select the correct answer. function h is a transformation of the parent tangent function such that h(x)=…

Explanation:

Step1: Analyze the transformation of the tangent - function

The parent tangent function is (y = \tan(x)). The given function (h(x)=-\tan(\frac{1}{2}x)) has two transformations: a vertical reflection (due to the negative sign) and a horizontal stretch. The period of the parent tangent function (y = \tan(x)) is (\pi). For the function (y = \tan(bx)), the period is (T=\frac{\pi}{|b|}). Here (b = \frac{1}{2}), so the period of (h(x)) is (T = \frac{\pi}{\frac{1}{2}}=2\pi).

Step2: Consider the vertical - reflection

The negative sign in front of the tangent function reflects the graph of (y = \tan(\frac{1}{2}x)) over the (x) - axis.

Step3: Evaluate the options

The graph of (y = \tan(x)) has vertical asymptotes at (x=(n+\frac{1}{2})\pi,n\in\mathbb{Z}). For (y = \tan(\frac{1}{2}x)), the vertical asymptotes are at (\frac{1}{2}x=(n + \frac{1}{2})\pi), or (x=(2n + 1)\pi,n\in\mathbb{Z}). After the vertical reflection, we look for a graph with a period of (2\pi) and is reflected over the (x) - axis compared to the standard tangent - function graph.

Answer:

(Without seeing the full - details of each option, assume the correct graph has a period of (2\pi) and is reflected over the (x) - axis. If we had to choose based on general knowledge of tangent - function transformations, we would pick the graph that has vertical asymptotes at (x=\pm\pi,\pm3\pi,\cdots) and is reflected over the (x) - axis compared to the basic (y = \tan(x)) graph. But since the options are not clearly labeled with text descriptions, we can't give a specific option like A, B, C, D. If you can provide more details about the options, a more precise answer can be given.)