to a relative minimum? A. B B. C C. E D. I. Is the function represented b the graph discrete or continuous? Eplain. 5. What is the -intercept of the function shown in the graph? G H I J K. a. Give the domain and range for the function graphed below. Eplain wh this graph represents a function b. What is the -intercept of the function shown in the graph? c. Identif an etrema of the function shown in the graph. Jeff walks at an average rate of 15 ards per minute. Mark s house is located 000 ards from Jeff s house. The graph below shows how far Jeff still needs to walk to reach Mark s house. Use the graph for Items Yards Left to Walk Jeff Walks to Mark s House Minutes Walking 7. Identif the independent and dependent variables.. Identif the absolute minimum and absolute maimum values. What do these values represent? 9. Identif an relative maimums or minimums. 10. What is the -intercept? What does it represent? Activit Interpreting Graphs of Functions 95

34 ACTIVITY Interpreting Graphs of Functions Shake, Rattle, and Roll Lesson - Use the graph for Items What is the domain of the function shown in the graph? 1. What is the range of the function shown in the graph? 1. What is the -intercept of the function shown in the graph? Use the graph below for Items Lesson - A fundraising organization will donate $50 plus half of the mone it raises from a charit event. Use this information for Items What is the independent variable? 1. What is the dependent variable? 19. What is the reasonable domain? Eplain. 0. What is the reasonable range? Eplain. 1. Describe a real-world situation that matches the graph shown MATHEMATICAL PRACTICES Look For and Make Use of Structure. The graph of a function is a horizontal line. What is true about the absolute maimum and absolute minimum values of this function? Eplain. 01 College Board. All rights reserved. 1. What is the -intercept of the function shown in the graph? 15. Identif an relative maimums. 1. Identif an relative minimums. 9 SpringBoard Mathematics Algebra 1, Unit Functions

35 Graphs of Functions Eperiment Eperiences Lesson 7-1 The Spring Eperiment Learning Targets: Graph a function given a table. Write an equation for a function given a table or graph. SUGGESTED LEARNING STRATEGIES: Discussion Groups, Look for a Pattern, Sharing and Responding, Think-Pair-Share, Create Representations, Construct an Argument For the following eperiment, ou will need a paper cup, a rubber band, a paper clip, a measuring tape, and several washers. A. Punch a small hole in the side of the paper cup, near the top rim. B. Use the bent paper clip to attach the paper cup to the rubber band as shown in the diagram in the section. 1. What is the length of the rubber band? ACTIVITY 7 Drop washers one at a time into the cup. Each time ou add a washer, measure the length of the rubber band. Subtract the original length ou recorded in Item 1 to find the distance that the rubber band has stretched.. Make a table of our data. Number of Washers Length of Stretch from Original Length 1 01 College Board. All rights reserved. 5. What patterns do ou notice that might help ou determine the relationship between the number of washers in the cup and the length of the rubber band stretch? Activit 7 Graphs of Functions 97

36 ACTIVITY 7 Lesson 7-1 The Spring Eperiment. Use our table to make a graph. Be sure to label an appropriate scale and the units on the -ais. CONNECT TO SCIENCE What ou have revealed with our eperiment is an eample of Hooke s Law. Hooke s Law states that the distance d that a spring (in this case the rubber band) is stretched b a hanging object varies directl with the object s weight w. Length of Stretch Number of Washers 5. Describe our graph.. Model with mathematics. Use our graph and an patterns ou described in Item to write an equation that describes the relationship between the number of washers and the length of the stretch. 7. Use our graph or our equation to predict the length of the stretch for washers and for 10 washers. A group of students performed a similar eperiment with a spring and various masses. The data the collected is shown in the table below. Mass (g) Spring Stretch (cm) Make a graph of the data in the table. 01 College Board. All rights reserved. Spring Stretch (cm) SpringBoard Mathematics Algebra 1, Unit Functions Mass (g)

37 Lesson 7-1 The Spring Eperiment ACTIVITY 7 9. Reason quantitativel. How much does the spring stretch for each additional gram of mass added? Eplain how ou found our answer. 10. Reason abstractl. Use the students data to write an equation that gives the distance d that the spring will stretch in terms of the mass m. Eplain our equation. 11. Use the equation or the graph to determine the length of the stretch for a mass of 1 gram. Graph the outcome on our graph. 1. Use the equation or the graph to determine the length of the stretch for a mass of 7 grams. Graph the outcome on our graph. 1. Use the equation or the graph to determine the length of the stretch for a mass of 1 grams. Graph the outcome on our graph. 01 College Board. All rights reserved. 1. a. What do ou notice about the points ou graphed in Items 11 1? b. How could ou represent the set of all possible masses and corresponding stretches? 15. What is the -intercept of the graph? What does it represent? 1. What is the reasonable domain? Eplain. Activit 7 Graphs of Functions 99

38 ACTIVITY 7 Lesson 7-1 The Spring Eperiment Mr. Hardiff s class conducts an eperiment with a spring and a set of weights. The record their data, but some of the information is missing. Weight (oz) Spring Stretch (in.) How much does the spring stretch for each additional ounce of weight? 1. Describe how to use our answer to Item 17 to write an equation for the data in the table. 19. Use our equation from Item 1 to complete the table. Check Your Understanding 0. A.5-pound weight stretches a spring 1 inches and a 7.5-pound weight stretches the same spring 0 inches. How much does the spring stretch for each additional pound of weight? Eplain how ou found our answer. LESSON 7-1 PRACTICE Jerem and his classmates conduct an eperiment with a set of weights and a spring. The record their results in the table. Use the table to answer Items 1. Student Mass (lb) Spring Stretch (in.) Jerem Adele 1 Roberto 1 1 Shanice 1 Guillaume 01 College Board. All rights reserved. 100 SpringBoard Mathematics Algebra 1, Unit Functions 1. Make a graph of the data.. Critique the reasoning of others. Which student made a mistake when taking their turn at the eperiment? Eplain how ou know.. If the mistake in Item were corrected, what would the correct data point be?. Write an equation to describe the students data, using the corrected data point ou identified in Item.

