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1 Printed from the Mathematica Help Browser of.: Transformation of Functions In this section, we will explore three types of transformations:.) Shifting.) Reflections (or flips).) Stretches and compressions Additionally, each of these three types of transformations can impact a function horizontally or vertically. Thus, there are six basic transformations that must be learned. And, of course, they can be combined... of Shifts: Horizontal and Vertical The follow graphs show examples of what is meant by shifting. In[]:= OffPlot::"plnr"; OffGraphics::"gptn"; OffParametricPlotD::"plld" Needs"Graphics`Arrow`" Needs"VisualLA`" Wolfram Research, Inc. All rights reserved.
2 Printed from the Mathematica Help Browser In[65]:= fx_ : x Sin x; ShowGraphicsArray Plotfx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Original Function", Plotfx, fx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Vertical Shift", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.6, f.6,.6, f.6, Plotfx, fx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Horizontal Shift", PlotStyle GrayLevel0.8, Red, Epilog Arrow0, f0,, f0 ; Original Function Vertical Shift Horizonta of Wolfram Research, Inc. All rights reserved.
3 Printed from the Mathematica Help Browser Vertical Shifts To shift vertically,we will add a constant A from the function (A negative in the case where we want subtraction). That is, we will consider graphs of fx A for A of Example : Vertical Shifts For example, if f x is as given below, consider f x. In[6]:= TablePlotfx, fx A, x,,, PlotRange,, PlotLabel "Vertical Shift", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.6, f.6,.6, f.6 A, A, 0,, 0.05; Vertical Shift of Example : Vertical Shifts If fx is as given below, consider fx. In[7]:= TablePlotfx, fx A, x,,, PlotRange,, PlotLabel "Vertical Shift", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.6, f.6,.6, f.6 A, A, 0,, 0.; Wolfram Research, Inc. All rights reserved.
4 Printed from the Mathematica Help Browser Vertical Shift of Summary: Vertical Shifts Given a function fx and a constant A, then: The graph of fx A is a shifting of fx up by A if A0and down by A if A0 7 of Horizontal Shifts To shift horizontally, we will subtract a constant A from the function (A negative in the case where we want addition). That is, we will consider graphs of fx A for A 8 of Example : Horizontal Shifts If fx is as given below, consider fx. In[8]:= Table Plotfx, fx A, x,,, PlotRange,, PlotLabel "Horizontal Shift", PlotLabel "Horizontal Shift", PlotStyle GrayLevel0.8, Red, Epilog Arrow0, f0, A, f0, A, 0,, 0.; Wolfram Research, Inc. All rights reserved.
5 Printed from the Mathematica Help Browser 5 Horizontal Shift of Example : Horizontal Shifts If fx is as given below, consider fx fx. In[9]:= Table Plotfx, fx A, x,,, PlotRange,, PlotLabel "Horizontal Shift", PlotLabel "Horizontal Shift", PlotStyle GrayLevel0.8, Red, Epilog Arrow0, f0, A, f0, A, 0,, 0.05; Horizontal Shift of Summary: Horizontal Shifts (Counter-intuitive) Given a function fx and a constant A, then: The graph of fx A is a shifting of fx right by A if A0and left by A if A Wolfram Research, Inc. All rights reserved.
6 6 Printed from the Mathematica Help Browser of Reflections: About the y-axis and about the x-axis The follow graphs show examples of what is meant by shifting. In[6]:= gx_ : x ; ShowGraphicsArray Plotgx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Original Function", Plotgx, gx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Flip about the y axis", PlotStyle GrayLevel0.8, Blue, Plotgx, gx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Flip about the x axis", PlotStyle GrayLevel0.8, Red ; Original Function Flip about the y axis Flip abou 0.5. of Wolfram Research, Inc. All rights reserved.
