Question

graph two periods of the given tangent function.
y = - 4 tan (\frac{1}{2}x)
choose the correct graph of two periods of y = - 4 tan (\frac{1}{2}x) below.

Explanation:

Step1: Recall the period formula for tangent function

The period of the tangent function $y = A\tan(Bx)$ is $T=\frac{\pi}{|B|}$. For the function $y=-4\tan(\frac{1}{2}x)$, $B = \frac{1}{2}$, so $T=\frac{\pi}{\frac{1}{2}}=2\pi$.

Step2: Analyze the amplitude - like property

The coefficient $A=-4$ means the graph of $y = \tan(\frac{1}{2}x)$ is vertically stretched by a factor of 4 and reflected about the $x$ - axis.

Step3: Check key - points and asymptotes

The tangent function $y = \tan x$ has vertical asymptotes at $x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}$. For $y=-4\tan(\frac{1}{2}x)$, the vertical asymptotes are given by $\frac{1}{2}x=(2n + 1)\frac{\pi}{2}$, or $x=(2n + 1)\pi$. When $x = 0$, $y=-4\tan(0)=0$.

Answer:

Analyze each option based on the period of $2\pi$, the vertical stretch by a factor of 4, reflection about the $x$ - axis, and the location of asymptotes. Without seeing the actual details of the graphs in the options, we know that the correct graph should have a period of $2\pi$, be reflected about the $x$ - axis and be vertically stretched. If we assume the standard behavior of tangent - type graphs, we can eliminate options based on these properties. Since the function is $y=-4\tan(\frac{1}{2}x)$ with period $2\pi$ and vertical stretch and reflection, we need to find the graph that has asymptotes at $x=\pm\pi,\pm3\pi,\cdots$ and passes through the origin and has the correct orientation (reflected). Based on the general form and properties, we would choose the graph that matches these characteristics. If we assume the options are well - formed, we look for the graph with the correct period, reflection, and vertical stretch.

(Note: Since the actual visual details of the graphs in the options are not fully described in text, a more definite answer in terms of option letter cannot be given without more information about the appearance of each graph. But the above steps show how to analyze and choose the correct graph.)