Question

How do you find the amplitude and period of $y=\dfrac{5}{2}\sin 7x$ ?

Answer 1

Now to find the amplitude and the period we will first write the function in the form $a\sin \left( b\left( x-c \right) \right)$ Now we know that for the function in this form a is the amplitude and $\dfrac{2\pi }{b}$ is the period of the function. Hence we can compare the functions and find a and b and write the amplitude and period of the given function.

Complete step by step solution:
Now we are given the function $y=\dfrac{5}{2}\sin 7x$ and we want to find the amplitude and period of the function.
Now the given function is a trigonometric function and we know that trigonometric functions are periodic in nature. Now periodic function means the function which repeats its nature after a fixed interval. Now the length of such a smallest interval after which the function repeats is called period of the function.
Now let us understand the concept of amplitude. Amplitude of the function is nothing but the maximum height reached by the function from its axis.
Now for any function of the form $a\sin \left( b\left( x-c \right) \right)$ we have the amplitude of the function is a, the period of the function is $\dfrac{2\pi }{\left| b \right|}$ and the phase shift of the function is c.
Now let us first write the given function in the form of $a\sin \left( b\left( x-c \right) \right)$ , hence we get
$\Rightarrow y=\dfrac{5}{2}\sin \left( 7\left( x-0 \right) \right)$
Hence comparing the function with $a\sin \left( b\left( x-c \right) \right)$ we get, $a=\dfrac{5}{2}$ and b = 7.
Hence the amplitude of the function is $a=\dfrac{5}{2}$ and period of the function is $\dfrac{2\pi }{7}$ .

Note: Now note that the period of the function is the length of smallest interval after which the function repeats its value. Hence we have if $\dfrac{2\pi }{7}$ is the period of the given function then we have$\dfrac{5}{2}\sin \left( 7x \right)=\dfrac{5}{2}\sin \left( 7x+\dfrac{2n\pi }{7} \right)$ where n is any integer.