Find the amplitude, period, phase shift, & zeros then sketch a graph of the function: y=tan(1/2x-pi/2), 0<x<4pi
For a function written in standard form: y = a tan k(x-b)
amplitude = undefined
period = pi/k and
phase shift = b
In the present case
y = tan (1/2 x - pi/2) = tan 1/2 [x-pi]
The concept of amplitude does not really apply to "tangent or tan" function, since the value is "undefined" whenever the denominator (cosine function) is 0.
Period: tangent functions have a period of "pi" with an asymptote separating the periods. However in this case the period = pi/(1/2) = 2 pi.
Phase shift: As mentioned earlier, phase shift is given by the term 'b' in standard equation form and here b= pi
Zeros in the interval, 0<x<4 pi: The zeros of a tangent function occurs at n*pi, i.e. 0, pi, 2pi,3pi,.....The zeros of (x-pi)/2 would be at
(x-pi)/2 = n*pi
or, x = 2n*pi - pi
and in the given interval 0<x<4pi,
the possible zeros are: 0,pi corresponding to x = pi and 3pi
As for the plot of the given function, it would be a standard tangent graph, with 'undefined' values at x =0, 2pi and 4pi; 0 value at pi and 3pi and as already stated the period is 2 pi.
hope this helps.