Question

Describe the transformations and sketch the graphs of the following trigonometric functions f(x)=-2 \cos(2x)+3\ b. g(x)=3 \sin (x-\pi)-1

Transformations of functions
ANSWERED
asked 2021-08-08

Describe the transformations and sketch the graphs of the following trigonometric functions. \(\displaystyle{a}.{f{{\left({x}\right)}}}=-{2}{\cos{{\left({2}{x}\right)}}}+{3}\\ {b}.{g{{\left({x}\right)}}}={3}{\sin{{\left({x}-\pi\right)}}}-{1}\)

Expert Answers (1)

2021-08-09

Step 1
We know \(\displaystyle{y}={A}{\sin{{\left({B}{x}-{C}\right)}}}+{D}\) is general equation of a sin function.
where \(\displaystyle{\left|{A}\right|}\) is amplitude, \(\displaystyle{\frac{{{2}\pi}}{{{\left|{B}\right|}}}}\) is time period, \(\displaystyle{\frac{{{C}}}{{{B}}}}\) is Phase shift and D is vertical shift
(a)
\(\displaystyle{f{{\left({x}\right)}}}=-{2}{\cos{{\left({2}{x}\right)}}}+{3}\)
\(\displaystyle={2}{\sin{{\left({\frac{{\pi}}{{{2}}}}-{2}{x}\right)}}}+{3}\)
\(\displaystyle={2}{\sin{{\left(-{2}{x}+{\frac{{\pi}}{{{2}}}}\right)}}}+{3}\)
Amplitude \(\displaystyle={2}\)
Time Period \(\displaystyle={\frac{{{2}\pi}}{{{2}}}}\)
\(\displaystyle=\pi\)
Phase shift \(\displaystyle={\frac{{{\frac{{\pi}}{{{2}}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{\pi}}{{{4}}}}\)
Vertical shift \(\displaystyle={3}\)
image

Step 2
(b)
\(\displaystyle{g{{\left({x}\right)}}}={3}{\sin{{\left({x}-\pi\right)}}}-{1}\)
Amplitude \(\displaystyle={3}\)
Time Period \(\displaystyle={\frac{{{2}\pi}}{{{1}}}}\)
\(\displaystyle={2}\pi\)
Phase shift \(\displaystyle={\frac{{\pi}}{{{1}}}}\)
\(\displaystyle=\pi\)
Vertical shift \(\displaystyle=-{1}\)
image Answer: Compare given equation with general equation of a \(\sin\) function i.e. \(\displaystyle{y}={A}{\sin{{\left({B}{x}-{C}\right)}}}+{D}\)
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