Question

graph the function over a two - period interval.
y = - 1 - 3 tan 2(x - \frac{3\pi}{2})
choose the correct graph below.
○ a.
○ b.
○ c.

Explanation:

Step1: Recall the period formula for tangent function

The general form of the tangent function is $y = A\tan(Bx - C)+D$, and its period is $T=\frac{\pi}{|B|}$. For the function $y=-1 - 3\tan2(x-\frac{3\pi}{2})$, where $B = 2$, so the period $T=\frac{\pi}{2}$.

Step2: Identify the phase - shift

The phase - shift of the tangent function $y = A\tan(Bx - C)+D$ is given by $\frac{C}{B}$. Here, $C = 3\pi$ and $B = 2$, so the phase - shift is $\frac{3\pi}{2}$.

Step3: Analyze the vertical shift and amplitude

The vertical shift is $D=-1$ and the amplitude of the tangent function is not applicable in the traditional sense (since the tangent function has no maximum or minimum values), but the coefficient $A=-3$ affects the steepness and reflection. The negative sign in front of 3 reflects the graph of $y = \tan(2(x-\frac{3\pi}{2}))$ about the $x$ - axis and the factor of 3 stretches it vertically.

Step4: Find key points

The tangent function $y=\tan x$ has vertical asymptotes at $x=(n+\frac{1}{2})\pi,n\in\mathbb{Z}$. For $y=-1 - 3\tan2(x-\frac{3\pi}{2})$, the vertical asymptotes are found by setting $2(x-\frac{3\pi}{2})=(n+\frac{1}{2})\pi$. Solving for $x$ gives $2x-3\pi=(n + \frac{1}{2})\pi$, then $2x=(n+\frac{1}{2})\pi + 3\pi=(n+\frac{7}{2})\pi$, and $x=\frac{(2n + 7)\pi}{4},n\in\mathbb{Z}$.
When $x=\frac{3\pi}{2}$, $y=-1$.

Step5: Compare with graphs

By analyzing the period, phase - shift, vertical shift, and the behavior of the tangent function, we can determine the correct graph.

Answer:

(Without seeing the actual characteristics of the graphs A, B, and C in detail, we can't give a definite letter - choice. But the above steps can be used to analyze and choose the correct graph among them. If we assume we have analyzed based on the above properties and found the correct one) A. Option A text, B. Option B text, C. Option C text (Replace with the actual descriptions of the graphs from the question)