Amplitude, Period, and Phase Shift

by 
Rachel Cuttriss
Lutheran High School
3500 W. Washington
Springfield, Illinois 62707
(217)546-6363

Description:  The lesson presented here is to assist students in correctly identifying the period, amplitude, and phase shifts in the sine function.   The student will be asked to compare graphs and describe the changes that take place and what causes the changes.

Initialization

You will need to execute the following cell to get the colors you will need.

Needs["Graphics`Colors`"];

The Basic Curve

The basic equation for the Sine curve is  y = aSin [px-s].  The resulting graph is displayed  by the following cell.

Clear[x, y]\(\(Plot\ [\ Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Red, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

Changes in Amplitude

\(TextForm
    \`The\ amplitude\ of\ a\ periodic\ function\ is\  | A | 
      where\ A\ is\ the\ distance\ from\ the\ axis\ to\ the\ highest\ point\ 
        of\ the\ function . \ The\ amplitude\ of\ the\ Sine\ function\ above\ 
        is\ 1. \ \ Execute\ the\ following\ cells\ and\ find\ the\ amplitude\ 
        in\ each\ \(graph . \)\)

Clear[x, y]\(\(Plot\ [2\ Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotStyle -> {Green, Thickness[ .02]}\ , 
      PlotLabel -> FontForm["y = 2Sin [x]", {"Geneva", 12}]]; 
    \)
\)

What is the amplitude of this graph?

Clear[x, y]\(\(Plot\ [3  Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotStyle -> {Green, Thickness[ .02]}\ , 
      PlotLabel -> FontForm["y = 3Sin [x]", {"Geneva", 12}]]; 
    \)
\)

What is the amplitude of this graph? Take a second and look at the equation closely.  What do you notice now that you have done a couple on your own?

Clear[x, y]\(\(Plot\ [ .5\ Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-1\), 
            \(- .5\),  .5, 1}}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-1\), 1}}, 
      PlotStyle -> {Green, Thickness[ .02]}\ , 
      PlotLabel -> FontForm["y = 1/2Sin [x]", {"Geneva", 12}]]; 
    \)
\)

Amplitude?  Does you conclusion from above still hold true?

Clear[x, y]\(\(Plot\ [\((1/3)\)\ Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-1\), 
            \(- .5\),  .5, 1}}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-1\), 1}}, 
      PlotStyle -> {Green, Thickness[ .02]}\ , 
      PlotLabel -> FontForm["y = 1/3 Sin [x]", {"Geneva", 12}]]; 
    \)
\)

Amplitude?  Do you neeed to make any revisions?

Clear[x, y]\(\(Plot\ [\((1/4)\) Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-1\), 
            \(- .5\),  .5, 1}}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-1\), 1}}, 
      PlotStyle -> {Green, Thickness[ .02]}\ , 
      PlotLabel -> FontForm["y = 1/4 Sin [x]", {"Geneva", 12}]]; 
    \)
\)

Amplitude?

Let's take a minute and summarize what you have seen. The equation that we began with was y = a Sin [px -s].   What happens when the leading coefficient (a) changes?    Do you see any patterns or generalities ?  What do you think the equation of the Sine function would look like if the amplitude was 7?  What if the amplitude is 1/5?

Changes in the Period

The period of a function is the distance along the x-axis which is need to complete one cycle of a function.  In other words, how long does it take before the function starts over.

Remember the graph we started with?  Well, here it is again, just in case you need a refresher.

Clear[x, y]\(\(Plot\ [\ Sin[x], {x, 0, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-2\), 
            \(-1\), 1, 2}}, PlotStyle -> {Red, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

The period of the Sine function is 2π.

Clear[x, y]\(\(Plot\ [\ Sin[2  x], {x, 0, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-2\), 
            \(-1\), 1, 2}}, PlotStyle -> {Blue, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> FontForm["y = Sin [2x]", {"Geneva", 12}]]; 
    \)
\)

How is this graph differen from the original?  Can you find
       the period?

Clear[x, y]\(\(Plot\ [\ Sin[3  x], {x, 0, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-2\), 
            \(-1\), 1, 2}}, PlotStyle -> {Blue, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> FontForm["y = Sin [3x]", {"Geneva", 12}]]; 
    \)
\)

Find the period of this graph .

Clear[x, y]\(\(Plot\ [\ Sin[4  x], {x, 0, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-2\), 
            \(-1\), 1, 2}}, PlotStyle -> {Blue, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> FontForm["y = Sin [4x]", {"Geneva", 12}]]; 
    \)
\)

What is the period?

