Question

How do you find the amplitude and period of the periodic of the function?

Answer 1

Hint: Now we know that amplitude of the function is the height of the function from the axis. Similarly the period of the function is the length of interval after which the function repeats its value. Hence using the definition we can find the amplitude and period of periodic functions.

Complete step by step solution:
Now let us first understand the amplitude and period of the function.
These are terms defined for wave function. Now since we know that all trigonometric functions are wave functions we have the amplitude and period of these functions.
Now amplitude is nothing but the maximum height that a function can reach from the axis. To calculate the amplitude of the function we check the distance of the crest and the axis. Now note that for the functions of the form $a\sin \left( b\left( x+c \right) \right)$ or $a\cos \left( b\left( x+c \right) \right)$ the amplitude is given by a. Note that for infinite functions we cannot calculate amplitude.
Now let us understand the period of the function.
Now if a function repeats its values after an interval then the function is known as periodic function. The period of function is defined as the length of smallest interval after which the function repeats its value. Hence the period of the function can be measured by calculating the distance between two crests or trough. For trigonometric functions of the form $a\sin \left( b\left( x-c \right) \right)$ the period is given by $\dfrac{2\pi }{\left| b \right|}$ . Note that we can take any trigonometric ratio in place of sin.

Note: Now note that we can also define phase shift for a wave function. Phase shift is the horizontal shift of a function from the original function. Similarly we can define the vertical shift of the function. Also note that the frequency of the function is defined as $f=\dfrac{1}{t}$ where t is the period of the function.