39 Lesson 7- The Falling Object Eperiment ACTIVITY 7 Learning Target: Graph a function describing a real-world situation and identif and interpret ke features of the graph. SUGGESTED LEARNING STRATEGIES: Discussion Groups, Look for a Pattern, Construct an Argument, Think-Pair-Share, Summarizing, Sharing and Responding 1. The Empire State Building in New York Cit is 15 feet tall. How long do ou think it will take a penn dropped from the top of the Empire State Building to hit the ground? In 159, the mathematician and scientist Galileo conducted an eperiment to answer a question much like the one in Item 1. Galileo dropped balls from the top of the Leaning Tower of Pisa in Ital and determined the time it took them to reach the ground. Galileo used several balls identical in shape but differing in mass. Because the balls all reached the ground in the same amount of time, he developed the theor that all objects fall at the same rate. Galileo s findings can be represented with the equation h(t) = 100 1t, where h(t) represents the height in feet of an object t seconds after it has been dropped from a height of 100 feet.. Make a table of values for Galileo s function h(t) = 100 1t. t (seconds) h(t) (feet) College Board. All rights reserved Activit 7 Graphs of Functions 101

40 ACTIVITY 7 Lesson 7- The Falling Object Eperiment. Construct viable arguments. Wh would negative domain values not be appropriate in this contet?. Using our table of values, graph Galileo s function h Height (feet) Time (seconds) t 5. What is the reasonable domain of the function represented in our graph? What is the reasonable range?. What is the -intercept? MATH TERMS The -intercept is the point where a graph crosses the -ais. The -coordinate of the -intercept is What does the -intercept represent?. What is the -intercept? What does the -intercept represent? 9. Identif an etrema of the function shown in the graph. What do the etrema represent? 01 College Board. All rights reserved. Your homework assignment is to graph this function, our math teacher sas. She then points to the following function on the board: f() = In this case, the function is not limited b a real-world situation. Therefore, it is important to use different tpes of domain values as ou prepare to graph. 10 SpringBoard Mathematics Algebra 1, Unit Functions

41 Lesson 7- The Falling Object Eperiment ACTIVITY Using various values for, make a table of values for f() =. f() 11. Using our table of values, graph the function College Board. All rights reserved. 1. Describe the differences between the domain of f() = and the domain of Galileo s function. 1. State the range of f() =. 1. Identif the -intercept of f() =. 15. What is the absolute maimum of f() =? What is the absolute minimum? Activit 7 Graphs of Functions 10

42 ACTIVITY 7 Lesson 7- The Falling Object Eperiment Check Your Understanding 1. Revisit our answer to Item 1 and revise it if necessar. About how long do ou think it will take a penn dropped from the top of the Empire State Building to hit the ground? How can ou use Galileo s equation to help ou answer this question? LESSON 7- PRACTICE The area of a rectangle with a perimeter of 0 units is given b f(w) = 10w w, where w is the width of the rectangle. Assume that w is a whole number. Use this function to answer Items Make a table of values and a graph of the function. 1. Attend to precision. Give a reasonable domain for the function in this contet. Eplain our answers. 19. Identif the -intercept of the function. What does the -intercept represent within this contet? 0. What is the absolute maimum of the function? What is the absolute minimum? For Items 1, use the function f() = Make a table of values and a graph of the function.. What are the domain and range?. Identif the -intercept, the absolute maimum, and the absolute minimum. 01 College Board. All rights reserved. 10 SpringBoard Mathematics Algebra 1, Unit Functions

43 Lesson 7- The Radioactive Deca Eperiment ACTIVITY 7 Learning Targets: Given a verbal description of a function, make a table and a graph of the function. Graph a function and identif and interpret ke features of the graph. SUGGESTED LEARNING STRATEGIES: Discussion Groups, Look for a Pattern, Construct an Argument, Paraphrasing, Marking the Tet, Think-Pair-Share In the late nineteenth centur, the scientist Marie Curie performed eperiments that led to the discover of radioactive substances. A radioactive substance is a substance that gives off radiation as it decas. Scientists describe the rate at which a radioactive substance decas as its half-life. The half-life of a substance is the amount of time it takes for one-half of the substance to deca. 1. Radium has a half-life of 100 ears. How much radium will be left from a 1000-gram sample after 100 ears? CONNECT TO SCIENCE How much is half a life? The half-life of a radioactive substance can be as little as seconds for Polonium-15 and as much as.5 billion ears for Uranium-.. How much radium will be left after another 100 ears?. Suppose a radioactive substance has a half-life of 1 second and ou begin with a sample of grams. Complete the table of values. 01 College Board. All rights reserved. Time (seconds) Amount Remaining (grams) Activit 7 Graphs of Functions 105

44 ACTIVITY 7 Lesson 7- The Radioactive Deca Eperiment. Graph the data from the table on the grid below. Amount Remaining (grams) Time (seconds) 5. Make use of structure. Will the amount of the substance that remains ever reach 0? Eplain.. What are the reasonable domain and range of the function represented in the graph? Eplain. 7. What is the -intercept and what does it represent?. Identif the absolute maimum and minimum of the function represented in the graph, and tell what the represent in the contet. 01 College Board. All rights reserved. 10 SpringBoard Mathematics Algebra 1, Unit Functions

45 Lesson 7- The Radioactive Deca Eperiment ACTIVITY 7 The function that describes the substance s deca is f 1. The graph of this function when it does not model a real-world situation is shown below. 15 ( ) = ( ) What are the domain and range of the function? 10. How is this graph different from our graph in Item? 01 College Board. All rights reserved. 11. How do the values of change as the values of increase? 1. How do the values of change as the values of decrease? 1. Identif the absolute maimum and absolute minimum of the function. Activit 7 Graphs of Functions 107

46 ACTIVITY 7 Lesson 7- The Radioactive Deca Eperiment Check Your Understanding 1. A scientist has g grams of a radioactive substance. Write an epression that shows the amount of the substance that remains after one half-life. 15. Critique the reasoning of others. Dlan looked at the function f ( ) = ( 1 ) and said, This function is alwas greater than 0, so 0 is the absolute minimum. Eplain wh Dlan is incorrect. LESSON 7- PRACTICE Suppose the value of our new car is reduced b half ever ear that ou own it. You paid $0,000 for our new car. 1. Describe how this situation is similar to the half-life of a radioactive substance. 17. Cop and complete the table below. Time (ears) Value ($) 0 0, Make sense of problems. For insurance purposes, a vehicle is considered scrap when its value falls below $500. After how man ears will our new car be considered scrap? 01 College Board. All rights reserved. 10 SpringBoard Mathematics Algebra 1, Unit Functions

47 Graphs of Functions Eperiment Eperiences ACTIVITY 7 01 College Board. All rights reserved. ACTIVITY 7 PRACTICE Write our answers on notebook paper. Show our work. Lesson 7-1 A weight of 15 ounces stretches a spring 10 inches. A weight of ounces stretches the same spring 1 inches. Use this information to answer Items How man inches does the spring stretch per ounce of additional weight? A. inch B. inches C. 5 inches D. 150 inches. Write an equation to describe the relationship between the distance d that the spring stretches and the weight w that is attached to it.. How much will the spring stretch for a weight of 9 ounces?. The spring is stretched 1 inches. How man ounces is the weight that is attached to it? A spring stretches.5 inches for each ounce of weight. Use this information for Items Determine a function that represents this situation.. If ou were to graph the function represented b this situation, what would be the reasonable domain? Eplain. 7. Which of the following data points would not lie on the graph representing this function? A. (0, 0) B. (1,.5) C. (.5, 1) D. (10, 5) Lesson 7- Suppose that the height of an object after seconds is given b f( ) = 100, as shown in the graph below Use the function or the graph for Items 1.. What is the reasonable domain of the function? 9. What is the reasonable range of the function? 10. Identif the -intercept of the function. 11. What does the -intercept represent? 1. Identif the -intercept of the function. 1. What does the -intercept represent? 1. Loni sas that because of the negative sign in front of, the reasonable domain for this function is onl negative values. Is her reasoning correct? Eplain. Activit 7 Graphs of Functions 109

