GraphingCalculator 4;
Window 44 0 860 1406;
PaneDivider 413;
FontSizes 18;
2D.Scale 1 1 5 5;
2D.BottomLeft -9.9375 -43.3125;
2D.GraphPaper 0;
Text "Elementary linear transformations of functions.  Compare f(x) to cf(bx+a)+d.  Note the pattern – transformations to the ARGUMENT of f (a,b) result in HORIZONTAL transformations of the graph, while changes OUTSIDE f (c,d) result in VERTICAL tranformations.    (a,d are horizontal/vertical <I>shifts</I>; b,c are horizontal/vertical changes of <I>scale</I>.)  

Note that f(bx+a)=f(b(x+a/b)) is f(bx) shifted by a/b ... in other words, f(bx+a) is f(x) <I>first</I> scaled by b, <I>then</I> shifted by a/b.   You can see this by first changing b to scale f(x) horizontally.  Adjusting a then shifts that shape rigidly right or left by a/b.

HORIZONTAL <I>SHIFT</I>:  f(x+a) – right (a<<0) or left (a>0)  [default a=0]";
Color 2;
MathPaneSlider 32;
Expr a=slider([-5,5,40]);
Text "HORIZONTAL <I>SCALING</I>:  f(bx) – stretch (b<<1) or shrink (b>1)  [default b=1]  
[reflection across y-axis if b<<0]";
Color 3;
MathPaneSlider 27;
Expr b=slider([-5,5,40]);
Text "VERTICAL <I>SCALING</I>:  cf(x) – stretch (c>1) or shrink (c<<1)  [default c=1]
[reflection across x-axis if c<<0]";
MathPaneSlider 24;
Expr c=slider([-5,5,40]);
Text "VERTICAL <I>SHIFT</I>:  f(x)+d – down (d<<0) or up (d>0)  [default d=0]";
Color 6;
MathPaneSlider 20;
Expr d=slider([-5,5,40]);
Text "Functions to play with:";
MathPaneSlider 24;
Expr p=slider([0,10,40]);
Color 8;
Expr function(h,x)=cos([x/p])*[1+x]*e^[-(0.01*x^2)];
Color 4;
Expr function(f,x)=cos([x])*[1+x];
Color 7;
Expr function(g,x)=cos([x])*[1+x]*e^[-(0.01*x^2)];
Color 4;
Expr function(s,x)=cos(x);
Color 2;
Expr function(L,x)=x;
Text "The function and its transformation:";
Color 5;
Expr function(F,x)=function(h,x);
Color 7;
Expr y=function(F,x);
Color 3;
Expr y=c*function(F,b*x+a)+d;
Color 17;
Expr y=function(F,x+a);
Color 17;
Expr y=function(F,b*x);
Color 17;
Expr y=c*function(F,x);
Color 17;
Expr y=function(F,x)+d;
Text "Point on graph of F(x) and the corresponding point on its transform.  The first pair is F(0), the second, F(j), where j can be freely chosen, such that F(j)=F(bx+a), so that the mapping between points on the graphs of F(x) and F(bx+a) is (x,F(x)) -> ((x-a)/b,F(x)).";
Color 17;
Expr vector(0,function(F,0));
Color 17;
Expr vector(-a/b,c*function(F,0)+d);
Color 17;
Expr vector(-a/b*t,c*function(F,0)+d);
Color 6;
Expr -a/b;
Text "";
Color 5;
MathPaneSlider 140;
Expr j=slider([-20,20]);
Color 17;
Expr vector(j,function(F,j));
Color 17;
Expr vector((j-a)/b,c*function(F,j)+d);
Color 17;
Expr vector(j*[1-t]+(j-a)/b*t,c*function(F,j)+d);
Text "Part of displacement due to horizontal scaling of graph (as j -> j/b):";
Color 17;
Expr vector(j*[1-t]+j/b*t,c*function(F,j)+d);
Text "Rigid shift of graph by a/b (as j/b -> (j-a)/b):";
Color 17;
Expr vector(j/b*[1-t]+(j-a)/b*t,c*function(F,j)+d);
Text "

<size=15>Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";