Shake, Rattle and Roll

Transcription

1 00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar and fun to ride. Wooden roller coasters shake and rattle as part of the thrill of the ride. Below is the graph of the heights reached b the cars of the wooden roller coaster, Thunderball, over its first 0 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) _SB_A_-_SE.indd 77 Shake, Rattle and Roll Thunderball Roller Coaster Graph Distance Along the Track (feet) The domain gives all values of the independent variable: distance along the track in feet. These values are graphed along the horizontal or -ais. The domain can be written in set notation as: {all real values of : 0 0} Read this notation as: the set of all real values of, between 0 and 0, inclusive. The range gives the values of the dependent variable: height above the ground in feet. The values are graphed on the vertical or -ais. The range can be written in set notation as: {all real values of : 0 0 Read this notation as: the set of all real values of, between 0 and 0, inclusive. The graph above shows data that are continuous. The points in the graph are connected, indicating that domain and range are sets if 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. MINI-LESSON: Reading Set Notation M Notes ACADEMIC VOCABULARY The independent variable is the input of a function. The dependent variable is the output of a function. Its value depends on the value of the independent variable. ACTIVITY. MATH TERMS An The ordered vertical pair line shows test is the a visual relationship check to see between if a graph two appears values, to be a function. written in a For specific a function, order ever using vertical parentheses line drawn in n the otation coordinate and a comma plane will separating intersect the the graph two values. in at most one point. This is equivalent to having each domain element associated with one and onl one element of the range. Unit Linear Functions 77 /7/09 0:8: AM Give practice in reading set notation, where { } is read the set of all values and : or is read such that and then an restrictions on the set are given. a. {real : = } [ the set of all real values of, such that is less than or equal to ] b. {odd numbers z : < z < } [ the set of all odd numbers z, such that z is between and ] c. {integer < 7} [ the set of all integer values of, such that is less than 7 ] d. {even numbers < or > } [ the set of all even numbers, such that is less than and greater than ] ACTIVITY. Domain and Range of Continuous Functions Activit Focus Domain and range of continuous functions Set notation Vertical line test Additional Materials Graphing calculators (optional) Whiteboards or chart paper Chunking the Activit Roller Coaster Eample Mini-lesson on reading set notation #a b Mini-lesson on vertical line test Eample Tr These Technolog Time EXAMPLE Shared Reading, Marking the Tet, Visualization, Interactive Word Wall The roller coaster eample introduces the vertical line test. Demonstrating was to visualize a vertical line moving from left to right along the -ais can assist students as the begin to appl this test for themselves. Some methods that ou can demonstrate are: pass a ruler along the -ais perpendicular to it displa the graph on an overhead slide then pass a second acetate over the top of the first with a vertical line drawn on it Technolog such as a projected graph or an overhead graphing calculator provides opportunities to help students with their visualization of the vertical line test. Unit Linear Functions 77

