11.) ( y = \tan left( \frac { 1 } { 2 } x \right) + 1 ) amp = ____ pd = ____ p. shift = ____ v. shift = ____

Explanation:

Step1: Amplitude of tangent function

For the tangent function (y = A\tan(Bx - C)+D), the amplitude is not defined for the tangent function (since the range of (y = \tan x) is ((-\infty,\infty))). But if we consider the general form, for (y=\tan(\frac{1}{2}x)+1) (where (A = 1), (B=\frac{1}{2}), (C = 0), (D=1)), the "amplitude - like" concept (the coefficient of the tangent function) is (1).

Step2: Period of the tangent function

The period of the tangent function (y=\tan(Bx - C)+D) is given by (T=\frac{\pi}{|B|}). Here (B=\frac{1}{2}), so (T=\frac{\pi}{\frac{1}{2}}=2\pi).

Step3: Phase - shift of the tangent function

The phase - shift of the function (y = A\tan(Bx - C)+D) is given by (x=\frac{C}{B}). Since (C = 0) and (B=\frac{1}{2}), the phase - shift is (0) (no phase - shift).

Step4: Vertical - shift of the tangent function

For the function (y=A\tan(Bx - C)+D), the vertical - shift is (D). Here (D = 1), so the vertical - shift is (1) unit up.

Answer:

amp (=1), pd (=2\pi), p. shift (=0) (no phase - shift), v. shift (=\text{up }1)