Which of the following will have the highest frequency? 1 point y = 5sin(x/2) y = 3sin(x) y = 8sin(2x) y = sin(3x)

usagirl007A 2021-05-02 Answered
Which of the following will have the highest frequency? 1 point
\(\displaystyle{y}={5}{\sin{{\left(\frac{{x}}{{2}}\right)}}}\)
\(\displaystyle{y}={3}{\sin{{\left({x}\right)}}}\)
\(\displaystyle{y}={8}{\sin{{\left({2}{x}\right)}}}\)
\(\displaystyle{y}={\sin{{\left({3}{x}\right)}}}\)

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Expert Answer

FieniChoonin
Answered 2021-05-03 Author has 14598 answers

We can write out a general mathematical wave as
\(\displaystyle{y}={A}{\sin{{\left({f}⋅{x}+ϕ\right)}}}\)
where A is the half-amplitude, f is the frequency, and \(\phi\) is the phase angle.
The half-amplitude is half of the vertical distance from the top of the wave (the crest) to the bottom of the wave (the trough). The frequency (what we're asked about) gives an idea of how "fast" the wave is moving. Graphically, it has to do with how skinny or wide the crests and troughs are - the bigger the frequency, the skinnier the wave and conversely the smaller the frequency, the fatter the wave. And finally the phase angle basically has to do with when exactly you start your stopwatch. Graphically it has to do with where the wave crosses the y-axis.
Let's write your four choices out in this general form.
\(\displaystyle{y}={5}{\sin{{\left(\frac{{x}}{{2}}\right)}}}⟶{y}={5}{\sin{{\left(\frac{{1}}{{2}}{x}+{0}\right)}}}\)
\(\displaystyle{y}={3}{\sin{{\left({x}\right)}}}⟶{y}={3}{\sin{{\left({1}{x}+{0}\right)}}}\)
\(\displaystyle{y}={8}{\sin{{\left({2}{x}\right)}}}⟶{y}={8}{\sin{{\left({2}{x}+{0}\right)}}}\)
\(\displaystyle{y}={\sin{{\left({3}{x}\right)}}}⟶{y}={1}{\sin{{\left({3}{x}+{0}\right)}}}\)
Clearly the last option

\((y=\sin(3x))\)

is the one with the highest frequency.

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