7 Printed from the Mathematica Help Browser 7 Example 5: Reflections: About the y-axis To reflect or flip about the y-axis, we negate the argument of the function. That is, we graph fx In[]:= dq Pi5; fctcurveq_ : ParametricPlotDt Cosq, t Sinq, gt, t 5, 0, 0, 0, 0, t, t, 0, 5, PlotRange 5, 5, 5, 0, 0,, Boxed False, Axes False, ViewPoint.00,.7, 0.779, DisplayFunction Identity; arrowshaftq_ : ParametricPlotDCosu, Sinu,, u, 0, q dq, PlotRange 5, 5, 5, 0, 0,, Boxed False, Axes False, ViewPoint.00,.7, 0.779, DisplayFunction Identity; arrowheadq_ : DrawVectorDCosq, Sinq,, Cosq dq, Sinq dq,, HeadLength 0., DisplayFunction Identity; TableShowfctcurveq, arrowshaftq, arrowheadq, DisplayFunction $DisplayFunction, q, 0., Pi, dq; of Example 6: Reflections: About the x-axis In[8]:= To reflect or flip about the xaxis, we negate function. That is, we graph fx In[7]:= gx_ : x ; dq Pi5; fctcurveq_ : ParametricPlotD t, gt Cosq, Sinq gt, t5, 0, 0, 0, 0, t5, t, 0, 5, Boxed False, Axes False, ViewPoint.00,.7, 0.779, DisplayFunction Identity; arrowshaftq_ : ParametricPlotD, Cosu, Sinu, u,.5 Pi, q dq, Boxed False, Axes False, ViewPoint.00,.7, 0.779, DisplayFunction Identity; arrowheadq_ : DrawVectorD, Cosq, Sinq,, Cosq dq, Sinq dq, HeadLength 0., DisplayFunction Identity; TableShowfctcurveq, arrowshaftq, arrowheadq, DisplayFunction $DisplayFunction, PlotRange 5, 5, 5, 5, 5, 5, q,.5 Pi, 0.5 Pi, dq; Wolfram Research, Inc. All rights reserved.
8 8 Printed from the Mathematica Help Browser of Summary: Reflections (or Flips) Given a function fx, then: The graph of fx is a reflection (or flip) of f about the y-axis and the graph of fx is a reflection of f about the x-axis. 5 of Stretches and Compressions: Vertical and Horizontal The follow graphs show examples of what is meant by stretching and compressing Wolfram Research, Inc. All rights reserved.
9 Printed from the Mathematica Help Browser 9 In[6]:= hx_ : 0.5x x x x ; ShowGraphicsArray Plothx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Original Function", Plothx, hx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Vertical Stretch", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.58, h.58,.58, h.58, Arrow.5, h.5,.5, h.5, Arrow.5, h.5,.5, h.5, Plothx, hx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "Horizontal Compression", PlotStyle GrayLevel0.8, Red, Epilog Arrow.5, h.5,.5, h.5, Arrow.5, h.5,.5, h.5 ; Original Function Vertical Stretch Horizontal Co 6 of Wolfram Research, Inc. All rights reserved.
10 0 Printed from the Mathematica Help Browser Vertical Stretches and Compressions (Squashes) To stretch or compress vertically, we will multiply the f by a positive constant A. That is, we will consider graphs of A fx for A0 and A(why A ) 7 of Example 7: Vertical Stretches If f x is as given below, consider f x. In[5]:= TablePlothx, A hx, x,,, PlotRange,, PlotLabel "Vertical Stretch", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.58, h.58,.58, A h.58, Arrow.5, h.5,.5, A h.5, Arrow.5, h.5,.5, A h.5, A,,, 0.; Vertical Stretch of Example 8: Vertical Compressions (Squashes) If fx is as given below, consider fx. In[6]:= TablePlothx, A hx, x,,, PlotRange,, PlotLabel "Vertical Compression", PlotStyle GrayLevel0.8, Blue, Epilog Arrow.58, h.58,.58, A h.58, Arrow.5, h.5,.5, A h.5, Arrow.5, h.5,.5, A h.5, A,,, 0.05; Wolfram Research, Inc. All rights reserved.
11 Printed from the Mathematica Help Browser Vertical Compression of Summary: Vertical Stretches and Compressions Given a function fx and a constant A, then: The graph of A fx is a stretching of fx about the x-axis if Aand a compression if 0 A. 0 of Horizontal Stretches and Compressions (Squashes) To stretch or compress horizontally, we will multiply the argument of f by a positive constant A. That is, we will consider graphs of fa x for A0 and A (why A ) of Example 9: Horizontal Stretches If fx is as given below, consider f x. In[7]:= Table Plothx, ha x, x, 5, 5, PlotRange,, PlotLabel "Horizontal Stretch", PlotStyle GrayLevel0.8, Red, Epilog Arrow.5, h.5,.5 A, h.5, Arrow.5, h.5,.5, h.5, A,, 0.5, 0.05; A Wolfram Research, Inc. All rights reserved.
