{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 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 255 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 255 1 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 255 0 1 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 255 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 255 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 257 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 5 "S lope" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "A n exploration of the slope of a line." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Reso urce section first. Although there will be no output immediately, thes e definitions are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1080 "slope_plot := proc( a, b, m, del ta, n)\n local x1,x2, y1,y2,i, p1, p2, p3, a1, a2, T, T2, T3;\n\n x2 := a; y2 := b; a1 := x2-1; a2 := round(x2 + n*delta + 1 );\n for i from 1 to n do\n x1 := x2; x2 := evalf(x1 + delta); \n y1 := y2; y2 := evalf(y1 + m*delta);\n \+ p1[i] := plot( [[x1,y1], [x2,y1]],x = a1..a2, color = blue):\n \+ p2[i] := plot( [[x2,y1], [x2,y2]],x = a1..a2, color = \+ green):\n \n od:\n if( m > 0 ) then y1 := min(b,0); else y1 := max(b,0); fi; \n p3 := plot( [[a,b ], [x2, y2]],x = a1..a2, y = y1..y2, \n color = gold, thi ckness= 3):\n T := textplot([ x2 - delta, b + 2*m*delta, slope = m ] \+ , \n align=\{ABOVE,RIGHT\});\n T2 := textplot([ x2 - delta, b + m*delta, rise=delta*m], \n align = \{ABOVE,RIGHT \});\n T3 := textplot([ x2 - delta, b, run=delta], \n align=\{ ABOVE,RIGHT\} );\n plots[display](seq( p1[i], i = 1..n), seq( p2[i], \+ i = 1..n), p3,T, T2, T3, scaling =constrained);\n end:\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1208 "RiseRunPlot := proc(expr, a, delta )\n local t, b, m, A, B, C, wid, xi, yi, xl, xr, yl, yr, xt,yt, Run, Rise, TRu, TRi;\n \+ \n xt := 1.3* max( abs(a), a + delta);\n xl := a - .6*delta; \+ xr := a + 1.8*delta;\n \n m := subs( x = 1, solve( expr, \+ y)-solve( subs( x = 0, expr), y));\n\n b := solve( subs( x = a, ex pr), y):\n yt := 1.3* max( abs(b), b + delta*m);\n \n yl := \+ -abs(yt); yr := abs(yt); \n\n if ( a < 0 ) then xr := xr/3; else x l := xl/3; fi;\n if ( b < 0 ) then yr := yr/3; else yl := yl/3; fi ;\n wid := a/10;\n A := plot( solve( expr, y), x = xl..xr, \n thickness = 2, color = blue ):\n xi := \+ plottools[disk]([ a, b], wid, color=red ):\n yi := plottools[disk] ([ a+delta, b+delta*m], wid, color=red ):\n Run := plot( [[ a,b], \+ [a + delta, b]], color = green, linestyle = 2 );\n Rise := plot( [ [a + delta, b],[a + delta, b + delta*m] ], \n c olor = green, linestyle = 2 );\n TRu := textplot( [ a + delta/2, b - 1, delta ] );\n TRi := textplot( [ a + delta + 1, b + delta*m/2 , delta*m ] );\n plots[display]( A, xi, yi, Run, Rise, TRu, TRi, p lot(0, x=xl..xr), scaling=constrained );\n end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "1. The Concept of Slope, Rise, and Run" }}{PARA 0 "" 0 "" {TEXT -1 92 "\nEvery line has a quality called slope. It is a measurement of ho w steep the line is uphill." }}{PARA 0 "" 0 "" {TEXT -1 71 "\nLets loo k at some examples to see how the slope of a line is measured." }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 270 "This line begins at the point (-4,2). If we go over 5 un its, the line goes up 10 units. We can see that this is true no matter where we choose the point to start measuring over and up. In all case s, we get the same ratio when we divide rise by run, 2. This is the sl ope." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "slope_plot ( -4, 2 , 2, 5, 6 ); \n# (x1, y1), slope, delta x, n" }}}{PARA 0 "" 0 "" {TEXT -1 111 "\nIf we choose a smaller run, we get a smaller rise . However, the ratio of rise to run is the same as before, 2." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "slope_plot ( -4, 2 , 2, 3, 1 0);" }}}{PARA 0 "" 0 "" {TEXT -1 111 "\nEven if we take a smaller run, we get a smaller rise, but the slope (the ratio of rise to run) is th e same, 2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "slope_plot ( - 4, 2 , 2, 1, 30 );" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }} {PARA 0 "" 0 "" {TEXT -1 161 "\nLets look at a different line to see i f this same phenomenon occurs again. In each of the cases below, the s lope is -3/4, no matter how small we choose the run." