Families of functions
depending on parameters

You can copy the cell below, change the values of a and/or omega, and get a look at the different graphs.  Be sure to run the first cell (<<Graphics`Colors`) so that you can get color graphics.

<< Graphics`Colors`

y(t) = (e^(-at)) sin(ωt)

The following cells are set up to let you plot E^(-at)*Sin[omega*t] using three different values of "a".  You can look at all three graphs together using the "Show" command.

It might be interesting to plot the graph of y=E^(-a*t) and compare it with the graphs of y below.

<< Graphics`Colors`
Clear[a1, omega, y, t, firstplot]
a1 = 1;
omega = 2 π;
y[t_] = E^(-a1*t)*Sin[omega*t];
firstplot  = 
        Plot[y[t], {t, 0, 10*Pi/omega}, PlotRange -> {-1, 1}, 
          PlotStyle -> Blue];

Clear[a2, omega, y, t, secondplot]\(a2 = 2; 
omega = 2\ π; 
y[t_] = E^\((\(-a2\)*t)\)*Sin[omega*t]; 

    secondplot\  = 
      Plot[y[t], {t, 0, 10*Pi/omega}, PlotRange -> {\(-1\), 1}, 
        PlotStyle -> Red]; \)

Clear[a3, omega, y, t, thirdplot]\(a3 = 0.5; 
omega = 2\ π; 
y[t_] = E^\((\(-a3\)*t)\)*Sin[omega*t]; 
    
thirdplot\  = 
      Plot[y[t], {t, 0, 10*Pi/omega}, PlotRange -> {\(-1\), 1}, 
        PlotStyle -> Green]; \)

Show[firstplot,  secondplot,  thirdplot];
Print["\!\(\*
StyleBox[\(a\),
FontColor->RGBColor[0, 0, 1]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 0, 1]]\)",a1,"     \!\(\*
StyleBox[\(a\),
FontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[1, 0, 0]]\)",a2,"     \!\(\*
StyleBox[\(a\),
FontColor->RGBColor[0, 1, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 1, 0]]\)",a3];
Print["omega = ", omega, " (for all graphs)"];

The following cells are set up to let you plot E^(-at)Sin[omega*t] using three different values of "omega".  You can look at all three graphs together using the "Show" command.

Clear[a, omega1, y, t, firstplot]\(a = 0.5; 
omega1 = 2\ π; 
y[t_] = E^\((\(-a\)*t)\)*Sin[omega1*t]; 
    
firstplot\  = 
      Plot[y[t], {t, 0, 10*Pi/omega1}, PlotRange -> {\(-1\), 1}, 
        PlotStyle -> Blue]; \)

Clear[a, omega2, y, t, secondplot]\(a = 0.5; 
omega2 = π; 
y[t_] = E^\((\(-a\)*t)\)*Sin[omega2*t]; 

    secondplot\  = 
      Plot[y[t], {t, 0, 10*Pi/omega2}, PlotRange -> {\(-1\), 1}, 
        PlotStyle -> Red]; \)

Clear[a, omega3, y, t, thirdplot]\(a = 0.5; 
omega3 = 10; 
y[t_] = E^\((\(-a\)*t)\)*Sin[omega3*t]; 

    thirdplot\  = 
      Plot[y[t], {t, 0, 10*Pi/omega3}, PlotRange -> {\(-1\), 1}, 
        PlotStyle -> Green]; \)

Show[firstplot,  secondplot,  thirdplot, 
        PlotRange -> {{0, 2*Pi/Min[omega1, omega2, omega3]}, {-1, 1}}];
Print["\!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[0, 0, 1]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 0, 1]]\)",N[omega1],"     \!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[1, 0, 0]]\)",N[omega2],"     \!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[0, 1, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 1, 0]]\)",N[omega3]];
Print[" a = ", a];

Clear[extraplot, y];
extraplot = 
          Plot[E^(-a*t), {t, 0, 2*Pi/Min[omega1, omega2, omega3]}, 
            DisplayFunction -> Identity];
Show[firstplot,  secondplot,  thirdplot, extraplot, 
          PlotRange -> {{0, 2*Pi/Min[omega1, omega2, omega3]}, {-1, 1}}]
          ;
Print["\!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[0, 0, 1]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 0, 1]]\)",N[omega1, 2],"     \!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[1, 0, 0]]\)",N[omega2],"     \!\(\*
StyleBox[\(omega\),
FontColor->RGBColor[0, 1, 0]]\)\!\(\*
StyleBox[\(=\),
FontColor->RGBColor[0, 1, 0]]\)",N[omega3]];
Print[" a = ", a, " (for all graphs)"];
Print["black line:  y = ", e^(-a*t)];

y(x) = (e^(-((((x - a))^2)/b)))