12 Printed from the Mathematica Help Browser Horizontal Stretch of Example 0: Horizontal Compressions (Squashes) If f x is as given below, consider f x. In[8]:= TablePlothx, ha x, x, 5, 5, PlotRange,, PlotLabel "Horizontal Compression", PlotStyle GrayLevel0.8, Red, Epilog Arrow.5, h.5,.5 A, h.5, Arrow.5, h.5,.5, h.5, A,,, 0.; A Horizontal Compression of Wolfram Research, Inc. All rights reserved.
13 Printed from the Mathematica Help Browser Summary: Horizontal Stretches and Compressions (Counter-intuitive) Given a function fx and a constant A, then: The graph of fa x is a stretching of fx about the y-axis if 0 Aand a compression if A. of Conclusion of Basic Transformations We explored the three types of transformations: 5 of A Quiz on Transformations In[69]:= OffPlot::"plnr"; OffSolve::"ifun"; OffLessEqual::"nord"; $TextStyle FontSize ; $FormatType TraditionalForm; gx_ If x, x, If x, x; QQQ ; Fori, i, i, Forj, j 5, j, AppendToQQQ, i, j; GFCT_, LABEL_:"" : ModulePLOT, PLOT PlotFCT, x, 5, 5, PlotPoints 00, DisplayFunction Identity; ShowGraphicsPoint QQQ, PLOT, Axes True, AxesLabel x, "", PlotLabel LABEL, AspectRatio, DisplayFunction Identity; Wolfram Research, Inc. All rights reserved.
14 Printed from the Mathematica Help Browser In[78]:= ShowGgx, y, DisplayFunction $DisplayFunction; y x 6 of An Introduction to Symmetry Mathematicians are notoriously lazy. By this, I do not mean that we are unwilling to work hard (far be it). But rather, that we are unwilling to work hard if there is an easier way. Symmetry is a nice example. Symmetry allows us to understand a whole situation with half the work by simply recognizing the symmetry. Two famous symmetries are known as odd and even symmetry. 7 of Odd and Even Symmetry The following graphs show no symmetry, odd, and even symmetry Wolfram Research, Inc. All rights reserved.
15 Printed from the Mathematica Help Browser 5 In[60]:= ShowGraphicsArray Plot x, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "No Symmetry", Plotx Sin x, x,,, PlotRange 7, 7, DisplayFunction Identity, PlotLabel "Odd Symmetry", PlotStyle Blue, Plot0.5Cos x.5, x,,, PlotRange,, x DisplayFunction Identity, PlotLabel "Even Symmetry", PlotStyle Red ; No Symmetry Odd Function 6 Even Symmet 6 8 of Odd Functions Odd functions are symmetric about the origin. To see odd symmetry, rotate the graph about the origin by 80 degrees and see if your resultant graph is identical to the original.. Examples of odd functions include: Wolfram Research, Inc. All rights reserved.
16 6 Printed from the Mathematica Help Browser In[59]:= ShowGraphicsArray Plotx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "yx", Plotx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "yx ", PlotStyle Blue, PlotSignx x 6, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "y x ", PlotStyle Red, Plot, x,,, PlotRange,, DisplayFunction Identity, x PlotLabel "y ", PlotStyle Green, x PlotSinx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "ysinx ", PlotStyle Purple ; yx y x Wolfram Research, Inc. All rights reserved.
17 Printed from the Mathematica Help Browser 7 ysinx 9 of Odd Functions: The Test For odd functions, fx fx. (Why?) Examples.) fx x x.) gx x x.) hxx x 0 of Even Functions Even functions are symmetric about the y-axis. resultant graph is identical to the original. To see even symmetry, flip the graph about the y-axis and see if your Examples of even functions include: Wolfram Research, Inc. All rights reserved.
18 8 Printed from the Mathematica Help Browser In[67]:= ShowGraphicsArray Plot, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "y", Plotx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "yx ", PlotStyle Blue, PlotAbsx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "yx", PlotStyle Red, PlotCosx, x,,, PlotRange,, DisplayFunction Identity, PlotLabel "ycosx ", PlotStyle Purple ; y yx Wolfram Research, Inc. All rights reserved.
19 Printed from the Mathematica Help Browser 9 of Even Functions: The Test For odd functions, f x f x. (Why?) Examples.) fx x.) gx xx.) hx x x of Summary: Symmetry Two famous kinds of symmetry of functions are odd and even symmetry. Odd functions are symmetric about the origen and satisfy fx fx. Even functions are symmetric about the y-axis and satisfy fx fx. Not all functions are symmetric and many symmetric problems can be solved without reference to symmetry. However, the mathematician who can account for symmetry shows promise Wolfram Research, Inc. All rights reserved.
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