0. Code

>	restart; with(plots):

Warning, the name changecoords has been redefined

>

slope_plot := proc( a, b, m,  delta, n)   local     x1,x2, y1,y2,i, p1, p2, p3, a1, a2, T, T2, T3;   x2 := a;    y2 := b;     a1 := x2-1;    a2 := round(x2 + n*delta + 1);     for i from 1 to n do        x1 := x2;        x2 := evalf(x1 + delta);        y1 := y2;        y2 := evalf(y1 + m*delta);        p1[i] := plot( [[x1,y1],  [x2,y1]],x = a1..a2, color = blue):        p2[i] := plot(  [[x2,y1], [x2,y2]],x = a1..a2, color =                                              green):               od:   if( m > 0) then y1 := min(b,0); else y1 := max(b,0); fi;   p3 := plot(  [[a,b], [x2, y2]],x = a1..a2, y = y1..y2,               color = gold, thickness= 3):   T := textplot([ x2 - delta, b + 2*m*delta, slope = m ] ,                      align={ABOVE,RIGHT});   T2 := textplot([ x2 - delta, b + m*delta, rise=delta*m],         align = {ABOVE,RIGHT});   T3 := textplot([ x2 - delta, b, run=delta],         align={ABOVE,RIGHT} );   plots[display](seq( p1[i], i = 1..n), seq( p2[i], i = 1..n),                                       p3,T, T2, T3, scaling=constrained);   end:

>

RiseRunPlot  :=  proc(expr, a, delta )      local t, b, m, A, B, C, wid, xi, yi, xl, xr, yl, yr, xt,yt, Run, Rise, TRu, TRi;            xt := 1.3* max( abs(a), a + delta);      xl := a - .6*delta;       xr := a + 1.8*delta;            m := subs( x = 1, solve( expr, y)-solve( subs( x = 0, expr), y));      b := solve( subs( x = a, expr), y):      yt := 1.3* max( abs(b), b + delta*m);          yl := -abs(yt); yr := abs(yt);      if ( a < 0 ) then xr := xr/3; else xl := xl/3; fi;      if ( b < 0 ) then yr := yr/3; else yl := yl/3; fi;      wid := a/10;      A := plot( solve( expr, y), x = xl..xr,                              thickness = 2, color = blue ):      xi := plottools[disk]([ a, b], wid, color=red ):      yi := plottools[disk]([ a+delta, b+delta*m], wid, color=red ):      Run := plot( [[ a,b], [a + delta, b]], color = green, linestyle = 2 );      Rise := plot( [[a + delta, b],[a + delta, b + delta*m] ],                         color = green, linestyle = 2 );      TRu := textplot( [ a + delta/2, b - 1, delta ] );      TRi := textplot( [ a + delta + 1, b + delta*m/2, delta*m ] );      plots[display]( A, xi, yi, Run, Rise, TRu, TRi, plot(0, x=xl..xr), scaling=constrained );  end proc:

>

1. The Concept of Slope, Rise, and Run

Every line has a quality called slope. It is a measurement of how steep the line is uphill.

Lets look at some examples to see how the slope of a line is measured.

2. Spectrum of Slopes

Slope is a measurement of the steepness of a line. Positive slopes are uphill (as we move left to right).

>	display(      plot( {tan(k*Pi/(24))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = blue ), textplot( [-3, 3, `Positive Slopes`])  );

More specifically, if the slope is 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. The blue lines indicate lines with slopes greater  than one . On the other hand, slopes which are positive but less than one - that is, between 0 and 1, are slower rising.   The green lines indicate lines with slope less  than one.

>	display(   plot( {tan( (12+k)*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = blue ),   plot( {tan(k*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = green ), plot( x, x = -4..4, y = -4..4, color = black, thickness = 3 ), textplot( [-3, 3, `Positive Slopes`]) );

Lines which are going downhill have negative slope.

>	display(   plot( {-tan(k*Pi/(24))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = red ), textplot( [ 3, 3, `Negative Slopes`]) );

The black line indicates the line of slope -1, where the rise is equal to negative the run. Lines which decrease faster are colored red , and l ines which decrease more slowly are colored pink .

>	display(   plot( {-tan( (12+k)*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = red ),   plot( {-tan(k*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = magenta ), plot( -x, x = -4..4, y = -4..4, color = black, thickness = 3 ), textplot( [ 3, 3, `Negative Slopes`]) );

Here is a representation of all the types of slopes, positive and negative.

>

display(   plot( {tan( (12+k)*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = blue ),   plot( {tan(k*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = green ),   plot( x, x = -4..4, y = -4..4, color = black, thickness = 3 ),   plot( {-tan( (12+k)*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = red ),   plot( {-tan(k*Pi/(48))*x $ k = 1..11 }, x = -4..4, y = -4..4, color = magenta ),   plot( -x, x = -4..4, y = -4..4, color = black, thickness = 3 ));

3. Rise and Run

To compute the slope of a line, we compute :
           - the run, which is the change in x        = x2 - x1
           - the rise, which is the change in y        = y2 - y1
           - then take the ratio of rise over run

What is the slope of this line?

>	RiseRunPlot( y = 3*x + 2, 2, 4);

Now we draw the line through these two points. There is only one unique line which passes through two points.

>	RiseRunPlot( 3*x + 5*y + 15 = 0, 2, 3);

Determine the slope

>	RiseRunPlot( 3*x - 2*y = 60, 0, 5);

Determine the slope.

>	RiseRunPlot( y = .5*x + 1, -6, 7);

4. Computing Slope

Here is the formula for slope

>	m := (y2 - y1) / ( x2 - x1);

We can substitute in values....

>	subs( { x1 = 3, y1 = 13, x2 = 11, y2 = 29 }, m);

Here is another case, where the slope is negative.

>	subs( { x1 = -3, y1 = 5, x2 = 7, y2 = 4   }, m);

Here are other examples

>	subs( {x1 = 3, y1 = 5, x2 = 7, y2 = 4   }, m);

>	subs( {x1 = 51, y1 = 156, x2 = 129, y2 = 340   }, m);

We can get the slope of a horizontal line in this way.

>	subs( {x1 = 4, y1 = 5, x2 = 10, y2 = 5   }, m);

If we try to find the slope of a vertical line, we get an error because we are dividing by zero.

>	subs( {x1 = 6, y1 = 2, x2 = 6, y2 = 5   }, m);

Error, numeric exception: division by zero