In Lesson 7.1, you learned that you can write rules for some of the coding grids. y x 1

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1 LEON 7. PLANNING LEON OUTLINE One da: 0 min Investigation min haring min Eample min min ATERIAL Closing Eercises Function or Not? (T), optional Calculator Note 1J LEON 7. Functions and Graphs In Lesson 7.1, ou learned that ou can write rules for some of the coding grids. You can also write rules, often in the form of equations, to transform numbers into other numbers. One simple eample is Add one to each number. You can represent this rule with a table, an equation, a graph, or even a diagram. Table Equation Graph Diagram 1 Domain Range This rule turns 7 into 8, 7 into 6,.8 into 11.8, and into 1. When ou eplored relations in previous chapters, ou used recursive routines, graphs, and equations to relate input and output data. To tell whether a relationship between input and output data is a function, there is a test that ou can appl to the relation s graph on the -plane. TEACHING The vertical line test is one method to determine whether agraph represents a function. Continue to use and insist that students use complete sentences when writing up the mathematics. One tep how the Function or Not? transparenc. Ask students which graphs represent functions, reviewing the fact that a function has onl one output value for ever input value. Then ask what the graphs of functions have in common that distinguish them from graphs of nonfunctions. As students think together, encourage them as necessar to use straightedges to help eplain their ideas to each other. The panish painter Pablo Picasso ( ) was one of the originators of the art movement Cubism. Cubists were interested in creating a new visual language, translating realism into a different wa of seeing. LEON OBJECTIVE This painting is titled Guitare et Journal. Learn a definition of function Learn about properties and geometric representations of functions NCT TANDARD CONTENT Number Algebra Geometr easurement Data/Probabilit PROCE Problem olving Reasoning Communication Connections Representation 396 CHAPTER 7 Functions

2 tep 1 Tables 1 tep 1 and 3 represent functions because each -value has onl one -value paired with it. Tables and do not represent functions because an -value can have multiple -values, for eample, (, ) and (, ) in Table and (1, 1) and (1, 3) in Table. Investigation Testing for Functions In this investigation ou will use various kinds of evidence to determine whether relations are functions. Each table represents a relation. Based on the tables, which relations are functions and which are not? Givereasonsforouranswers. 1 Table 1 Table Table 3 Table Guiding the Investigation tep tatements 1 and 3 represent functions because each -value produces a unique -value. tatements and do not represent functions because there are -values that result in more than one -value. tep 3 You might have students sketch the graphs and label two points that demonstrate that Graphs and are not functions. For eample, if students choose the points (, ) and (, ) for Graph, the can see that the vertical line will intersect the graph at both points. tep tep 3 Graphs 1 tep 3 and 3 represent functions because a vertical line never intersects the curve more than once. Graphs and do not represent functions because a vertical line can contact more than one point on the graph. Each algebraic statement below represents a relation. Based on the equations, which relations are functions and which are not? Give reasons for our answers. tatement 1 tatement tatement 3 tatement Each graph below represents arelation.oveaverticalline,suchastheedgeof aruler,fromsidetoside on thegraph.based on the graph and our vertical line, which relations are functions and which are not? Give reasons for our answers. Graph 1 Graph Graph 3 tep If a graph tep Use our results in tep 3 to write a rule eplaining how ou can determine represents a function, whether a relation is a function, based onl on its graph. then a vertical line will never intersect the graph at more than one point. Graph A function is a relation between input and output values. Each input has eactl one output. The vertical line test helps ou determine if a relation is afunction.ifall possible vertical lines cross the graph once or not at all, then the graph represents a function. The graph does not represent a function if ou can draw even one vertical line that crosses the graph two or more times. HARING IDEA Ask students to share their ideas on how to tell whether a graph represents a function. Connect the vertical line test to the definition of function: For each input value, there s onl one output value. [Ask] What does the function 1 do to numbers like 7? [adds 1] What does it do to t? [t 1] To a? [a 6] [Ask] Is the inequalit a function? [ome students ma think of the graph on a number line, which is a horizontal line and passes the vertical line test. Remind students that we re dealing with two variables and that the two-variable equivalent of is 0. The graph of this inequalit is a half-plane, which does not pass the vertical line test.] Assessing Progress You can assess students abilit to create a table, graph, or equation representation of a function from one of the other forms and to plot points and graph functions on a calculator. LEON 7. Functions and Graphs 397

