Question

How do you find the amplitude and period of $y=\cos 6x$?

Answer 1

Now we know that the given function is a trigonometric wave function. Now first we will write the function in the form $a\cos \left( b\left( x-c \right) \right)$. Now we know that for the function of the form $a\cos \left( b\left( x-c \right) \right)$ the amplitude is equal to a and similarly the phase shift of the function is equal to $\dfrac{2\pi }{b}$ . Hence we can easily find the required amplitude and period.

Complete step by step solution:
Now we are given with a trigonometric function $\cos 6x$ .
Now we will first write the given function in the form $a\cos \left( b\left( x-c \right) \right)$ .
Hence we get, $y=1\cos \left( 6\left( x-0 \right) \right)$ .
Now the amplitude is nothing but the maximum height that a function can reach from its axis.
Similarly period of the function is nothing but the length of smallest interval after which the function repeats itself.
Now we know for the function of this type the amplitude is given by a and the period is given by $\dfrac{2\pi }{\left| b \right|}$
Here by comparing we have a = 1.
Hence the amplitude of the function is 1.
Now similarly we have the period of the function is $\dfrac{\pi }{6}$ .
Hence the amplitude of the function is 1 and the period of the function is $\dfrac{\pi }{6}$ .

Note: Now note that for a wave function we have two more quantities. The phase shift of a wave function is the horizontal shift of the function from the original function. Similarly the vertical shift of the function is the vertical shift from the original function. In the function of form $a\cos \left( b\left( x-c \right) \right)+d$ the phase shift is given by c and the vertical shift is given by d.