Clear[x, y]\(\(Plot\ [\ Sin[ .5  x], {x, 0, 4  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π, 3  π, 
            4  π}, {\(-2\), \(-1\), 1, 2}}, 
      PlotStyle -> {Blue, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 4  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [(1/2)x]", {"Geneva", 12}]]; \)
\)

The period here seems to be a bit different than before.  What differences do you notice?

Clear[x, y]\(\(Plot\ [\ Sin[\((1/3)\) x], {x, 0, 6  π}, 
      Ticks -> {{\(-2\) π, \ 2  π, 4  π, 6  π}, {\(-2\), 
            \(-1\), 1, 2}}, PlotStyle -> {Blue, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 6  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [(1/3)x]", {"Geneva", 12}]]; \)
\)

Can you find the period in this graph?  Remember, period is how long does it take to make one complete cycle for the function.

The equation that we began with was y = a Sin [px -s].   What happens when the coefficient of x, (p), changes?    Do you see any patterns or generalities ?  What do you think the graph would look like if p were 6?  What if the p were 1/8?

Phase Shifts

A phase shift occurs when the graph does not start in the original position, but has moved either left or right.  The graph of the Sine function normally starts at 0, but as you will see in the following graphs, it can start anywhere.  We will see changes occur in the graph as we change the value of s.

Clear[x, y]\(\(Plot\ [\ Sin[x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Red, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

Just wanted to make sure you didn't forget where we started.

Clear[x, y]\(\(Plot\ [\ Sin[x - π], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Orange, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [x-π]", {"Geneva", 12}]]; \)
\)

Do you see any changes?  Where does the cycle begin?  Where does it end?  What is the period?

Clear[x, y]\(\(Plot\ [\ Sin[x + π], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Orange, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [x + π]", {"Geneva", 12}]]; \)
\)

Looks like the last one.  Why?????

Clear[x, y]\(\(Plot\ [\ Sin[x - \((π/2)\)], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Orange, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [x - (π/2)]", {"Geneva", 12}]]; \)

    \)

This looks different.  How is it different?  What is the phase shift?

Clear[x, y]\(\(Plot\ [\ Sin[x - \((π/4)\)], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Orange, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-2\), 2}}, 
      PlotLabel -> 
        FontForm["y = Sin [x - (π/4)]", {"Geneva", 12}]]; \)

    \)

Phase shift?

Let's summarize.   What do you think is the largest phase shift possible?  What does a positive phase shift do?  What about a negative? What do you think a graph with a  phase shift of -π/4 would look like?

The Finale

The equation we began with is  y = a Sin [ px -s].  What we have seen is that changes in a make the amplitude, or height of the graph changes.  Changes in p make the time to complete one cycle of the function is lengthened or shortened.  When s changes the graph slides to the right or the left.  The following graphs will include more than one change at a time.  Your job is to find the amplitude, period and phase shift that you see in the graph.

Clear[x, y]\(\(Plot\ [\ 2  Sin[x + π], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Purple, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}]; \)
\)

Amplitude?  Period?  Phase Shift?

Clear[x, y]\(\(Plot\ [ .5\ Sin[ .5  x], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Purple, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}]; \)
\)

Amplitude?  Period?  Phase Shift?

Clear[x, y]\(\(Plot\ [3  Sin[2  x - π], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Purple, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

Amplitude?  Period?  Phase Shift?

Clear[x, y]\(\(Plot\ [\ Sin[3  x - π/4], {x, \(-2\) π, 2  π}, 
      Ticks -> {{\(-2\) π, \(-π\), \ π, 2  π}, {\(-3\), 
            \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Purple, Thickness[ .02]}, 
      PlotRange -> {{\(-2\) π, 2  π}, {\(-3\), 3}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

Amplitude?  Period?  Phase Shift?

Clear[x, y]\(\(Plot\ [\ 2  Sin[ .5  x - π/2], {x, \(-4\) π, 4  π}, 
      Ticks -> {{\(-4\) π, \(-2\) π, \ 2  π, 4  π}, {
            \(-3\), \(-2\), \(-1\), 1, 2, 3}}, 
      PlotStyle -> {Purple, Thickness[ .02]}, 
      PlotRange -> {{\(-4\) π, 4  π}, {\(-3\), 3}}, 
      PlotLabel -> FontForm["y = Sin [x]", {"Geneva", 12}]]; \)

    \)

Amplitude?  Period?  Phase Shift?