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

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 $$\displaystyle={3}$$

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 $$\displaystyle=-{1}$$
Answer: Compare given equation with general equation of a $$\sin$$ function i.e. $$\displaystyle{y}={A}{\sin{{\left({B}{x}-{C}\right)}}}+{D}$$