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "slope_plot ( -10, 15 , -1/2, 8, 5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "slope_plot ( -10, 15 , -3/4, 4, 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "slope_plot ( -1 0, 15 , -3/4, 2, 20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "2. Spectrum of Slopes" }}{PARA 0 "" 0 "" {TEXT -1 105 "Slope is a \+ measurement of the steepness of a line. Positive slopes are uphill (as we move left to right)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "display( \n plot( \{tan(k*Pi/(24))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = blue ),\ntextplot( [-3, 3, `Positive Slopes`]) );" }}}{PARA 0 "" 0 "" {TEXT -1 216 "\n\nMore specifically, if the slope i s 1, then the line rises an equal amount as it runs. This is indicated by the black (45 degree line). If the slope is greater than one, then the line increases faster than it runs. " }{TEXT 261 42 "The blue lin es indicate lines with slopes " }{TEXT 264 7 "greater" }{TEXT 265 9 " \+ than one" }{TEXT 263 2 ". " }{TEXT -1 111 "On the other hand, slopes w hich are positive but less than one - that is, between 0 and 1, are sl ower rising. " }{TEXT 262 42 "The green lines indicate lines with slo pe " }{TEXT 266 4 "less" }{TEXT 267 10 " than one." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 278 "display( \n plot( \{tan( (12+k)*Pi/(48))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = blue ),\n plot( \{tan( k*Pi/(48))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = green ),\np lot( x, x = -4..4, y = -4..4, color = black, thickness = 3 ),\ntextplo t( [-3, 3, `Positive Slopes`])\n);" }}}{PARA 0 "" 0 "" {TEXT -1 53 "\n \nLines which are going downhill have negative slope." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "display( \n plot( \{-tan(k*Pi/(24))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = red ),\ntextplot( [ 3, 3, `Negative Slopes`])\n);" }}}{PARA 0 "" 0 "" {TEXT -1 94 "\n\nThe blac k line indicates the line of slope -1, where the rise is equal to nega tive the run. " }{TEXT 268 43 "Lines which decrease faster are colored red" }{TEXT -1 6 ", and " }{TEXT 270 1 "l" }{TEXT 269 48 "ines which \+ decrease more slowly are colored pink" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "display( \n plot( \{-tan( (12+k)*Pi/(48 ))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = red ),\n plot( \{- tan(k*Pi/(48))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = magenta ),\nplot( -x, x = -4..4, y = -4..4, color = black, thickness = 3 ),\n textplot( [ 3, 3, `Negative Slopes`])\n);" }}}{PARA 0 "" 0 "" {TEXT -1 77 "\n\nHere is a representation of all the types of slopes, positi ve and negative." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "display ( \n plot( \{tan( (12+k)*Pi/(48))*x $ k = 1..11 \}, x = -4..4, y = -4 ..4, color = blue ),\n plot( \{tan(k*Pi/(48))*x $ k = 1..11 \}, x = - 4..4, y = -4..4, color = green ),\n plot( x, x = -4..4, y = -4..4, co lor = black, thickness = 3 ),\n plot( \{-tan( (12+k)*Pi/(48))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = red ),\n plot( \{-tan(k*Pi/( 48))*x $ k = 1..11 \}, x = -4..4, y = -4..4, color = magenta ),\n plo t( -x, x = -4..4, y = -4..4, color = black, thickness = 3 ));" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "3. Rise and Run" }}{PARA 0 "" 0 " " {TEXT -1 257 "\nTo compute the slope of a line, we compute :\n \+ - the run, which is the change in x = x2 - x1\n \+ - the rise, which is the change in y = y2 - y1\n - th en take the ratio of rise over run\n\nWhat is the slope of this line? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RiseRunPlot( y = 3*x + 2 , 2, 4);" }}}{PARA 0 "" 0 "" {TEXT -1 110 "\nNow we draw the line thro ugh these two points. There is only one unique line which passes throu gh two points." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "RiseRunPlo t( 3*x + 5*y + 15 = 0, 2, 3);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "\nDeter mine the slope" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "RiseRunPlo t( 3*x - 2*y = 60, 0, 5);" }}}{PARA 0 "" 0 "" {TEXT -1 22 "\nDetermine the slope. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "RiseRunPlot( y = .5*x + 1, -6, 7);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "4. Computing Slope" }}{PARA 0 "" 0 "" {TEXT -1 30 "\nHere is the formula for slope" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m := (y2 - y1) / ( x2 - x1);" }}}{PARA 0 "" 0 "" {TEXT -1 32 "\nWe can substitute in values...." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "subs( \{ x1 = 3, y1 = 13, x2 = 11, y2 = 29 \}, m );" }}}{PARA 0 "" 0 "" {TEXT -1 51 "\nHere is another case, where the \+ slope is negative." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "subs( \+ \{ x1 = -3, y1 = 5, x2 = 7, y2 = 4 \}, m);" }}}{PARA 0 "" 0 "" {TEXT -1 24 "\nHere are other examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs( \{x1 = 3, y1 = 5, x2 = 7, y2 = 4 \}, m);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "subs( \{x1 = 51, y1 = 156, x 2 = 129, y2 = 340 \}, m);" }}}{PARA 0 "" 0 "" {TEXT -1 56 "\n\nWe ca n get the slope of a horizontal line in this way." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs( \{x1 = 4, y1 = 5, x2 = 10, y2 = 5 \}, m);" }}}{PARA 0 "" 0 "" {TEXT -1 49 "\nIf we try to find the slope of a vertical line, " }{TEXT 260 48 "we get an error because we are divi ding by zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs( \{x1 \+ = 6, y1 = 2, x2 = 6, y2 = 5 \}, m);" }}}{PARA 3 "" 0 "" {TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }