Families of functions

Introduction

This Mathematica CDF notebook is licensed under a MouseOver .  It will be revised from time to time.  I hope anyone interested will feel free to improve this work and to use it in their own publications and coursework. 

Each section shows a family of functions with one or two parameters.  The interactive displays show interesting phenomena you might have taken much longer to discover with pencil and paper.  In some cases I have created questions about these phenomena, some that beginning calculus students should be able to do and others that are harder.

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Family 1: Parameter inside and outside parentheses

Manipulate[Plot[{a (x+b),a x+b},{x,-4,4},PlotRange->{{-4,4},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->1],{{a,1},-2,2,Appearance-> "Labeled"},{{b,1},-2,2,Appearance-> "Labeled"}]

Family 2: Parameter before and after squaring

Manipulate[Plot[{a (x+b)^2,a x^2+b},{x,-4,4},PlotRange->{{-4,4},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->1],{{a,1},-2,2,Appearance-> "Labeled"},{{b,1},-2,2,Appearance-> "Labeled"}]

For which pairs (a,b) do the two graphs intersect?  Ans: All.

If for a particular (a,b) the graphs intersect, what are the points of intersection?

Reduce[a (x+b)^2==a x^2+b,{a,b,x}]

For which pairs (a,b) are the two graphs identical?  Ans: b=0, a anything

Family 3: Frequency and Amplitude

Manipulate[Plot[{a Sin[x],Sin[a x]},{x,-2Pi,2Pi},PlotRange->{{-4,4},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->1],{{a,1},-4,4,Appearance-> "Labeled"}]

For which a do the two graphs intersect?  Ans : all a. (They all intersect at x = 0).

Reduce[0==Sin[a Pi],{a}]

Show that the two graphs intersect infinitely often, whatever a is.

For which a are the two graphs identical?  Ans : a = 1.

Find all the intersection points for any given real number a.  (I don’t know how to do this in general.  Mathematica has an algorithm for any particular rational a -- see below.)

Mathematica can' t produce a general solution:

Reduce[a Sin[x]==Sin[a x],{a,x},Rationals]

But it can do particular solutions if the parameters are rational :

Reduce[2Sin[x]==Sin[2 x],{x},Reals]

Reduce[-5Sin[x]==Sin[-5 x],{x},Reals]

Reduce[(6/5)Sin[x]==Sin[(6/5) x],{x},Reals]

Reduce[(-14/3)Sin[x]==Sin[(-14/3) x],{x},Reals]

Remove the semicolon and evaluate this, and you will get a very long answer:

Reduce[(14/41)Sin[x]==Sin[(14/41) x],{x},Reals];

Below is a nice try that doesn' t work :

Reduce[(m/n)Sin[x]==Sin[(m/n) x],{m,n,x},Integers]

Family 4: Sin[a x] and its derivative

Manipulate[Plot[{Sin[a x],a Cos[a x]},{x,-2Pi,2Pi},PlotRange->{{-4,4},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->1],{{a,1},-4,4,Appearance-> "Labeled"}]

Family 5: Exp[a x] and its derivative

Manipulate[Plot[{Exp[a x],a Exp[a x]},{x,-3,3},PlotRange->{{-3,3},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->4/3],{{a,2},-4,4,Appearance-> "Labeled"}]

Family 6: Doing Sine twice

Manipulate[Plot[{Sin[Sin[a x]],Sin[a Sin[x]]},{x,-10,10},PlotRange->{{-7,7},{-1.5,1.5}},PlotStyle->{Blue,Red},ImageSize->Large,AspectRatio->3/14],{{a,1},-4,4,Appearance-> "Labeled"}]

Family 7: ArcTan and its derivative

Manipulate[Plot[{ArcTan[a x],a/(1+a^2 x^2)},{x,-4,4},PlotRange->{{-4,4},{-4,4}},PlotStyle->{Blue,Red},AspectRatio->1],{{a,1},-4,4,Appearance-> "Labeled"}]

Family 8: Cubic

Manipulate[Plot[x^3+a x^2+b x,{x,-4,4},PlotRange->{{-4,4},{-8.5,8.5}},PlotStyle->Blue,AspectRatio->17/8],{{a,1},-3,3,Appearance-> "Labeled"},{{b,1},-2,2,Appearance-> "Labeled"}]

Family 8: Rational Function

Manipulate[Plot[(x+a)/(x-b),{x,-4,4},PlotRange->{{-4,4},{-4,4}},PlotStyle->Blue,AspectRatio->1],{{a,1},-3,3,Appearance-> "Labeled"},{{b,1},-2,2,Appearance-> "Labeled"}]

Family 9: Messing with Fresnel’s function

Manipulate[Plot[{Sin[(a x+b)^2],(a x+b)^2},{x,-7,7},PlotRange->{{-5,5},{-1.2,8}},PlotStyle->{Blue,Red},AspectRatio->9.2/10],{{a,1},-2,2,Appearance-> "Labeled"},{{b,1},-2,2,Appearance-> "Labeled"}]

Family 10: Quartic

Manipulate[Plot[x^4+a x^3+b x^2+c x,{x,-3,3},PlotRange->{{-3,3},{-6,6}},PlotStyle->Blue,AspectRatio->2,ImageSize->Medium],{{a,1},-1.5,1.5,Appearance-> "Labeled"},{{b,1},-1.5,1.5,Appearance-> "Labeled"},{{c,1},-1.5,1.5,Appearance-> "Labeled"}]