Clear[a1, b1, f1, x, firstplot]\(a1 = 0; 
b1 = 2; 
f1[x_] = E\^\(-\(\((x - a1)\)\^2\/b1\)\); 

    firstplot\  = \ 
      Plot[f1[x], {x, a1 - 1.5*b1, a1 + 1.5*b1}, PlotRange -> {0, 1}, 
        PlotStyle -> Blue]; \)

Clear[a2, b2, f2, x, secondplot]\(a2 = 0; 
b2 = 5; 
f2[x_] = E\^\(-\(\((x - a2)\)\^2\/b2\)\); 

    secondplot\  = \ 
      Plot[f2[x], {x, a2 - 1.5*b2, a2 + 1.5*b2}, PlotRange -> {0, 1}, 
        PlotStyle -> Red]; \)

Show[firstplot, secondplot];
Print["\!\(\*
StyleBox[\(BLUE\),
FontColor->RGBColor[0, 0, 1]]\)\!\(\*
StyleBox[\(:\),
FontColor->RGBColor[0, 0, 1]]\)  a = ",a1,"   b = ",b1];
Print["\!\(\*
StyleBox[\(RED\),
FontColor->RGBColor[1, 0, 0]]\)\!\(\*
StyleBox[\(:\),
FontColor->RGBColor[1, 0, 0]]\)   a = ",a2,"   b = ",b2];

Tami's Commentary

The Exponentially Damped Sine Wave.

The first function might describe a linear spring with energy loss due to friction.  How quickly energy is lost is modeled by the parameter "a".  This energy loss is called "damping" and a spring which loses a lot of energy is called "highly damped."

Notice that you can interpret the function as being a sine wave with an amplitude that changes (decays exponentially, to be precise) with time:
Instead of    y = A sin(wt)
we have    y = A(t) sin(wt),    where A(t) = e^(-at).

The Normal Distribution

The second function is the "normal distribution" or "Gaussian distribution" and is frequently
used to describe quantities in statistics and physics.  In statistics, it can describe the distribution of a random variable.  For example, you may have heard that the grades on an
exam should be distributed in a "bell curve", which comes from a model probably using
a normal distribution (notice the bell shape of the curves above).  In physics the Gaussian distribution  may be a model for random noise or various other quantities.  In biology, it has been used to describe certain populations, and has been used to describe how the retina responds to incoming light.  It gets used all the time in modeling, and you will probably be able to tell me many more examples before you finish college.

The most important quality of this bell-shaped distribution is where it is centered, which is called the "mean."  In the bell-curve grades model this is just the average grade, and in the other models it has essentially the same interpretation.  The other important quality of this curve is the width, or "standard deviation."  In the grades model, this tells you how spread
out the grades are.  If almost everyone gets about 75 points on the exam, then the mean will be a 75 and the curve will be narrow (sharply peaked).  If the grades range randomly over the
whole interval from 50 to 100, then the mean could still be 75, but the standard deviation would be much greater -- meaning that it is more likely that your grade will be considerably different from the mean of 75.

Homework Help

\(Clear[p, a, b, x]; 
a\  = \ \(-1\); 
p[x_]\  = \ x^3 - a*x; 


    Plot[p[x], {x, \(-5\), 5}]; 
Print["a = ", a]; \)

Clear[T, t, a, k, b, Derivative];
a  =  3;
b  =  2;
k  =  7;
T[t_]  =  a*(1 - E^(-k*t)) + b;
Print["T[t] = ", a*(1 - E^(-k*t)), " + ", b];
Print["
T'[t] = ",D[T[t], t]];
Plot[T[t], {t, 0, 2}, PlotRange -> {0, 1.1*(a + b)}];

\(Clear[f, x, k]; 
k\  = \ 7; 
f[x_]\  = \ x\  - \ k*Sqrt[x]; 

    Print["f[x] = ", f[x]]; 
Print["f'[x] = ", D[f[x], x]]; 

    Plot[f[x], {x, 0, 1.5*k^2}]; \)

\(Clear[f, x, k]; 
f[x_]\  = \ x\  - \ k*Sqrt[x]; 

    Print["f[x] = ", f[x]]; 
Print["f'[x] = ", D[f[x], x]]; \)

\(Clear[y, x, b]; 
b\  = \ 1; 
y[x_]\  = \ x*E^\((\(-b\)*x)\); 

    Plot[y[x], {x, \(-1\)/10*b, 6/b}]; \)

\(Clear[f, x, A, B]; 
A\  = \ 1; 
B\  = \ 0; 

    f[x_]\  = \ A/\((x + B)\); 

    Plot[f[x], {x, \(-B\) - 2, \(-B\) + 2}, PlotRange -> {\(-50\), 50}]; 

    Print["A = ", A, "    B = ", \ B]; \)