Question

1. Graph the function f(x)=2 sin x-2 (Graph 1)

Answer 1

Explanation:

The function is $f(x)=2 \sin x-2$

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, $y=A \sin (B(x+C))+D$

where A is the amplitude = $|a|$

Period is $\frac{2 \pi }{B}$,

Phase shift is C,

Vertical shift is D.

Hence, from the function $f(x)=2 \sin x-2$, we have,

Amplitude = $A=2$

Period = $\frac{2 \pi }{B}=\frac{2\pi}{1} =2\pi$

Phase shift = 0

Vertical shift = -2

Now, substituting the values for x, we get the corresponding values for y.

For $x=0$ ⇒ $f(x)=-2$

For $x=\frac{\pi}{2}$ ⇒ $f(x)=0$

For $x=\pi$ ⇒ $f(x)=-2$

For $x=\frac{3 \pi}{2}$ ⇒ $f(x)=-4$

For $x=2\pi$ ⇒ $f(x)=-2$

Hence, plotting these points we get the graph which is attached below:

Now, we shall graph the function $f(x)=-\cos (2 x)+1$

Similarly, we shall determine the amplitude, period, phase shift and vertical shift.

Amplitude = $A=1$

Period = $\frac{2 \pi }{B}=\frac{2\pi}{2} =\pi$

Phase shift = 0

Vertical shift = 1

Now, substituting the values for x, we get the corresponding values for y in the function $f(x)=-\cos (2 x)+1$

For $x=0$ ⇒ $f(x)=0$

For $x=\frac{\pi}{4}$ ⇒ $f(x)=1$

For $x=\frac{\pi}{2}$ ⇒ $f(x)=2$

For $x=\frac{3 \pi}{4}$ ⇒ $f(x)=1$

For $x=\pi$ ⇒ $f(x)=0$

Hence, plotting these points we get the graph which is attached below: