{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 65 "MAT 155 Project II F all 2002" }}{PARA 257 "" 0 "" {TEXT -1 43 " Functi ons and Graphing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 267 32 "If you missed the first project ," }{TEXT -1 245 " you must complete it in order to ba able to do this one. Ask your teacher if you should do it now or if you should sit n ext to someone who was here last week. In the future, if you miss a l ab, you must make up the work before coming to class." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 74 "Before starting this \+ project you should have reviewed P1-P3 for homework. " }{TEXT -1 174 " If you have not read these sections, please refer to them as necessar y. In future projects it will be assumed that you have already done t he reading before coming to class." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 17 "Saving this file:" }} {PARA 0 "" 0 "" {TEXT -1 54 "Before you begin working you should save \+ this file as " }{TEXT 260 14 "yourname2.mws " }{TEXT -1 76 "by selecti ng \"File\" on the far left of the top bar of Maple V and choosing \" " }{TEXT 257 7 "Save As" }{TEXT -1 171 "\". A window will appear and \+ you will fill in the file name. You should regularly save this file \+ using the \"Save As\". You should also save the file with a second n ame " }{TEXT 261 15 "yourname2b.mws " }{TEXT -1 20 "using the \"Save A s\" " }{TEXT 262 26 "so that you have a backup " }{TEXT -1 281 "in cas e your file is damaged. The computer will tell you that the file alre ady exists and you will click on \"Yes\" that you agree to overwrite i t. If you just use the \"Save\" instead of the \"Save As\" then the f ile will be saved with whichever name is on the top bar of this window ." }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT 259 4 "Put " }{TEXT -1 9 "your name" }{TEXT 258 25 " on the top of this file " } {TEXT -1 111 "by clicking your mouse right below the title and then ty ping your name. Then return here using the down arrow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 31 "Problem 1: Defining functions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "A function is a mapping from a domain to a range, it assi gns one value for every input x." }}{PARA 0 "" 0 "" {TEXT -1 149 "For \+ example the square root function maps x to the square root of x. We c an also define a function which maps x to (x+3)/5. To do so you must \+ type:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:= x -> (x+3)/5;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Hit enter on the line so that Maple learns the definition. Now you can find the value of f at a few different value s of x. For example:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1 );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "To get a dec imal expansion from a fraction, use the evalf command:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(f(1));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "You could also fi nd f at Pi=3.14159...." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "f(Pi);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(f(Pi));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Now define a function g(x)=(x+3 0)(x-20). Don't forget to use a star (above the 8) for the multiplica tion." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Evaluate the function at 20 and at -30. Try some ot her x's as well. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 265 "" 0 "" {TEXT -1 193 "It is time to save your work. Got to the to p using the arrow key if you need to review how to save the file. Rem ember you must save it twice, once as yourname2.mws and once as yourna me2b.mws." }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 260 "" 0 "" {TEXT -1 31 "Problem 2: Graphing Functions: " }}{PARA 0 "" 0 "" {TEXT -1 95 "To graph a function on a computer you must tell it what x values you are interested in viewing." }}{PARA 0 "" 0 "" {TEXT -1 64 "Here is the command for graphing a function f fro m x= -10 to 10." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f(x ), x=-10..10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Notice that the y axis is scaled differently than th e x axis." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "N ow try to graph the function g from x=-10 to 10. " }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 "You'l l notice that g is never 0 in this graph. But in Problem 1 you saw th at it was 0 at 20 and at -30. So perhaps a graph that includes x=20 a nd x=-30 would be more interesting. Plot g again using a different do main of x values so that you see where it crosses the x axis. It shou ld cross twice. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Now graph sin(x) from -10 to 10. \+ Where does sin(x) cross 0? " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Find the value of sin(x) at \+ x=Pi. Remember that in calculus we use radians not degrees and that P i radians is 180 degrees. Review radians for homework (A.3 in the te xt) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Now find an x such that sin(x) =1 by locating one of the high points on the graph . " }{TEXT 286 8 "Use the " }{TEXT 283 14 "trace feature " }{TEXT 285 20 "to do this by taking" }{TEXT -1 1 " " }{TEXT 284 328 "the mous e and clicking on the high point with the arrow tip. A pair of number s will appear on the upper left corner of your screen. The first is t he x value of the point you clicked on and the second is the y value. \+ The second should be the number one. Write down the first. Then ver ify that the sine of that number is one." