91Ó°ÊÓ

The original period of a tangent function is \(\pi\) (or 180 degrees). The coefficient in front of \(x\) in the function changes the frequency of the function which affects the period. Using the formula for period \(P=\frac{\pi}{|b|}\), where \(b\) is the coefficient of \(x\), we find that the period of the graph of this function will be \(P = \frac{\pi}{\frac{1}{2}} = 2\pi\). The coefficient in front of the tangent function, -2, doesn't change the period or frequency, but it does change the amplitude of the function. However, for tangent and cotangent functions, the amplitude is undefined because the graph oscillates from negative infinity to positive infinity. The -2 will cause the graph to be 'steeper' or more vertical at its intercepts, which will be reflected in a horizontal dilation of the graph of the function.

The phase shift refers to the horizontal shift of the function along the x-axis, and vertical shift refers to the shift of the function along the y-axis. For our given function \(y=-2 \tan \frac{1}{2} x\), since there is no phase shift or vertical shift value in the function, we can say that there is no phase shift and vertical shift for this function.

Using the information about amplitude, period, phase shifts and vertical shifts, draw the graph of the function. For our function, we know that the period is \(2\pi\) so we mark these on the x axis at regular intervals. Since there are no phase shifts or vertical shifts, the graph will start from the origin. Repeat the offsets and as we know tangent function rises to positive infinity and falls to negative infinity so the graph at these points will go towards infinity. For the function \(y=-2 \tan \frac{1}{2} x\), the graph will be steeper because of the -2 coefficient. After sketching the basic shape of one period of the function, repeat the shape for the second period.