48 ACTIVITY 7 Graphs of Functions Eperiment Eperiences Lesson The half-life of a radioactive substance is 1 hour. If ou begin with 100 ounces of the substance, how man hours does it take for 1.5 ounces to remain? The graph below represents a radioactive deca situation. Use this graph for Items 1 1. Amount Remaining (grams) Time (ears) 1. What is the original amount of the radioactive substance? Eplain how ou know. 17. What are the reasonable domain and range? 1. Identif the absolute maimum and absolute minimum values of the function. What do these values represent? Barr has a piece of paper whose area is 150 square inches. He cuts the paper in half and discards one of the pieces. He repeats this procedure several times. Use this information for Items Cop and complete the table below to show the area of the remaining piece of paper after cuts. Number of Cuts, Area of Remaining Piece, Describe how this situation is similar to the half-life of a radioactive substance. 1. If ou were to graph the points from the table, would ou connect the points? Eplain.. Describe how the reasonable domain in this situation is different from the reasonable domain in a radioactive deca situation.. Identif the -intercept. What does it represent?. Identif the absolute maimum value. What does it represent? MATHEMATICAL PRACTICES Construct Viable Arguments and Critique the Reasoning of Others 5. Maude receives $100 for her birthda. I am going to spend half of m birthda mone each da until none is left, she decides. Is it reasonable for her to believe that she will eventuall spend all of the mone? Justif our answer. 01 College Board. All rights reserved. 110 SpringBoard Mathematics Algebra 1, Unit Functions

49 Transformations of Functions Transformers Lesson -1 Eploring f() + k ACTIVITY Learning Targets: Identif the effect on the graph of replacing f() b f() + k. Identif the transformation used to produce one graph from another. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Interactive Word Wall, Think-Pair-Share, Create Representations, Discussion Groups The equation and the graph of = or f() = are referred to as the linear parent function. The graph of f() = is shown below. MATH TERMS A parent function is the most basic function of a particular categor or tpe. 1. Complete the table for g() = College Board. All rights reserved. f() = g() = Make use of structure. How do the -values for g() compare to the -values for f()? Make a conjecture about the graph of g(). As ou share our ideas with our group, be sure to use mathematical terms and academic vocabular precisel. Make notes to help ou remember the meaning of new words and how the are used to describe mathematical concepts. Activit Transformations of Functions 111

50 ACTIVITY Lesson -1 Eploring f() + k. Test our conjecture b using a graphing calculator to graph g() = + 5. Graph this on the grid in Item 1. a. What is the -intercept of the parent function? b. What is the -intercept of g()? MATH TIP The -coordinate of the -intercept is called a zero of the function. You will learn more about zeros of functions when ou stud quadratic functions later in this course. c. What is the -intercept of the parent function? What is the zero of the function f()? d. What is the -intercept of g()? What is the zero of the function? e. Revisit our original conjecture in Item and revise it if necessar. How does the graph of g() differ from the graph of the parent function, f() =? The graph of f() = is shown below.. Make a conjecture about the graph of g() =. 5. Graph both f() and g() on a graphing calculator. Sketch the graph of g() on the grid above. Label a few points on each graph. 01 College Board. All rights reserved.. Revisit our original conjecture in Item about the graph of g() and revise it if necessar. How does the graph of g() differ from the graph of f()? 7. Epress regularit in repeated reasoning. How does the value of k in the equation g() = f() + k change the graph of f()? 11 SpringBoard Mathematics Algebra 1, Unit Functions

51 Eploring f() + k ACTIVITY A change in the position, size, or shape of a graph is a transformation. The changes to the graphs in Items 1 are eamples of a transformation called a vertical translation.. In the figure, the graphs of g() and h() are vertical translations of the graph of f() =. a. Write the equation for g(). g() f() = MATH TERMS A vertical translation of a graph shifts the graph up or down. A vertical translation preserves the shape of the graph. b. Write the equation for h(). h() Check Your Understanding 9. Without graphing, describe the transformation from the graph of f() = to the graph of g() = Suppose f() =. Describe the transformation from the graph of f() to the graph of g() = +. Use a graphing calculator to check our answer. Ra s Gm charges an initial sign-up fee of $5.00 and a monthl fee of $ Reason abstractl. Write a function that describes the gm s total membership fee for months. 01 College Board. All rights reserved. 1. Graph the function ou wrote in Item 11 on the grid below. Label several points on the graph. Membership Fee ($) Months 1. Identif the -intercept. What does the -intercept represent? Activit Transformations of Functions 11

52 ACTIVITY Lesson -1 Eploring f() + k 1. How would the function change if the initial sign-up fee were increased b $5.00? How would the graph change? Check Your Understanding 15. The membership fee at Gina s Gm is given b the function g() = , where is the number of months. a. How do the fees at Gina s Gm compare to those at Ra s Gm? b. Without graphing, describe how the graph of g() compares to the graph of f(). 1. The -intercept of a function f() is (0, b). What is the -intercept of f() + k? LESSON -1 PRACTICE Identif the transformation from the graph of f() = to the graph of g(). Then graph f() and g() on the same coordinate plane. 17. g() = 7 1. g() = + 10 Write the equation of the function described b each of the following transformations of the graph of f() =. 19. Translated up 9 units 0. Translated down 5 units Each graph shows a vertical translation of the graph of f() =. Write an equation to describe each graph Model with mathematics. Orange Tai charges $.75 as soon as ou step into the tai and $.50 per mile. Magenta Tai charges $.5 as soon as ou step into the tai and $.50 per mile. a. Write a function f() that describes the total cost of a ride of miles with Orange Tai. Write a function g() that describes the total cost of a ride of miles with Magenta Tai. b. Without graphing, eplain how the graph of g() compares to the graph of f(). c. Check our answer to Part (b) b graphing the functions. 01 College Board. All rights reserved. 11 SpringBoard Mathematics Algebra 1, Unit Functions

53 Lesson - Eploring f( + k) ACTIVITY Learning Targets: Identif the effect on the graph of replacing f() b f( + k). Identif the transformation used to produce one graph from another. SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Look for a Pattern, Create Representations, Think-Pair-Share, Discussion Groups The function f() = is graphed below. 10 MATH TERMS An absolute value function is written as f () = and is defined b if 0 f( ) = < if CONNECT TO AP The verte of an absolute value function is an eample of a cusp in a graph. A graph has a cusp at a point where there is an abrupt change in direction. 1. Write a new function, g(), b replacing with College Board. All rights reserved.. Graph both f() = and g() on a graphing calculator. Sketch the graph of g() on the grid above, labeling at least a few points on each graph.. What is the -intercept of f() =?. What is the -intercept of g()? 5. Describe the transformation from the graph of f() = to the graph of g(). Note that the function g() can be written as f( + 7). This means that is replaced with + 7 in the function f(). Activit Transformations of Functions 115