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 264 "" 0 "" {TEXT -1 63 "It is time to save your work. Remember you must save it twice." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 26 "Problem 3: Graphing Lines" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Define a function f(x)=5x+7 a nd graph it. Don't forget to use the multiplication symbol. " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(5* x+7, x=-10..10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Recall that the slope of th e line can be found by identifying two points on the line (a,b) and (c ,d). " }{TEXT 280 8 "Use the " }{TEXT 279 14 "trace feature " }{TEXT 282 20 "to do this by taking" }{TEXT -1 1 " " }{TEXT 281 201 "the mous e and clicking on the line with the arrow tip. A pair of numbers will appear on the upper left corner of your screen. Write them down. Th en click on another point in the line and copy down. " }{TEXT -1 190 " The first pair can be (a,b) and the second can be (c,d). Compute the slope of the line by computing the change in y coordinates over chang e in x coordinates=(d-b)/(c-a). You should get 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "To graph a line with a gi ven slope, m, and a point (a,b), you can find a line through that poin t using the formula " }{TEXT 273 17 "y - b = m (x -a) " }{TEXT -1 185 "However, this formula is an equation not a function. A function \+ should always have y by itself on one side so that when you input an x value you get a y value. So you rewrite this as " }{TEXT 274 15 "y = m(x-a) + b " }{TEXT -1 144 "and fill in the numbers for the point (a, b) and the slope m. For example, the function whose graph is a line t hrough (1,3) with a slope 2 is: " }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 0 20 "f:= x -> 2*(x-1) +3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Graph this function for x in [-10,10]. \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Define a function whose graph is a line which goes throug h (5,30) with slope -3." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Graph the function for x in [-10,1 0]." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 121 "Can you guess where the graph crosses the x axis, and \+ graph it again below changing your x domain to include that point. " } {TEXT 275 64 " You do not have to copy over the entire plot command by hand. " }{TEXT 276 291 "Hold the left mouse button down as you trace the part of the command you want to copy so it should be highlighted. Then go to the \"Edit\" on the top bar and select \"copy\". Then cl ick below on [>, go back to \"Edit\" and select \"Paste\" and it will \+ appear. Then you can just change the x part." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 304 "If the slope is positive then the function is increasi ng (and the graph goes upwards from left to right) and if the slope is negative then the function is decreasing (and the graph does downward from left to right). Define a function whose graph is just a horizon tal line. If necessary try a few times." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 263 "" 0 "" {TEXT -1 63 "It is time to save your work. Remember you m ust save it twice." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 35 "Problem 4: Odd and Eve n Functions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Define f(x)=cos(x) and graph it with x in [-10,10]." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Define g(x)=x^2 and grap h it with x in [-10,10]." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Notice that both of these graphs can be reflected symmet rically across the y axis (like a mirror). Thus they are called " } {TEXT 269 14 "even functions" }{TEXT -1 108 ". That is f(x)=f(-x). G raph f(-x) and g(-x) to see that they are the same as the graphs of f( x) and f(-x)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Now graph sin(x) and sin(-x). " }{TEXT 277 69 "Again use the copy and the paste to avoid typing the commands twice. \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Is sin(x) a n even function? No, these graphs are not the same. In fact sin(- x) = " }{TEXT 270 2 "- " }{TEXT -1 26 "sin(x) ! So sin(x) is an " } {TEXT 271 13 "odd function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 63 "It is time to s ave your work. Remember you must save it twice." }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 13 "Exploration: " }{TEXT -1 531 "Every project ends with explorat ion. Before starting the exploration you should make sure that you ha ve completed all of the problems. Once you have completed them you ca n make up similar problems or do recommended exploration problems. If you miss a lab, you must catch up on the previous labs problems to le arn all the commands and ideas that you missed. The exploration need \+ not have been completed to continue with the next project. Your profes sor may, however, require you to complete the exploration for grading \+ purposes." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 33 "Recommended exploration problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 10 "A) Shifts:" }{TEXT -1 376 " Let f(x)=x^3. Then graph f(x), f(x+2), and f(x)+2 and compare \+ them. Now try to shift the graph of the function other ways. For exa mple shift it to the right by 3, shift it to the left by 4, shift it u p by 5, and shift it down by 6. Now change f to another function and \+ hit enter on all your lines to see how the graphs change. Just hit e nter below to get more prompts " }{TEXT 272 2 "[>" }{TEXT -1 18 ". \+ " }}}{PARA 0 "" 0 "" {TEXT 264 0 "" }}}{MARK "98 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }