Explanation:
The function is [tex]f(x)=2 \sin x-2[/tex]
To plot the function in the graph, first we shall determine the amplitude, period, phase shift and vertical shift.
The function is of the form, [tex]y=A \sin (B(x+C))+D[/tex]
where A is the amplitude = [tex]|a|[/tex]
Period is [tex]\frac{2 \pi }{B}[/tex],
Phase shift is C,
Vertical shift is D.
Hence, from the function [tex]f(x)=2 \sin x-2[/tex], we have,
Amplitude = [tex]A=2[/tex]
Period = [tex]\frac{2 \pi }{B}=\frac{2\pi}{1} =2\pi[/tex]
Phase shift = 0
Vertical shift = -2
Now, substituting the values for x, we get the corresponding values for y.
For [tex]x=0[/tex] ⇒ [tex]f(x)=-2[/tex]
For [tex]x=\frac{\pi}{2}[/tex] ⇒ [tex]f(x)=0[/tex]
For [tex]x=\pi[/tex] ⇒ [tex]f(x)=-2[/tex]
For [tex]x=\frac{3 \pi}{2}[/tex] ⇒ [tex]f(x)=-4[/tex]
For [tex]x=2\pi[/tex] ⇒ [tex]f(x)=-2[/tex]
Hence, plotting these points we get the graph which is attached below:
Now, we shall graph the function [tex]f(x)=-\cos (2 x)+1[/tex]
Similarly, we shall determine the amplitude, period, phase shift and vertical shift.
Amplitude = [tex]A=1[/tex]
Period = [tex]\frac{2 \pi }{B}=\frac{2\pi}{2} =\pi[/tex]
Phase shift = 0
Vertical shift = 1
Now, substituting the values for x, we get the corresponding values for y in the function [tex]f(x)=-\cos (2 x)+1[/tex]
For [tex]x=0[/tex] ⇒ [tex]f(x)=0[/tex]
For [tex]x=\frac{\pi}{4}[/tex] ⇒ [tex]f(x)=1[/tex]
For [tex]x=\frac{\pi}{2}[/tex] ⇒ [tex]f(x)=2[/tex]
For [tex]x=\frac{3 \pi}{4}[/tex] ⇒ [tex]f(x)=1[/tex]
For [tex]x=\pi[/tex] ⇒ [tex]f(x)=0[/tex]
Hence, plotting these points we get the graph which is attached below:
