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

Explanation:

Step1: Recall tangent - function transformation rules

The parent tangent function is (y = \tan(x)) with a period of (\pi). For the function (h(x)=-\tan(\frac{1}{2}x)), the coefficient (\frac{1}{2}) in the argument (x) changes the period. The period of (y = \tan(bx)) is (T=\frac{\pi}{|b|}), so for (h(x)=-\tan(\frac{1}{2}x)), the period (T = \frac{\pi}{\frac{1}{2}}=2\pi).

Step2: Analyze the reflection

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

Step3: Evaluate the options

The graph of the parent function (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). The graph of (h(x)=-\tan(\frac{1}{2}x)) has the same vertical - asymptote locations but is reflected about the (x) - axis compared to (y = \tan(\frac{1}{2}x)). The graph of (y = \tan(x)) passes through the origin ((0,0)). For (h(x)=-\tan(\frac{1}{2}x)), when (x = 0), (h(0)=-\tan(0)=0). The correct graph is one with a period of (2\pi) and is reflected about the (x) - axis compared to the standard tangent function graph.

Answer:

The graph that has a period of (2\pi) and is reflected about the (x) - axis (compared to the parent tangent function graph) is the correct one. Without seeing the exact details of the graphs in a clear - cut way, if we assume the standard properties of the tangent function and its transformations, we note that the period of (y =-\tan(\frac{1}{2}x)) is (2\pi) and it is reflected about the (x) - axis. If we assume the options follow the general form of tangent - function graphs, we need to look for a graph with vertical asymptotes at (x=(2n + 1)\pi,n\in\mathbb{Z}) and is reflected about the (x) - axis. If we had to make a choice based on the general description of the transformations, we would choose the graph that shows a period of (2\pi) and is upside - down compared to the regular tangent function graph. If we assume the graphs are labeled in a standard way, we would need to check which one has the correct period and reflection. Since we can't directly point to an option letter without clear visual inspection, the key characteristics to look for are: period (2\pi) and reflection about the (x) - axis.