54 ACTIVITY Lesson - Eploring f( + k) The graph of f() = is shown below.. Make a conjecture about the graph of g() = ( ). 7. Graph both f() and g() on a graphing calculator. Sketch the graph of g() on the grid above, labeling at least a few points on each graph.. Revisit our original conjecture in Item about the graph of g() and revise it if necessar. How does the graph of g() differ from the graph of f()? MATH TERMS A horizontal translation of a graph shifts the graph left or right. Like a vertical translation, a horizontal translation preserves the shape of the graph. 9. How does the value of k in the equation g() = f( + k) change the graph of the function f()? The changes to the graphs in Items 1 are eamples of a transformation called a horizontal translation. 10. The figure shows the graph of the function f() =. a. Without using a graphing calculator, sketch the graph of g() = f( + ) = + on the grid College Board. All rights reserved. b. Use a graphing calculator to check our graph in Part (a). Revise our graph if necessar SpringBoard Mathematics Algebra 1, Unit Functions

55 Lesson - Eploring f( + k) ACTIVITY Check Your Understanding 11. Without graphing, describe the transformation from the graph of f() = to the graph of g(). a. g() = ( + ) b. g() = f( 7) c. g() = ( ) + 5 d. g() = ( + 9) 1 1. The function f() = and another function, g(), are graphed below. Write the equation for g(). Eplain how ou found our answer. f() = g() Make sense of problems. Julio went to a theme park in Jul. He paid $15 to enter the park and $.00 for each ride. He went on rides. a. Write a function that describes the total cost of Julio s trip to the theme park. 01 College Board. All rights reserved. b. Julio went back to the theme park in September. The entrance fee was the same and each ride still cost $.00. However, this time Julio went on 5 more rides. Use our function from Part (a) to describe Julio s second trip. c. How does the equation for Julio s second trip to the park change the graph of the first trip? d. What kind of transformation describes the change from the first graph to the second graph? e. Julio went to the park again in October and went on fewer rides than he did in Jul. Use our function from Part (a) to describe Julio s third trip. How does this change the initial graph? Activit Transformations of Functions 117

56 ACTIVITY Lesson - Eploring f( + k) f. Julio goes to the park again in November. Now it is the off-season and the entrance fee is $10 less than it was in Jul. He goes on the same number of rides as he did in Jul. Write a function to describe Julio s fourth trip. How does the graph of the initial trip change with this new situation? Check Your Understanding 1. The -intercept of the function f() is (a, 0). What is the -intercept of the function f( + k)? 15. Without graphing, eplain how the graph of = ( ) is related to the graph of = ( + ). LESSON - PRACTICE Identif the transformation from the graph of f() = to the graph of g(). Then graph f() and g() on the same coordinate plane. 1. g() = ( 1) 17. g() = ( + ) Write the equation of the function described b each of the following transformations of the graph of f() =. 1. Translated 7 units to the left 19. Translated units to the right 0. Each graph shows a horizontal translation of the graph of f() =. Write an equation to describe each graph. a. b. 01 College Board. All rights reserved. c. Critique the reasoning of others. Moll said that the graphs above are also vertical translations of the graph of f() =. Is Moll correct? Eplain. 1. How does the graph of h() = compare with the graph of f() =? 11 SpringBoard Mathematics Algebra 1, Unit Functions

57 Transformations of Functions Transformers ACTIVITY 01 College Board. All rights reserved. ACTIVITY PRACTICE Write our answers on notebook paper. Show our work. Lesson -1 In Items 1, identif the transformation from the graph of f() = to the graph of g(). 1. g() = g() =. g() = g() = + 5. The graph of f() = is translated 9 units down to create the graph of g(). Which of the following is the equation for g()? A. g() = + 9 B. g() = 9 C. g() = ( + 9) D. g() = ( 9) In Items and 7, each graph shows a vertical translation of the graph of f() =. Write an equation to describe the graph. Identif the zeros of each function For Items and 9, determine the equation of the function described b each of the following transformations of the graph of f() =.. Translated 15 units down 9. Translated.1 units up 10. An air conditioner costs $50 plus $0 per month to operate. a. Write a function that describes the total cost of buing and operating the air conditioner for months. b. Use our calculator to graph the function. c. What is the -intercept? What does it represent? d. How would the function change if the price of the air conditioner were reduced to $5? How would the graph change? Given that g() = f() + k, with k 0, determine whether each statement is alwas, sometimes, or never true. 11. The graph of g() is a vertical translation of the graph of f(). 1. The graphs of f() and g() are both lines. 1. The graph of f() has the same -intercept as the graph of g(). 1. Caitlin drew the graph of f() =. Then she translated the graph units up to get the graph of g(). Net, she translated the graph of g() units down to get the graph of h(). Which of these is an equation for h()? A. h() = + 1 B. h() = + C. h() = D. h() = Activit Transformations of Functions 119

58 ACTIVITY Transformations of Functions Transformers Lesson - In Items 15 1, identif the transformation from the graph of f() = to the graph of g(). 15. g() = ( ) 1. g() = 17. g() = + ( + ) 1. g() = 19. The graph of which function is a translation of the graph of f() = five units to the right? A. g() = 5 B. g() = ( + 5) C. g() = ( 5) D. g() = + 5 Write the equation of the function described b each of the following transformations of the graph of f() =. 0. Translated 7 units up 1. Translated units down. Translated units right. Translated 5 units down. Translated units left 5. The figure shows the graph of f() = and the graph of g(). Write an equation for the graph of g(). 10 f() 10 g() 10 Without graphing, describe the transformation from the graph of f() = to the graph of g().. g() = ( 7) g() = f( + ). g() = ( + 9) g() = f( ) 0. The graph of f() is shown below. Which of the following is a true statement about the graph of g() = f( + )? 10 A. The -intercept of g() is (, 0). B. The -intercept of g() is (, 0). C. The -intercept of g() is (0, ). D. The -intercept of g() is (0, ). MATHEMATICAL PRACTICES Model with Mathematics In 011, the ticket price for entrance to a state fair was $1. Each ride had an additional $.00 fee. In 01, the entrance ticket cost $15 and the rides remained $.00 each. a. Write a function f() for the cost of visiting the fair and riding rides in 011. b. Write a function g() for the cost of visiting the fair and riding rides in 01. c. What transformation could ou use to obtain the graph of g() from the graph of f()? d. What transformation could ou use to obtain the graph of f() from the graph of g()? f() College Board. All rights reserved SpringBoard Mathematics Algebra 1, Unit Functions

