Question

How do you find the amplitude, period and phase shift for $y=\sin \left( 1.5x \right)$?

Answer 1

Hint: Now we are given a wave function of the form $y=A\sin \left( B\left( x-C \right) \right)$ . Comparing the given equation we will find the values of A, B, and C. Now the amplitude of the function is given by A, the period of the function is given by $\dfrac{2\pi }{B}$ and the phase shift of the function is equal to C. Hence we can find all three terms.

Complete step by step solution:
Now for wave function we have three main terms Amplitude, period and phase shift.
Let us first understand the terms Amplitude, period and phase shift.
For any function amplitude is nothing but height if the function from the center line. Amplitude can also be calculated by calculating the distance between lowest point and highest point and dividing it by 2.
Now the period is horizontal distance between two trough or two crests. Period denotes after how much time will a function repeats its value.
Now phase shift is how far the function is shifted horizontally from its usual position.
Similarly vertical shift is how far a function is shifted from its usual position.
Now for any function of the type $y=A\sin \left( B\left( x-C \right) \right)$ we have,
The amplitude as A, the period as $\dfrac{2\pi }{B}$, and the phase shift as C.
Now consider the given function $y=\sin \left( 1.5x \right)$.
Now comparing the given function with $y=A\sin \left( B\left( x-C \right) \right)$ we have A = 1, B = 1.5 and $C=0$
Hence the amplitude of the given function is 1.
The phase shift of the function is 0.
And the period of the function is $\dfrac{2\pi }{1.5}$

Note: Now note that we have another terms related to wave function which is called frequency. Frequency is how far the function goes per unit time. Now we have that time period is the time for function to complete 1 cycle. Hence we have $f=\dfrac{1}{t}$ where f is frequency and t is time period.