Question

how to find amplitude and period of a function

Answer 1

Amplitude :- maximum displacement from mean position of particle is known as amplitude . Let a function y = f(x) is a periodic Function. And we try to find what is amplitude of it ?
First of all you should find out range of y = f(x) , for better understanding , I assumed range ∈[a , b] then, amplitude will be (b - a)/2 .

For example :- y = 3sinx + 4 cosx ,is a periodic function and find amplitude of it ?
You should find range of it
you see y belongs to [ -5, 5]
because y = 3sinx + 4cosx = 5sin(x + arctan(4/3))
and you know, sine lies -1 to 1
So, 5sin(x + arctan(4/3)) lies -5 to 5
Hence, range ∈ [ -5, 5]
Now, amplitude = (5 + 5)/2 = 5

Period :- a function y = f(x) , repeating a specific curve in a definite interval of then imterval of x is known as period of function y = f(x)
Example :- y = sinx is periodic Function . And it's period is 2π. Because graph of y= sinx repeating in every 2π interval .

For finding Period use F(x) = f(x + T)
Where T is period of function F(x)
Example :- y = sin2x , find period ?
sin2x = sin(2x + 2T)
sin(2π + 2x) = sin(2x + 2T)
2π + 2x = 2x + 2T
T = π ,
Hence, period of function is π.

well this concepts have so much demerits .
so , you should memorize some important terms
1. Sine, cosine , cosecent , secent ⇒ period 2π
2. Tangent, cotangent ⇒ period π
3. Period of exponential function is 1
4. Period of constant is 1
5. Period of sin(ax + b) , cos(ax + b), sec(ax + b), cosec(ax + b) = 2π/a

And many more concepts are appear for period ..you should check 11th book for more details.