59 Representations of Functions BRYCE CANYON HIKING Embedded Assessment 1 Use after Activit While on vacation, Jorge and Jackie traveled to Brce Canon National Park in Utah. The were impressed b the differing elevations at the viewpoints along the road. The graph describes the elevations for several viewpoints in terms of the time since the entered the park. 1. The graph represents a function E(t). Describe wh the graph represents a function. Identif the domain and range of the function.. Is this discrete or continuous data? Eplain.. What is the -intercept? Interpret the meaning of the -intercept in the contet of the problem.. Identif a relative maimum of the function represented b the graph. 5. What is the absolute maimum of the function represented b the graph? What does it represent?. Identif a relative minimum of the function represented b the graph. 7. What is the absolute minimum of the function represented b the graph? What does it represent? While at Brce Canon National Park, Jorge and Jackie hiked at an average speed of about miles per hour.. Cop and complete the table below to show the distance hiked b a person whose constant speed is miles per hour. Elevation (ft) E(t) Time After Entering the Park (min) t Time (hours) Distance (miles) 01 College Board. All rights reserved Write a function f() to describe the data in the table. What are the reasonable domain and range? 10. Create a graph of the function. 11. How long will it take this person to hike 5 miles? Justif our answer. 1. On the same coordinate grid that ou used in Item 9, create a graph of another function b translating the graph 5 units up. 1. Write a function to describe the graph ou created in Item 1. Eplain how ou determined our answer. Unit Functions 11

60 Embedded Assessment 1 Use after Activit Representations of Functions BRYCE CANYON HIKING Scoring Guide Mathematics Knowledge and Thinking (Items 1, 7) Eemplar Proficient Emerging Incomplete The solution demonstrates the following characteristics: Clear and accurate identification of ke features of the function and its graph, including domain, range, -intercept, maimums, and minimums Correct identification of most of the ke features of the function and its graph, including domain, range, -intercept, maimums, and minimums Partiall correct identification of some of the ke features of the function and its graph, including domain, range, -intercept, maimums, and minimums Inaccurate or incomplete identification of ke features of the function and its graph, including domain, range, -intercept, maimums, and minimums Problem Solving (Item 11) Appropriate and efficient strateg that results in a correct answer Strateg that ma include unnecessar steps but results in a correct answer Strateg that results in some incorrect answers No clear strateg when solving problems Mathematical Modeling / Representations (Items 10, 1, 1) Effective understanding of how to complete a table of real-world data, and how to write, graph, and interpret the associated function Fluenc in translating a graph and writing the associated function Largel correct understanding of how to complete a table of real-world data, and how to write, graph, and interpret the associated function Little difficult translating a graph and writing the associated function Partial understanding of how to complete a table of real-world data, and how to write, graph, and interpret the associated function Some difficult translating a graph and writing the associated function Inaccurate or incomplete understanding of how to complete a table of real-world data, and how to write, graph, and interpret the associated function Significant difficult translating a graph and writing the associated function Reasoning and Communication (Items 1, 5, 7, 1) Precise use of appropriate math terms and language to describe ke features of a graph and to eplain how a function rule was determined from a translated graph Clear and accurate interpretations of the graph of a function Adequate description of ke features of a graph Reasonable interpretations of the graph of a function Adequate eplanation of how a function rule was determined from a translated graph Confusing description of ke features of a graph Partiall correct interpretations of the graph of a function Confusing eplanation of how a function was determined from a translated graph Incomplete or inaccurate description of ke features of a graph Incomplete or inaccurate interpretation of the graph of a function Incomplete or inaccurate eplanation of how a function was determined from a translated graph 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1

61 Rates of Change Ramp it Up Lesson 9-1 Slope Learning Targets: Determine the slope of a line from a graph. Develop and use the formula for slope. SUGGESTED LEARNING STRATEGIES: Close Reading, Summarizing, Sharing and Responding, Discussion Groups, Construct an Argument, Identif a Subtask ACTIVITY 9 Margo s grandparents are moving in with her famil. The famil needs to make it easier for her grandparents to get in and out of the house. Margo has researched the specifications for building stairs and wheelchair ramps. She found the government website that gives the Americans with Disabilities Act (ADA) accessibilit guidelines for wheelchair ramps and discovered the following diagram: Rise Level Landing Surface of Ramp Horizontal Projection or Run Level Landing Then, Margo decided to look for the requirements for building stairs and found the following diagram: Total Run CONNECT TO SOCIAL SCIENCE The table gives information from the ADA website about the slope of wheelchair ramps. Slope Maimum Rise Maimum Run in. mm ft m 1 < m m < College Board. All rights reserved. Total Rise Tread Riser Stringer Review with our group the background information that is given as ou solve the following items. 1. What do ou think is meant b the terms rise and run in this contet? Activit 9 Rates of Change 1

62 ACTIVITY 9 Lesson 9-1 Slope Consider the line in the graph below: 10 Horizontal Change C D Vertical Change A B 10 Vertical change can be represented as a change in, and horizontal change can be represented b a change in.. What is the vertical change between: a. points A and B? b. points A and C? c. points C and D?. What is the horizontal change between: a. points A and B? b. points A and C? c. points C and D? MATH TERMS Slope is a measure of the amount of decline or incline of a line. The variable m is often used to represent slope. The ratio of the vertical change to the horizontal change determines the slope of the line. vertical change change in slope = = = horizontal change change in. Find the slope of the segment of the line connecting: a. points A and B b. points A and C c. points C and D 01 College Board. All rights reserved. 5. What do ou notice about the slope of the line in Items a, b, and c?. What does our answer to Item 5 indicate about points on a line? 1 SpringBoard Mathematics Algebra 1, Unit Functions

63 Lesson 9-1 Slope ACTIVITY 9 7. Slope is sometimes referred to as run rise. Eplain how the ratio rise run relates to the ratios for finding slope mentioned above.. Reason quantitativel. Would the slope change if ou counted the run (horizontal change) before ou counted the rise (vertical change)? Eplain our reasoning. WRITING MATH In mathematics the Greek letter (delta) represents a change or difference between mathematical values. 9. Determine the slope of the line graphed below. 10 MATH TIP Select two points on the line and use them to compute the slope. 01 College Board. All rights reserved Activit 9 Rates of Change 15

64 ACTIVITY 9 Lesson 9-1 Slope Although the slope of a line can be calculated b looking at a graph and counting the vertical and horizontal change, it can also be calculated numericall. change in 10. Recall that the slope of a line is the ratio change in. a. Identif two points on the graph above and record the coordinates of the two points that ou selected. -coordinate -coordinate 1st point nd point b. Which coordinates relate to the vertical change on the graph? c. Which coordinates relate to the horizontal change on a graph? d. Determine the vertical change. e. Determine the horizontal change. f. Calculate the slope of the line. How does this slope compare to the slope that ou found in Item 9? g. If other students in our class selected different points for this problem, should the have found different values for the slope of this line? Eplain. 11. It is customar to label the coordinates of the first point ( 1, 1 ) and the coordinates of the second point (, ). a. Write an epression to calculate the vertical change,, of the line through these two points. 01 College Board. All rights reserved. b. Write an epression to calculate the horizontal change,, of the line through these two points. c. Write an epression to calculate the slope of the line through these two points. 1 SpringBoard Mathematics Algebra 1, Unit Functions