3 Function Not a function Function Not a function Function You have learned man forms of linear equations. In the eample ou will see whether all lines represent functions. EXAPLE Part b Remind students that the slope-intercept form of a linear equation is similar to the intercept form, but the constant term appears last. Writing the equation 9 as 0 9 emphasizes that no matter what value is substituted for, the first term will be 0. o, for the input value of 9, there are infinitel man output values. et an eample for raising thoughtful questions. Other good questions here are What about horizontal lines? Are the graphs of functions? EXAPLE olution Name the form of each linear equation or inequalit, and use a graph to eplain wh it is or is not a function. a. 1 3 b. 0. c. 3 d. 3 6 e. ( 8) f. 7 g. 9 h. 1 Each equation is written in one of the forms ou have learned in this course. If ou graph the equations, ou can see that all of them ecept the graphs for parts g and h pass the vertical line test. o all the equations represent functions ecept for the ones in parts g and h. a. This equation is in the intercept b. This equation is in the slopeform a b. intercept form m b. c. This equation is a direct variation d. This equation is in the standard in the form k. form a b c. 398 CHAPTER 7 Functions

4 e. This equation is in the point-slope f. This equation is a horizontal line form 1 b 1. in the form k. g. This equation is a vertical line in h. The boundar of this inequalit, the form k. 1, is in the standard form a b c. Closing the Lesson A graph using rectangular coordinates represents a function if and onl if no vertical line through the graph intersects it at more than one point. [Link] In later courses students will encounter graphs using polar coordinates where functional relationships do not pass the vertical line test. Continue to emphasize that a function is a relationship for which ever input produces one output. The graphs of 9 and 1 fail the vertical line test. In both cases ou can match infinitel man output values of to a single input value of. o, 9 and 1 do not represent functions. In fact, graphs of all vertical lines and linear inequalities fail the vertical line test, and are therefore not functions. All nonvertical lines are functions. As ou work more with functions, ou will be able to tell if a relationship is a function without having to consider its graph on the -plane. If the graph is shown, use the vertical line test. Otherwise, see if there is more than one output value for an single input value. Carpenters use a tool called a level to determine if support beams are trul vertical. LEON 7. Functions and Graphs 399

5 BUILDING UNDERTANDING tudents practice determining whether graphs represent functions. Remind students to use complete sentences when writing out their assignment. AIGNING HOEWORK Essential 1, 6, 11 Performance assessment 9,, 1, 1 Portfolio Journal 6, 8, 1 Group 8, 13 Review 17 Helping with the Eercises EXERCIE Practice Your kills 1. Use the equations to find the missing entries in each table. a b a You will need our graphing calculator for Eercise. Domain Range. On the same set of aes, plot the points in the table and graph the equation in Eercise 1a. 3. On the same set of aes, plot the points in the table and graph the equation in Eercise 1b.. Use the tables and graphs in Eercises 1 3 to tell whether the relationships in Eercise 1 are functions. a Answers will var. In the table, ever input value produces eactl one output value. Both graphs in Eercises and 3 pass the vertical line test. Both rules are functions Reason and Appl The graph at right describes another student s distance from ou. What are the walking instructions for the graph? Does it represent a function? a 6. Find whether each graph below represents a function. Does it pass the vertical line test? a. es b. Distance (m) Time (s) 6 8 Eercise This eercise depends on the previous three eercises. Be sure students justif their answers. Eercise [Ask] Will ever set of walking instructions represent a function? c. d. es e. No; each input value has infinitel man output values.. ample answer: tart at the m mark and stand still for s. Walk toward the m mark at m/s for 1s.tand still for another second. Walk toward the 8m mark at m/s for 1 s. Then stand still for 3 s. Yes, the graph represents a function. 6b. No; man input values have two different output values. 6c. No; there is a vertical segment. All the points on the vertical segment have the same input value but different output values. 00 CHAPTER 7 Functions

