Function Transformations

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Section A.3 Function Transformations

In pure mathematics settings, both the inputs and the outputs of functions are typically pure numbers, without dimensions, and the descriptions of functions like \(\cos x\) and \(e^x\) are chosen to make them look as simple as possible. But, in applied settings these functions must be tailored to model the physical situation, by inserting parameters that may have dimensions, such as \(A \cos(kx-\delta)\) and \(A e^{(kx-\delta)}\text{.}\) How do these parameters change the shape of the graphs?

Activity A.1. Visualizing Parameters that Transform Functions.

In the applet below, you can enter a function of your choice, of the form \(A f(kx-\delta)\text{,}\) where \(A\text{,}\) \(k\text{,}\) and \(\delta\) are three arbitrary real parameters. (The default function is \(A \cos(kx-\delta)\text{.}\)) Determine how (and why!) the three parameters change the function.

Figure A.3. An applet that allows you to change the values of \(A\text{,}\) \(k\text{,}\) and \(\delta\) in \(A f(kx-\delta)\text{.}\) The default function displayed is \(A \cos(kx-\delta)\text{.}\)

Solution.

Because \(A\) multiplies the overall value of the function (i.e. the range), increasing \(A\) increases the vertical extent of the graph by a factor of \(A\text{.}\) Because \(k\) multiplies the argument of the function (i.e. the domain), increasing \(k\) means the argument will increase faster, so the shape of function will be compressed horizontally. Because \(\delta\) is subtracted from the argument of the function (i.e. the domain), increasing delta makes the argument look less than it really is, so the whole graph moves horizontally, to the right.

Definition A.4. Amplitude, Wave Number, Angular Frequency, and Phase.

In the function \(A f(kx-\delta)\text{,}\) the parameter \(A\) is called the amplitude; the parameter \(k\) is called the wave number; and the parameter \(\delta\) is called the phase. If the variable in the function is time \(t\text{,}\) then the functional form is usually written \(A f(\omega t-\delta)\text{,}\) and the parameter \(\omega\) is called the angular frequency rather than wave number.