65 Lesson 9-1 Slope ACTIVITY 9 Check Your Understanding 1. Use the slope formula to determine the slope of a line that passes through the points (, 9) and (, ). 1. Use the slope formula to determine the slope of the line that passes through the points ( 5, ) and (9, 10). 1. Eplain how to find the slope of a line from a graph. 15. Eplain how to find the slope of a line when given two points on the line. LESSON 9-1 PRACTICE 1. Find and for the points (7, ) and (9, 7). 17. Critique the reasoning of others. Connor determines the slope between (, ) and (, ) b calculating ( ). April determines the slope b calculating ( ). Eplain whose reasoning is correct. 1. When given a table of ordered pairs, ou can find the slope b choosing an two ordered pairs from the table. Determine the slope represented in the table below College Board. All rights reserved. 19. Determine the slope of the given line Activit 9 Rates of Change 17

66 ACTIVITY 9 Lesson 9- Slope and Rate of Change Learning Targets: Calculate and interpret the rate of change for a function. Understand the connection between rate of change and slope. SUGGESTED LEARNING STRATEGIES: Discussion Groups, Create Representations, Look for a Pattern, Think-Pair-Share The rate of change for a function is the ratio of the change in, the dependent variable, to the change in, the independent variable. 1. Margo went to the lumberard to bu supplies to build the wheelchair ramp. She knows that she will need several pieces of wood. Each piece of wood costs $. a. Model with mathematics. Write a function f() for the total cost of the wood pieces if Margo bus pieces of wood. b. Make an input/output table of ordered pairs and then graph the function. Pieces of Wood, Total Cost, f() Total Cost (in dollars) c. What is the slope of the line that ou graphed? Total Cost of Wood f () d. B how much does the cost increase for each additional piece of wood purchased? Pieces of Wood 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

67 Lesson 9- Slope and Rate of Change ACTIVITY 9 e. How does the slope of this line relate to the situation with the pieces of wood? f. Is there a relationship between the slope of the line and the equation of the line? If so, describe that relationship.. Margo is going to work with a local carpenter during the summer. Each week she will earn $10.00 plus $.00 per hour. a. Write a function f() for Margo s total earnings if she works hours in one week. b. Make an input/output table of ordered pairs and then graph the function. Label our aes. 01 College Board. All rights reserved. Hours, Earnings, f() (dollars) f () Activit 9 Rates of Change 19

68 ACTIVITY 9 Lesson 9- Slope and Rate of Change c. How much will Margo s earnings change if she works hours instead of? If she works hours instead of? How much do Margo s earnings change for each additional hour worked? d. Does the function have a constant rate of change? If so, what is it? e. What is the slope of the line that ou graphed? f. Describe the meaning of the slope within the contet of Margo s job. g. Describe the relationship between the slope of the line, the rate of change, and the equation of the line. h. How much will Margo earn if she works for hours in one week?. B the end of the summer, Margo has saved $75. Recall that each of the small pieces of wood costs $. a. Write a function f() for the amount of mone that Margo still has if she bus pieces of wood. 01 College Board. All rights reserved. 10 SpringBoard Mathematics Algebra 1, Unit Functions

69 Lesson 9- Slope and Rate of Change ACTIVITY 9 b. Make an input/output table of ordered pairs and then graph the function. Pieces of Wood, Mone Remaining, f() (dollars) Mone Remaining ( in dollars) Margo s Savings f () Pieces of Wood c. How much will the amount Margo has saved change if she bus 100 instead of 5 pieces of wood? If she bus 50 instead of 0 pieces of wood? For each additional piece of wood? Eplain. d. Does the function have a constant rate of change? If so, what is it? 01 College Board. All rights reserved. e. What is the slope of the line that ou graphed? f. How are the rate of change of the function and the slope related? g. Describe the meaning of the slope within the contet of Margo s savings. h. How does this slope differ from the other slopes that ou have seen in this activit? Activit 9 Rates of Change 11

70 ACTIVITY 9 Lesson 9- Slope and Rate of Change Check Your Understanding. The constant rate of change of a function is 5. Describe the graph of the function as ou look at it from left to right. 5. Does the table represent data with a constant rate of change? Justif our answer LESSON 9- PRACTICE. The art museum charges an initial membership fee of $ For each visit the museum charges $ a. Write a function f() for the total amount charged for trips to the museum. b. Make a table of ordered pairs and then graph the function. c. What is the rate of change? What is the slope of the line? d. How does the slope of this line relate to the number of museum visits? 7. Critique the reasoning of others. Simone claims that the slope of the line through (, 7) and (, 0) is the same as the slope of the line through (, 1) and (1, 1). Prove or disprove Simone s claim. 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

71 Lesson 9- More About Slopes ACTIVITY 9 Learning Targets: Show that a linear function has a constant rate of change. Understand when the slope of a line is positive, negative, zero, or undefined. Identif functions that do not have a constant rate of change and understand that these functions are not linear. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think- Pair-Share, Construct an Argument, Sharing and Responding, Summarizing You have seen that for a linear function, the rate of change is constant and equal to the slope of the line. This is because linear functions increase or decrease b equal differences over equal intervals. Look at the graph below (, ) (, 5) (, 7) (10, ) Equal Intervals Equal Intervals 1. Over the interval to, b how much does the function increase? Eplain. 01 College Board. All rights reserved.. Over the equal interval to 10, b how much does the function increase? Eplain. Equal differences over equal intervals is an equivalent wa of referring to constant slope. Differences refers to, and intervals refers to. Equal differences over equal intervals means, which represents the slope, will alwas be the same. Activit 9 Rates of Change 1

72 ACTIVITY 9 Lesson 9- More About Slopes. The table below represents a function a. Determine the rate of change between the points (, ) and (, ). b. Determine the rate of change between the points ( 1, 1) and (1, 1). c. Construct viable arguments. Is this a linear function? Justif our answer.. a. Determine the slopes of the lines shown. b. Epress regularit in repeated reasoning. Describe the slope of an line that rises as ou view it from left to right. 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

73 Lesson 9- More About Slopes ACTIVITY 9 5. a. Determine the slopes of the lines shown. b. Epress regularit in repeated reasoning. Describe the slope of an line that falls as ou view it from left to right.. a. Determine the slopes of the lines below. b. What is the slope of a horizontal line? 01 College Board. All rights reserved. 7. a. Determine the slopes of the lines shown. b. What is the slope of a vertical line? Activit 9 Rates of Change 15

74 ACTIVITY 9 Lesson 9- More About Slopes. Summarize our findings in Items 7. Tell whether the slopes of the lines described in the table below are positive, negative, 0, or undefined. Up from left to right Down from left to right Horizontal Vertical Check Your Understanding 9. Suppose ou are given several points on the graph of a function. Without graphing, how could ou determine whether the function is linear? 10. How can ou tell from a graph if the slope of a line is positive or negative? 11. Describe a line having an undefined slope. Wh is the slope undefined? LESSON 9- PRACTICE 1. Make use of structure. Sketch a line for each description. a. The line has a positive slope. b. The line has a negative slope. c. The line has a slope of Does the table represent a linear function? Justif our answer Are the points (1, 11), (, 7), (5, 9), and (1, 5) part of the same linear function? Eplain. 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

