0. Code

>	restart; with(plots):

Warning, the name changecoords has been redefined

>	merge := proc( a1,a2,a3, b1,b2,b3,k,n) local c; c := COLOR(RGB, a1 + k*(b1-a1)/n,                 a2 + k*(b2-a2)/n,                 a3 + k*(b3-a3)/n ); return c; end:

>	ShadedPlot := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge(.15,.15,.15,.7,.7,.7,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotR := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge(1,.1,.1,1,.5,.5,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotG := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge( .3,.8,.2, .7,.9,.6,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotB := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge(.15,.15, .8, .6,.6, .8,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotE := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotP := proc( f)    display( [seq( plot( k*f(x), x = -3..3, filled = true,              color = merge( .8,.4,.8, .2,.2,.2,k,10)), k = 0..10 )] ); end proc:

>	ShadedPlotP2 := proc( f,a,b)    display( [seq( plot( k*f(x), x = a..b, filled = true,              color = merge( .85,.5,.85, .3,.2,.3,k,10)), k = 0..10 )] ); end proc:

  1. Exponential Functions

Here is a typical exponential function.

>	f := x -> 2^x;

>	for k from 6 to -4 by -1 do 'f'(k) = f(k); od;

Here is another

>	g := x -> 5^x; for k from 6 to -4 by -1 do 'g'(k) = g(k); od;

What is the same about these?

>	A := array( [seq(  [k, 2^k, 5^k], k = -5..5) ]): A[1,1] := `k`:   A[1,2] := `2^k`:   A[1,3] := `5^k`: print(A);

Let's look at even more examples, side by side.

>	A := array( [seq(  [k, 2^k,3^k,4^k,5^k], k = -5..5) ]): A[1,1] := `k`:   A[1,2] := `2^k`:   A[1,3] := `3^k`: A[1,4] := `4^k`: A[1,5] := `5^k`: print(A);

It looks like exponential functions are one when x = 0.

>	f(0); g(0);

It looks like exponential functions are the value being exponentiated when x = 1

>	f(1); g(1);

Also, when x = -1, we get reciprocals.

>	f(-1); g(-1);

  2. Graphs of Exponential Functions

The basic exponential function looks like this.

>	ShadedPlotB( exp(x) );

When the exponent is negative, it looks like this - the mirror image through the y axis.

>	ShadedPlotG( exp(-x) );

Here are some other variations obtained by negating or shifting exponential graphs.

>	ShadedPlotE( - exp(x) );

>	ShadedPlotP( - exp(-x) );

>	ShadedPlotR( 3 - exp(x) );

  3. Exponential & Log Graphs

Exponentials and logs are inverse functions of one another.

>	display(  plot( [exp(x),ln(x)], x = -5..7, y = -5..7, color = [red, blue], thickness = 3, scaling = constrained), plot( [[-5,-5],[7,7]], color = maroon, linestyle = 3));

>	display(  seq(plot( [k*exp(x),ln(x/k)], x = -3.5..4.5, y = -3.5..4.5,             color = [red, blue],thickness = 1, scaling = constrained),             k = 1..15), plot( [[-5,-5],[7,7]], color = maroon, linestyle = 3));

>	display(  seq(plot( [k+exp(x),ln(x-k)], x = -5..16, y = -5..16,             color = [red, blue],thickness = 1, scaling = constrained),             k = 1..15), plot( [[-5,-5],[16,16]], color = maroon, linestyle = 3));

  4. Surge Functions

Here is a family of functions called surge functions which are variations of exponential functions.

>	f := (x,a,b) -> a*x*exp(-b*x);

>	display( [seq( plot( k*f(x, .1, .1), x = 0..100, filled = true,              color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 0..10 )]);

>	display( [seq( plot( f(x, 1, (50-2.7*k)/50), x = 0..20, filled = true,              color = merge( .8,.6,.7, .5,.3,.4,k,10)), k = 0..16 )]);

>	plot(  { seq(f(x, 1, k/50), k = 1..40) }, x = 0..70, color = gold );

  5. Logistic Growth Functions

There are certain problems, in biology in particular, which involve this kind of exponential function. This kind of function has a bound on how high it can grow.

>	ShadedPlot( exp(x)/(1+exp(x)));

>	ShadedPlotP2( exp(x)/(1+exp(x)), -10, 10);

>	plot( {seq(exp(x*(.5*k))/(1 + exp(x*k)), k = 1..16)}, x = -1..1, color =khaki);

>	display( [seq( plot( exp(x*(.5*(10-k)))/(1 + exp(x*(10-k))),              x = -1..1, filled = true,              color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 1..10 )]);

  6. Other Exponential Functions

Here are some other wilder exponential functions.

>	plot( { seq( exp(x*k)/(1 + exp(k*x)), k= 1..20)}, x = -4..4, color = khaki);

>	plot( {seq(exp(x*k)/(1 + k*exp(x*k)), k= 1..16)}, x = -3..3, color = khaki);

>	plot( {seq(1 - exp(x*k)/(1 +  exp(x*k)), k = 1..20)}, x = -2..2, color = navy);

>	plot( {seq(exp(x*(.5*k))/(1 + k*exp(x*k)), k = 1..16)}, x = -1..1, color = khaki);

>	plot( {seq(exp(x*(.01*k))/(1 + (k)*exp(x*k)), k = 1..20)}, x = -3..3, color = khaki);

>	plot( {seq(exp(x*(.9*k))/(1 + sqrt(k)*exp(x*k)), k = 1..20)}, x =-3..3);

>	plot( {seq(1+k*exp(x*(.9*k))/(1 + sqrt(k)*exp(x*k)), k = 1..20)},x=-3..3);

>	plot( {seq(1+ (k^2)*exp(x*(.95*k))/(1 + sqrt(k)*exp(x*k)), k = 1..20)},x = -3..3);

>	display( [seq( plot( 1+ (k^2)*exp(x*(.95*k))/(1 + sqrt(k)*exp(x*k)),              x = -3..3, filled = true,              color = merge( .4,.2,.0, .8,.6,.2,k,10)), k = 1..20 )]);

>	display( [seq( plot( exp(x*(.9*(20-k)))/(1 + sqrt(20-k)*exp(x*(20-k))),              x = -2..2, filled = true,              color = merge( .6,.7,.6, .3,.5,.3,k,10)), k = 1..19 )]);

>

display( [seq( plot( exp(x*k)/(1 + exp(k*x)) ,              x = 0..2, filled = true,              color = merge( .8,.5,.6, .5,.3,.3,k,10)), k = 1..10 ),           seq( plot( exp(x*(10-k))/(1 + exp((10-k)*x)) ,              x = -2..0, filled = true,              color = merge(  .5,.3,.3,.8,.5,.6, k,10)), k = 1..9 )]);