2 ACTIVITY. Continued EXAMPLE () Since the roller coaster graph passes the vertical line test and is therefore a function, ou should solicit eamples of non-functions that can be drawn to illustrate how some relations (like a circle graph, vertical line, or an polgon) fail the vertical line test and are not functions. See mini-lesson below. Identifing the domain and range of this finite, continuous graph is the objective of this activit. Helping students practice reading and writing the set notation that describes the domain and range is another skill that students will need to practice. See mini-lesson on the previous page. Group Discussion, Debriefing In part a, ask students to come up with a real-world scenario that might be represented b this graph. If the have difficult, scaffold b offering four scenarios and asking them which one is represented. ACTIVITY. M Notes Domain: {all real values of : - < } Range: {all real values of : - } This graph represents a function because it passes the vertical line test. The data are continuous because the function includes all real values of between - and, and all real values of between - and, inclusive. The independent variable is t, the minutes since the bath began, and the dependent variable is d, the depth of the bath water, since the depth of the water depends on how man minutes the water has been running. Domain {all real values of t: 0 t } Range {all real values of d: 0 d 8} The data is continuous because the function includes all real values of t between 0 and, inclusive, and all real values of d between 0 and 8, inclusive. SUGGESTED LEARNING STRATEGIES: Group Discussion a. Use set notation to write the domain and range for the graph below. Does this graph appear to represent a function? Justif our answer. Are the data discrete or continuous? Wh? b. The graph below shows the relationship between t, the length of time of the bath (from the time water starts running through the time the tub is drained) and d, the depth of the water in the bath tub. The graph represents function d (bath water depth). What are the dependent and independent variables? Eplain. Use set notation to write the domain and range of function d. Are the data discrete or continuous and wh? Depth of bath water (in.) Bath Water Depth d t Minutes since bath began 00 College Board. All rights reserved. 78 SpringBoard Mathematics with Meaning Algebra 77-8_SB_A_-_SE.indd 78 MINI-LESSON: Vertical Line Test Provide additional eamples of graphs that will fail the vertical line test because the are not functions. Make coordinate graphs (without equations) of the items below on the board or overhead. Then ask students to determine, first for themselves and then b comparing answers with their neighbors, which of the following are functions. a. circle: b. vertical parabola: ( + ) = ( - ) ( + ) + ( - ) = c. hperbola: - = d. eponential function: = -e e. horizontal parabola: f. right triangle: (, ), (, ), (, ) ( + ) = ( - ) g. cubic: = ( + ) h. absolute value: = + /7/09 0:8: A7 00 College Board. All rights reserved. Items a, c, e, f, and h do not represent functions. 78 SpringBoard Mathematics with Meaning Algebra

3 ACTIVITY. ACTIVITY. Continued 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Questioning the Tet, Think Aloud EXAMPLE Give the domain and range of the function f () = ( - ) graphed below. Step : Step : Stud the graph. The sketch of this graph is a portion of the function represented b the equation f () = ( - ). Look for values for which the domain causes the function to be undefined. Look how the graph behaves near =. EXAMPLE Think Aloud, Question the Tet, Marking the Tet This eample uses the graph of a rational function. Students share what the notice or ask what the want to know about the graph. Reading graphs in mathematics to make meaning of the information represented is similar to reading tet in English class. Students ma not believe that can never take on the value of. Substituting into the equation f() = results in division ( - ) b zero producing an undefined result for the function. This eample coupled with the visual clue of the dotted line can help students recognize undefined values in the future. TRY THESE Debriefing 00 College Board. All rights reserved. Solution: The domain and range for f () = can be written: ( - ) Domain: {all real values of : } Range: {all real values of : > 0} TRY THESE a. Give the domain and range of the function f () = 8 + graphed below M Notes Notice the result when = is substituted into f (). f () = ( - ) = 0 Division b zero is undefined in mathematics. Domain Range {all real numbers} {all real values of : 9} Unit Linear Functions 79 M7-8_SB_A_-_SE.indd 79 /7/09 0:8: AM Unit Linear Functions 79

4 ACTIVITY. Continued Suggested Assignment CHECK YOUR UNDERSTANDING p. 8, # UNIT PRACTICE p., # Technolog Time Create Representations, Group Discussion All of the equations can be viewed in the standard window where 0 = = 0 and 0 = = 0. However, for tracing purposes a friendl window or decimal window will serve students better. On a TI 8 this would be [-9., 9.,, -0, 0, ]. Depending on how much practice students have had using the graphing calculator, ou ma need to assist students in entering the equations into the calculator. Functions ma need to be written out b kestrokes if students are not familiar with the calculator nomenclature. For eample, = + on some calculators is entered as = ( + ) and = is entered as = /. Allow students to work in partners or partners within foursomes so that the work can be collaborative and students can help each other with their questions. After students have had the opportunit to complete the chart and to compare answers with another pair, debrief the activit b asking students to share what the noticed about the results and the behavior of the graphs. As an etension, have students graph other linear, quadratic, and absolute value functions and ask how the restrictions on the domain and range of their new graphs compare with the restrictions on the previous set of functions. ACTIVITY. M Notes The domain is restricted to avoid situations where division b zero or taking the square root of a negative number would occur. TRY THESE () 80 SpringBoard Mathematics with Meaning Algebra 77-8_SB_A_-_SE.indd 80 SUGGESTED LEARNING STRATEGIES: Create Representations, Group Discussion b. Give the domain and range for the equation =. Eplain whether this equation represents a function and how ou determined this. Domain {all real numbers} Range {all real numbers} Answers ma var. Sample answer: Since the graph of this equation passes the vertical line test, this equation and its graph represent a function. Technolog Time Work with a partner to investigate the equations listed in the chart using graphing technolog. Ever equation given here is a function. Determine the domain and range for each function from the possibilities listed below the chart. Select the appropriate domain from choices 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. When the chart is complete, compare our answers with those from another group. Function Domain Range. = - + a. = - +. = 9 -. = +. = +. = Possible Domains: Possible Ranges: ) 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 ) all real, such that 0 e) all real, such that ) all real, such that 0 f) all real, such that Note on the Use of Graphing Calculators The purpose of this activit is to have students identif domain and range, not graph functions. If our students do not have graphing calculators or do not know how to graph the functions with a calculator, simpl displa the graphs on an overhead projector or on the board and have students identif the domain and range. c a d f b /7/09 0:8:7 A7 00 College Board. All rights reserved. 00 College Board. All rights reserved. 80 SpringBoard Mathematics with Meaning Algebra