6 7. Does each relationship in the form (input, output) represent a function? If the relationship does not represent a function, find an eample of one input that has two or more outputs. This is called a countereample. a. (cit, ZIP Code) b. (person, birth date) c. (last name, first name) a d. (state, capital) Yes; each state has onl one capital. 8. Here are the graphs of seven walks showing distance from a motion sensor. i. ii. iii. iv. v. vi. Eercise 7 [Language] ake sure students understand that a countereample is an eample that proves a statement false. You might give another eample. [Ask] Is the relation (musical note name, pitch) a function? [It depends. If several different notes are all called C, then the note name will have man different pitches, so it won t be a function. If the different C s have different names, like C and C,then it will be a function.] You can also etend this eercise and preview the Take Another Look activit at the end of the chapter b looking at inverse functions. [Ask] If we interchange the input and output values, do we still have a function? [(ZIP Code, cit), no; (birth date, person), no; ( first name, last name), no; (capital, state), es] 7a. No; Los Angeles, for eample, has more than one ZIP Code (90001, 9000,...). 7b. Yes; each person has onl one birth date. 7c. No; the same last name will correspond to man different first names. vii. a. Which graphs represent functions? i, ii, iii, iv, and vi b. For which graphs is it not possible to write walking instructions? v and vii c. What conclusion can ou make? ample conclusion: It is not possible to walk a graph that does not represent a function. LEON 7. Functions and Graphs 01

7 Eercise 9 [Ask] How can the table in part a be changed to make it a function? 9a. Not a function; the -value 3 has two different -values, and 8. 9b. A function; each -value corresponds to onl one -value. 9c. A function; each -value corresponds to onl one -value. Eercises and 11 Both noncontinuous and continuous functions are acceptable. You might have students present a variet of answers. Eercise 1 If students have difficult filling in the tables, especiall in 1c, encourage them to solve the equations for in terms of. In 1c, is a function of, but is not a function of. 1a Find whether each table of - and -values represents a function. Eplain our reasoning. a. Domain Range b. Domain Range c. Domain On graph paper, draw a graph that is a function and has these three properties: Domain of -values satisfing 3 Range of -values satisfing Includes the points (, 3) and (3, ) a 11. On graph paper, draw a graph that is not a function and has these three properties: Domain of -values satisfing 3 Range of -values satisfing Includes the points (, 3) and (3, ) inequalities are possible. 1. Complete the table of values for each equation. Let represent domain values, and let represent range values. Graph the points and find whether the equation describes a function. Eplain our reasoning. a. 3 a Range Graphs must pass the vertical line test, have the correct domain and range, and pass through the points (, 3) and (3, ). Graphs will not pass a vertical line test, but the should include points (, 3) and (3, ) and have the correct domain and range. Graphs of The graph is a line. This is a function; each -value is paired with onl one -value. 1b b This is a function; each -value is paired with onl one -value. 1c c d. The equation does not represent a function because there are two different -values for some -values. Tables will var. ample: (0, 0), (, 3), (, 3), (, 6). The graph is a line and represents a function because each -value is paired with onl one -value. 1d. 0 CHAPTER 7 Functions

8 Identif all numbers in the domain and range of each graph. a. b. c. a Consider the capital letters in our alphabet. a. Draw two capital letters that do not represent the graph of a function. Eplain. b. Draw two capital letters that do represent the graph of a function. Eplain. Review 6 6. If represents actual temperature and represents wind chill temperature, the equation 9 1. approimates the wind chill temperatures for a wind speed of 0 mi/h. Enter this equation into Y1 on our calculator and find the requested - and -values. a. What -value gives a -value of? Eplain how ou use the calculator table function to find this answer. b. Enter into Y on our calculator. Graph both equations. Eplain how to use the graph to answer a. 16. how how ou can use an undoing process to solve these equations. a. ( 7) b. 3 3 domain: 0 360; range: Eercise 13 The arrows on graph c indicate that the input values etend indefinitel. The output ma or ma not become zero or negative. 13a. domain: {,, 3,, 1, 0, 1,, 3,, }; range: {0, 1,,3,,} 13c. Answers ma var; possible domain: all numbers ; possible range: all numbers. 1a. Letters such as A, B, and C are not functions because a vertical line could intersect them at more than one point. 1b. Letters such as V and W are functions because no vertical line could intersect them at more than one point. Eercise Have copies of Calculator Note 1J available for students who need to review graphing two equations. a. When,. Answers will var. Zoom in on the table b changing the start values and the table increments (Tbl). b. Answers will var. The lines intersect at the solution point Find the solution to each sstem. a. 3 a , 9 11 (.,.6) b. ( ) ( ) , 1 13 (.31, 1.6) [0, 0,, 0,, ] 16a. ultipl b 3, add 8, divide b, add 7;. 16b. Invert the proportion to get 3. 3,multipl b., add 3; 9.7. LEON 7. Functions and Graphs 03