MATH 1113 PreCalculus

Day #23, Part I
Modifying the Sine and Cosine Graphs

REVIEW:  See Section 3.4, p. 216 in the textbook, about the effects of shifting and stretching general graphs.  We'll be applying the modifications, in particular, to the trig functions.

We'll first look at the effect of multiplying the output of a graph.  Given any function, f(x), if we compare the graph of y = f(x) to y = A*f(x), it is clear that the effect of multiplying AFTER applying the function simply multiplies the height by the corresponding amount, so that if A>1, we stretch the graph vertically, while if 0<A<1 we compress it vertically, and A<0, the stretching or compression is combined with a vertical flip.  See the following video of the cosine function:

Video

In the video, we see that multiplying the cosine function by 2 stretches the graph so that its new high and low points are 2 units from the x-axis.  This value (from the axis to the high or low points) is called the "amplitude" of the trig function, so the function:

f(x) = A*cos(x) or f(x) = A*sin(x) has amplitude |A|.

Here's an interactive GEOGEBRA applet that I hope works to demonstrate the effect of the constant "A" on the sine function.  To make the applet work, you'll most likely have to click on the warning bar at the top of the page and "allow ActiveX content."  The message will probably say that you need a newer version of Java, but before you go and update Java, try clicking on the yellow warning bar and "allow ActiveX content."  Let me know if you can't get it to work.

Geogebra applet:

Changes to the input: Period, Frequency and Phase Shift:

Period and Frequency: The natural sine, cosine, secant and cosecant functions have a period of 2*pi.  That means that they repeat themselves every 2*pi units, precisely because the terminal angle has made a complete circuit of the unit circle and the x and y coordinates are back to where they were 2*pi radians before.  (The tangent and cotangent functions have a period of pi, because of the interrelationship between the x and y coordinates and the fact that cos(x + pi) = -cos(x), and sin(x + pi) = - sin(x); therefore tan(x + pi) = sin(x + pi)/cos(x + pi) = -sin(x)/(-cos(x)) = sin(x)/cos(x).)

The period is the length of an oscillation (in units of time).  The frequency is the number of oscillations per unit time.  Therefore, the frequency is the reciprocal of the period: f = 1/P.

The period (and frequency) of a function are affected by the input -- that is, changing the x-value before applying the function.  The strange thing about changing the input of a function is that it appears to work backwards, until you realize that what you're actually transforming is not the function, but the input, that is, the x-axis.  Here's a video of what's actually going on.  (Most textbooks omit this crucial observation.)

Video:

Notice that what's actually happening is that the x-axis is stretching by a factor of 2, so that when the scale is returned to normal, it appears that the function graph has been compressed by a factor of 2.  This is why it seems that operating on the "inside" of a function does the "opposite" of what you might think it should -- just as (see below), when you subtract values on the inside, the graph moves right; what's actually happening is that the axis is moving left, and the function is staying put.

Therefore, when you have a function like f(x) = sin(b*x), increasing "b" increases the FREQUENCY (decreases the period).  In fact, the period is P = 2*pi/b, and the frequency is f = b/(2*pi).

Here's another Geogebra applet (boy, aren't you glad I went to Boston and learned how to make Geogebra applets!), demonstrating both the amplitude and the frequency modifications to a sine function.

Geogebra applet.

Phase Shift:  The phase shift is also a change to the input of a function.  It shifts the graph left or right by sliding the axis (input) right or left before applying the function.  Consequently, it also seems to work "backwards" from what you'd expect.  f(x+c) shifts the graph left for c > 0, since it is actually shifts the axes right.  Similarly, f(x-c) shifts the graph right for c > 0, since it actually shifts the axes left.  Here's a video:

Video:

There is also a vertical shift that works exactly as you'd expect it to (adding 3 to the outside of the function lifts the graph by 3 units, etc.

Here's a wonderful practice exercise to help, from the World-wide Interactive Mathematics Server, from the University of Nice, in France:

http://wims.unice.fr/wims/wims.cgi?session=DFFC32C676.1&lang=en&cmd=new&module=home&search_keywords=trigonometry&search_category=A

Page Last Modified: 27 February, 2007

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