5 00 College Board. All rights reserved. M7-8_SB_A_-_SE.indd 8 00 College Board. All rights reserved. CHECK YOUR UNDERSTANDING ACTIVITY. Write our answers on notebook paper. Show our work.. The graph below shows five points that our work. make up the function h. Give the domain. Give the domain and range for the and the range for the function h. function graphed below. Eplain wh this graph represents a function A student calculates how far awa a lightning strike is, based on when the thunder is heard. The student makes the table below using km/sec as the average speed of sound under rain conditions. If the thunder is onl heard when the lightning strike is within km of the listener, what are the domain and range for this model? Is this relation a function? How do ou know? Time until thunder is heard (sec) Distance from lightning strike (km). Give the domain and range of the function f () =. Yards Left to Walk. Jeff walks at an average rate of 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. Give the domain and range for this model. Is this model a function? Eplain Jeff Walks to Mark s House 8 0 Minutes Walking Unit Linear Functions 8 /7/09 0:8:0 AM ACTIVITY. Continued Suggested Assignment CHECK YOUR UNDERSTANDING p. 8, # 7 UNIT PRACTICE p., #,. domain: {all real values of : 0 9}; range: {all real values of : 0 }; This graph represents a function because it passes the vertical line test.. Since the time it takes to hear the thunder is dependent upon how far awa the lightning strikes, the domain will address the distances [d] and the range will address the time [t] in seconds since the lightning struck. Domain: {all real values of d: 0 d kilometers]. Range:{all real values of t : 0 t seconds} The pattern ehibits a constant change in time over distance, and if graphed, the pattern would look like a line. This relationship has onl one dependent value for ever independent value, so it is a function. If graphed, it would pass the vertical line test.. domain: {all real numbers} range: {all real numbers}. domain: {-, -,,, } range: {-, -,, }. domain {all real values of : 0 minutes}; range: {all real values of : ards}. This model is a function because no domain value is paired with more than one range value and its graph passes the vertical line test. Unit Linear Functions 8

6 ACTIVITY. Continued. Depending on how the letters are drawn, most letters will not pass the vertical line test. The letter V is one that could possibl pass the test. Another possibilit would be the letter W. Students should include a sketch of the letters chosen on grid paper to fit each categor along with a written justification that includes information about the vertical line test. 7. Answers ma var. Sample answer: Eamine a set of ordered pairs to see if an -value is repeated; use the vertical line test on a graph; eamine a mapping to see if one input value is mapped to more than one output value. Students preferences will var. ACTIVITY. CHECK YOUR UNDERSTANDING () Write. Capital our answers letters sketched on notebook in the paper. coordinate Show our work. 7. MATHEMATICAL plane ma or ma not be functions. Pick REFLECTION one letter that represents a function and two that do not. Use the vertical line test as part of the eplanation for our selections. Describe at least three different methods for determining if a relation is a function. Which method do ou prefer and wh? 00 College Board. All rights reserved. 00 College Board. All rights reserved. 8 SpringBoard Mathematics with Meaning Algebra 77-8_SB_A_-_SE.indd 8 /7/09 0:8: A 8 SpringBoard Mathematics with Meaning Algebra