Question

How do you find the amplitude, period and phase shift for $y=4\tan \left( 2x-\pi \right)$?

Answer 1

Now we know that amplitude of the function is nothing but maximum value of the function. Hence we will calculate the distance between axis and maximum point to find amplitude. Now for a function of the type $a\tan \left( b\left( x-c \right) \right)$ the period is given by $\dfrac{2\pi }{\left| b \right|}$ and the phase shift is given by c. Hence we will write the given function in the form $a\tan \left( b\left( x-c \right) \right)$ and find the required values.

Complete step by step solution:
Now the given function is a trigonometric wave function.
For wave function we can define three properties.
Now if the function repeats the same value after every interval then the function is said to be periodic. All trigonometric functions are hence periodic. Now the length of such a smallest interval after which the function repeats its values is called period of the function.
Now the maximum height a function can attend from the axis is called the amplitude of the function. The amplitude of the function is also the length of the highest point or lowest point from the axis of the graph.
Now the horizontal shift of the function from the original function is known as phase shift.
Now the given function is of the form $a\tan \left( b\left( x-c \right) \right)$ where a = 4, b = 2 and $c=\pi$ .
Now we know that the tan function ranges from $-\infty$ to $\infty$ . Hence we cannot calculate the amplitude of the function.
Now let us write the function in the form $a\sin \left( b\left( x+c \right) \right)$
Hence we get, $y=\tan \left( 2\left( x-\dfrac{\pi }{2} \right) \right)$
Now for a function of the type $a\tan \left( b\left( x-c \right) \right)$ the period is given by $\dfrac{2\pi }{\left| b \right|}$ and phase shift is given by c. Hence the period of the function is $\dfrac{2\pi }{2}=\pi$ and the phase shift is $\dfrac{\pi }{2}$ .

Note: Now note that in general for a function of the type $a\sin \left( b\left( x+c \right) \right)$ we have the amplitude of the function as a. Now note that this is because the maximum value of sin and cos is 1 and the maximum value will be multiplied by a and hence we will get the maximum height as a. But this will not be the case in infinite functions and hence do not write the amplitude as a in case of tan.