Question

What is the amplitude, period and the phase shift of $ f\left( x \right) = 3\sin \left( {2x + \pi } \right) $ ?

Answer 1

Hint: The given equation is a sine wave equation. The standard form of sine function is $ f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ . The amplitude of a sine wave is given by $ \left| a \right| $ , the period is given by $ \dfrac{{2\pi }}{{\left| b \right|}} $ and the phase shift is given by c . We are going to compare the given equation with the standard form and then obtain the values of a, b and c.

Complete step-by-step answer:
In this question, we are given a function of sine wave and we are supposed to find the amplitude, period and the phase shift of this function.
First of all, what is sine wave?
A sine wave is a geometric waveform that oscillates periodically and is defined by the function
 $ y = \sin x $ .
The standard form of sine function is given by:
 $ \Rightarrow f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ - - - - - - - (1)
Now, this sine function has some properties. Let us discuss these properties.
 $ \to $ Amplitude: Amplitude is the vertical distance between the sinusoidal axis and maximum or minimum value of the function.
The amplitude of this function is given by $ \left| a \right| $ .
 $ \to $ Period: Period is the length of one cycle of the curve.
The period of this function is given by $ \dfrac{{2\pi }}{{\left| b \right|}} $ .
 $ \to $ Phase shift: Phase shift is the angle in degrees or radian that the waveform has shifted from a certain point along the horizontal zero axis.
The phase shift of this function is given by $ c $ units to the right.
So, now know everything that we need to find.
Therefore, $ f\left( x \right) = 3\sin \left( {2x + \pi } \right) $
To get the value of b, take 2 common out, we get
 $ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $
Now, compare $ f\left( x \right) = 3\sin \left( {2\left( {x + \dfrac{\pi }{2}} \right)} \right) $ with equation $ f\left( x \right) = a\sin \left( {b\left( {x - c} \right)} \right) $ , we will get the values of a, b and c.
Therefore, $ a = 3,b = 2,c = - \dfrac{\pi }{2} $ .
Now, we can find the amplitude, period, phase shift easily.
 $ \to $ Amplitude $ = \left| a \right| = \left| 3 \right| = 3 $
 $ \to $ Period $ = \dfrac{{2\pi }}{{\left| b \right|}} = \dfrac{{2\pi }}{{\left| 2 \right|}} = \dfrac{{2\pi }}{2} = \pi $
 $ \to $ Phase shift $ = c = - \dfrac{\pi }{2} $

Note: If the value of phase shift is negative, the sine wave will shift in the right direction and if the value of the phase shift is positive, the sine wave will be shifted in the left direction. In our question, we got phase shift as $ - \dfrac{\pi }{2} $ so, the sine wave will be shifted in the right direction.