de-CH
utf-8
math graphie polynomials
Determine Frequency Modulation
t-03-02
number
50
randRange(2,7) randRange(1,10) 0.5 * a

Given are the graph of the sine function {\color{orange}f(x)=\sin(x)} and the graph of the function {\color{blue}g} with {\color{blue}g(x)= A\cdot\sin({\color{red}b}\cdot x)}.

Determine {\color{red}b}.

graphInit({ range: [[-1, 7],[-A-1, A+1]], scale: [ 25, 25 ], gridStep: [ 1* PI / 4, 1 ], tickStep: [ 2, 1 ], labelStep: [ 4, 1 ], unityLabels: true, xLabelFormat: piFraction }); label( [ 0, A+1 ], "y", "above" ); label( [ 7, 0 ], "x", "right" ); label( [ PI, 0 ], "\\pi", "below" ); style({stroke: "black", strokeWidth: 2}); plot(function(x) {return A*sin(b*x);}, [-1, 10], {stroke: "blue"}); plot(function(x) {return sin(x);}, [-1, 10], {strokeDasharray: ".",stroke: "orange"});

b

The factor {\color{red}b} causes scaling (stretching/compression) along the x-axis and is equal to the angular frequency \omega.

In the (blue) graph of {\color{blue}g} we observe b full periods in the intervall 0 till 2\pi.

Thus, the frequency is F = number of oscillations per unit = \dfrac{b}{2\pi}

and the angular frequency is

\omega = {\color{red}b} =2\pi \cdot F = 2\pi \cdot \dfrac{b}{2\pi} = b.