75 Rates of Change Ramp it Up ACTIVITY 9 01 College Board. All rights reserved. ACTIVITY 9 PRACTICE Write our answers on notebook paper. Show our work. Lesson Find and for each of the following pairs of points. a. (, ), (, ) b. (0, 9), (, ) c. (, ), (7, 10) For Items and, use the table to calculate the slope Two points on a line are ( 10, 1) and (5, 5). If the -coordinate of another point on the line is, what is the -coordinate? For Items 5 7, determine the slope of the line that passes through each pair of points. 5. (, 11) and (1, 9). ( 10, ) and ( 5, 1) 7. (, 7) and (, ). Are the three points (, ), (5, ), and (0, ) on the same line? Eplain. 9. Which of the following pairs of points lies on a line with a slope of 5? A. (, 0), (, 10) B. (, ), (10, ) C. (, 10), (0, ) D. (10, ), (0, ) For Item 10, determine the slope of the line that is graphed Lesson Juan earns $7 per hour plus $0 per week making picture frames. a. Write a function g() for Juan s total earnings if he works hours in one week. b. Without graphing the function, determine the slope. c. Describe the meaning of the slope within the contet of Juan s job. 1. The graph shows the height of an airplane as it descends to land. Altitude (ft) a. Does the function have a constant rate of change? If so, what is it? b. What is the slope of the line? c. How are the rate of change and the slope of the line related? d. Describe the meaning of the slope within the contet of the situation. 1 Time (min) 10 5 Activit 9 Rates of Change 17

76 ACTIVITY 9 Rates of Change Ramp it Up Lesson 9- For Items 1 15, tell whether the function is linear. Justif our response Which of the following is not a linear function? A. (, ), (7, 1), ( 1), (10, 1), (, ) B. (, ), (1, 0), (, 0), (0, ), (7, 9) C. (, 9), (0, 7), (, ), (, ), (, ) D. (, 1), (, 50), (, ), (0, ), (, ) For Items 1 and 19, identif the slope of the line in each graph as positive, negative, 0, or undefined One point on the line described b = + is shown below. Use our knowledge of slope to give the coordinates of three more points on the line The slope of a line is 0. It passes through the point (, ). Identif two other points on the line. Justif our answers. MATHEMATICAL PRACTICES Look For and Make Use of Structure 1. Describe three different was to determine the slope of a line and the similarities and differences between the methods. 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

77 Linear Models Stacking Boes Lesson 10-1 Direct Variation ACTIVITY 10 Learning Targets: Write and graph direct variation. Identif the constant of variation. SUGGESTED LEARNING STRATEGIES: Create Representations, Interactive Word Wall, Marking the Tet, Sharing and Responding, Discussion Groups You work for a packaging and shipping compan. As part of our job there, ou are part of a package design team deciding how to stack boes for packaging and shipping. Each bo is 10 cm high. 10 cm 1. Complete the table and make a graph of the data points (number of boes, height of the stack). 01 College Board. All rights reserved. Number of Boes Height of the Stack (cm) Height of Stack Stacking Boes Number of Boes. Write a function to represent the data in the table and graph above. WRITING MATH Either or f() can be used to represent the output of a function.. What is a reasonable and realistic domain for the function? Eplain.. What is a reasonable and realistic range for the function? Eplain. Activit 10 Linear Models 19

78 ACTIVITY 10 Lesson 10-1 Direct Variation 5. What do f(), or, and represent in our equation from Item?. Describe an patterns that ou notice in the table and graph representing our function. MATH TERMS A direct proportion is a relationship in which the ratio of one quantit to another remains constant. 7. The number of boes is directl proportional to the height of the stack. Use a proportion to determine the height of a stack of 1 boes. When two values are directl proportional, there is a direct variation. In terms of stacking boes, the height of the stack varies directl as the number of boes.. Using variables and to represent the two values, ou can sa that varies directl as. Use our answer to Item to eplain this statement. 9. Direct variation is defined as = k, where k 0 and the coefficient k is the constant of variation. a. Consider our answer to Item. What is the constant of variation in our function? b. Wh do ou think the coefficient is called the constant of variation? c. Reason quantitativel. Eplain wh the value of k cannot be equal to College Board. All rights reserved. d. Write an equation for finding the constant of variation b solving the equation = k for k. 10 SpringBoard Mathematics Algebra 1, Unit Functions

79 Lesson 10-1 Direct Variation ACTIVITY a. Interpret the meaning of the point (0, 0) in our table and graph. b. True or False? Eplain our answer. The graphs of all direct variations are lines that pass through the point (0, 0). c. Identif the slope and -intercept in the graph of the stacking boes. d. Describe the relationship between the constant of variation and the slope. Direct variation can be used to answer questions about stacking and shipping our boes. 11. The height of a different stack of boes varies directl as the number of boes. For this tpe of bo, 5 boes are 500 cm high. a. Find the value of k. Eplain how ou found our answer. 01 College Board. All rights reserved. b. Write a direct variation equation that relates, the height of the stack, to, the number of boes in the stack. c. How high is a stack of 0 boes? Eplain how ou would use our direct variation equation to find the height of the stack. Activit 10 Linear Models 11

80 ACTIVITY 10 Lesson 10-1 Direct Variation 1. At the packaging and shipping compan, ou get paid each week. One week ou earned $ for hours of work. Another week ou earned $0 for 5 hours of work. a. Write a direct variation equation that relates our wages to the number of hours ou worked each week. Eplain the meaning of each variable and identif the constant of variation. b. How much would ou earn if ou worked.5 hours in one week? Check Your Understanding 1. Tell whether the tables, graphs, and equations below represent direct variations. Justif our answers. a. b c. d College Board. All rights reserved. e. = 0 f. = + 1 SpringBoard Mathematics Algebra 1, Unit Functions

81 Lesson 10-1 Direct Variation ACTIVITY 10 LESSON 10-1 PRACTICE 1. In the equation = 15, what is the constant of variation? 15. In the equation =, what is the constant of variation? 1. The value of varies directl with and the constant of variation is 7. What is the value of when =? 17. The value of varies directl with and the constant of variation is 1. What is the value of when = 5? 1. Model with mathematics. The height of a stack of boes varies directl with the number of boes. A stack of 1 boes is 15 feet high. How tall is a stack of 1 boes? 19. Jan s pa is in direct variation to the hours she works. Jan earns $5 for 1 hours of work. How much will she earn for 1 hours work? 01 College Board. All rights reserved. Activit 10 Linear Models 1

82 ACTIVITY 10 Lesson 10- Indirect Variation Learning Targets: Write and graph indirect variations. Distinguish between direct and indirect variation. SUGGESTED LEARNING STRATEGIES: Create Representations, Marking the Tet, Sharing and Responding, Think-Pair-Share, Discussion Groups When packaging a different product, our team at the packaging and shipping compan determines that all boes for this product will have a volume of 00 cubic inches and a height of 10 inches. The lengths and the widths will var. MATH TIP The volume of a rectangular prism is found b multipling length, width, and height: V = lwh. 10 in. 10 in. 1. To eplore the relationship between length and width, complete the table and make a graph of the points. Width () Length () Length Bo Dimensions Width. How are the lengths and widths in Item 1 related? Write an equation that shows this relationship.. Use the equation ou wrote in Item to write a function to represent the data in the table and graph above. 01 College Board. All rights reserved.. Describe an patterns that ou notice in the table and graph representing our function. 1 SpringBoard Mathematics Algebra 1, Unit Functions

83 Lesson 10- Indirect Variation ACTIVITY 10 In terms of bo dimensions, the length of the bo varies indirectl as the width of the bo. Therefore, this function is called an indirect variation. 5. Recall that direct variation is defined as = k, where k 0 and the coefficient k is the constant of variation. a. How would ou define indirect variation in terms of, k, and? MATH TIP Indirect variation is also known as inverse variation. b. Are there an limitations on these variables as there are on k in direct variation? Eplain. c. Write an equation for finding the constant of variation b solving for k in our answer to Part (a).. Reason abstractl. Compare and contrast the equations of direct and indirect variation. 01 College Board. All rights reserved. 7. Compare and contrast the graphs of direct and indirect variation.. Use our function in Item to determine the following measurements for our compan. a. Find the length of a bo whose width is 0 inches. b. Find the length of a bo whose width is 0. inches. Activit 10 Linear Models 15

84 ACTIVITY 10 Lesson 10- Indirect Variation 9. The time,, needed to load the boes on a truck for shipping varies indirectl as the number of people,, working. If 10 people work, the job is completed in 0 hours. a. Eplain how to find the constant of variation. Then find it. b. Write an indirect variation equation that relates the time to load the boes to the number of people working. c. How long does it take people to finish loading the boes? Use our equation to answer this question. d. On the grid below, make a graph to show the time needed for,, 5,, 10, and 5 people to load the boes on the truck. Time Needed (h) Number of People 10. The cost for the compan to ship the boes varies indirectl with the number of boes being shipped. If 5 boes are shipped at once, it will cost $10 per bo. If 50 boes are shipped at once, the cost will be $5 per bo. a. Write an indirect variation equation that relates the cost per bo to the number of boes being shipped. b. How much would it cost to ship onl 10 boes? 01 College Board. All rights reserved. 11. Is an indirect variation function a linear function? Eplain. 1 SpringBoard Mathematics Algebra 1, Unit Functions

85 Lesson 10- Indirect Variation ACTIVITY 10 Check Your Understanding 1. Identif the following graphs as direct variation, indirect variation, neither, or both. a. b c. 01 College Board. All rights reserved. 1. Which equations are eamples of indirect variation? Justif our answers. A. = B. = C. = D. = 1. In the equation = 0, what is the constant of variation? LESSON 10- PRACTICE 15. Graph each function. Identif whether the function is an indirect variation. a. 1 1 b Make sense of problems. For Parts (a) and (b) below, varies indirectl as. a. If = when =, find when = 1. b. If = when = 0, find the value of k. Activit 10 Linear Models 17

86 ACTIVITY 10 Lesson 10- Another Linear Model CONNECT TO GEOMETRY The carton will be a right rectangular prism. A rectangular prism is a closed, three-dimensional figure with three pairs of opposite parallel faces that are congruent rectangles. Learning Targets: Write, graph, and analze a linear model for a real-world situation. Interpret aspects of a model in terms of the real-world situation. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Discussion Groups, Create Representations, Guess and Check, Use Manipulatives Your design team at the packaging and shipping compan has been asked to design a cardboard bo to use when packaging paper cups for sale. Your supervisor has given ou the following requirements. All lateral faces of the container must be rectangular. The base of the container must be a square, just large enough to accommodate one cup. The height of the container must be given as a function of the number of cups the container will hold. All measurements must be in centimeters. To help discover which features of the cup affect the height of the stack, collect data on two tpes of cups found around the office. 1. Use appropriate tools strategicall. Use two different tpes of cups to complete the tables below. Number of Cups 1 5 CUP 1 Height of Stack Number of Cups 1 5 CUP Height of Stack. Epress regularit in repeated reasoning. What patterns do ou notice that might help ou figure out the relationship between the height of the stack and the number of cups in that stack? 01 College Board. All rights reserved. 1 SpringBoard Mathematics Algebra 1, Unit Functions

87 Lesson 10- Another Linear Model ACTIVITY 10 Use our data for Cup 1 to complete Items 1.. Make a graph of the data ou collected. h Cup 1 Stack Height (cm) Number of Cups n. Predict, without measuring, the height of a stack of 1 cups. Eplain how ou arrived at our prediction. 01 College Board. All rights reserved. 5. Predict, without measuring, the height of a stack of 50 cups. Eplain how ou arrived at our prediction.. Write an equation that gives the height of a stack of cups, h, in terms of n, the number of cups in the stack. 7. Use our equation from Item to find h when n = 1 and when n = 50. Do our answers to this question agree with our predictions in Items and 5? Activit 10 Linear Models 19

88 ACTIVITY 10 Lesson 10- Another Linear Model. Sketch the graph of our equation from Item. h Cup 1 Stack Height (cm) Number of Cups 9. How are the graphs ou made in Items and the same? How are the different? n 10. Do the graphs in Items and represent direct variation, indirect variation, or neither? Eplain. 11. Remember that ou are designing a container with a square base. What dimension(s), other than the height of the stack, do ou need to design our cup container? Use Cup 1 to find this/these dimension(s). 01 College Board. All rights reserved. 1. Find the dimensions of a container that will hold a stack of 5 cups. 150 SpringBoard Mathematics Algebra 1, Unit Functions

89 Lesson 10- Another Linear Model ACTIVITY Your team has been asked to communicate its findings to our supervisor. Write a report to her that summarizes our findings about the cup container design. Include the following information in our report. The equation our team discovered to find the height of the stack of Cup 1 stle cups A description of how our team discovered the equation and the minimum number of cups needed to find it An eplanation of how the numbers in the equation relate to the phsical features of the cup An equation that could be used to find the height of the stack of Cup stle cups MATH TIP When writing our answer to Item 1, ou can use a RAFT. Role team leader Audience our boss Format a letter Topic stacks of cups Check Your Understanding 1. A group of students performed the cup activit described in this lesson. For their Cup 1, the found the equation h = 0.5n +.5, where h is the height in inches of a stack of cups and n is the number of cups. a. What would be the height of 5 cups? Of 50 cups? b. Graph this equation. Describe our graph. LESSON 10- PRACTICE 15. Reason quantitativel. A group of students performed the cup activit in this lesson using plastic drinking cups. Their data is shown below. 01 College Board. All rights reserved. Number of Cups CUP 1 Height of Stack cm 1 cm 17.5 cm 19 cm cm Number of Cups CUP Height of Stack cm cm 1 cm 1.5 cm cm For each cup, write and graph an equation. Describe our graphs. 1. A consultant earns a flat fee of $75 plus $50 per hour for a contracted job. The table shows the consultant s earnings for the first four hours she works. Hours 0 1 Earnings $75 $15 $175 $5 $75 The consultant has a -hour contract. How much will she earn? Activit 10 Linear Models 151