which is the graph of the function $y = 2\tan(x+\frac{3pi}{4})$?

Answer:

We need to analyze the key - features of the function (y = 2\tan(x+\frac{3\pi}{4})) to determine its graph.

1. Period: The period of the tangent function (y = A\tan(Bx - C)+D) is given by (T=\frac{\pi}{|B|}). For the function (y = 2\tan(x+\frac{3\pi}{4})), where (B = 1), the period (T=\pi).
2. Phase - shift: The general form of a tangent function is (y=A\tan(Bx - C)+D), and the phase - shift is given by (\frac{C}{B}). For the function (y = 2\tan(x+\frac{3\pi}{4})=2\tan(x-(-\frac{3\pi}{4}))), with (B = 1) and (C=-\frac{3\pi}{4}), the phase - shift is (-\frac{3\pi}{4}) (a shift to the left by (\frac{3\pi}{4}) units).
3. Vertical stretch: The coefficient (A = 2) causes a vertical stretch of the graph of (y=\tan(x)) by a factor of 2.

We know that the vertical asymptotes of the tangent function (y=\tan(x)) occur at (x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}). For the function (y = 2\tan(x+\frac{3\pi}{4})), the vertical asymptotes are found by setting (x+\frac{3\pi}{4}=\frac{\pi}{2}+k\pi). [ \begin{align*} x+\frac{3\pi}{4}&=\frac{\pi}{2}+k\pi\ x&=\frac{\pi}{2}-\frac{3\pi}{4}+k\pi\ x&=-\frac{\pi}{4}+k\pi,k\in\mathbb{Z} \end{align*} ]

When (x =-\frac{3\pi}{4}), (y = 2\tan(-\frac{3\pi}{4}+\frac{3\pi}{4})=2\tan(0)=0)

We can also check some other points. For example, when (x=-\frac{\pi}{4}), the function is undefined (vertical asymptote). When (x = \frac{\pi}{4}), (y=2\tan(\frac{\pi}{4}+\frac{3\pi}{4})=2\tan(\pi)=0)

To determine the correct graph among the options (not shown completely here), we look for a graph of a tangent - type function with a period of (\pi), a left - shift of (\frac{3\pi}{4}) units, and a vertical stretch by a factor of 2. The vertical asymptotes should be at (x =-\frac{\pi}{4}+k\pi,k\in\mathbb{Z}) and the function should pass through points such as ((-\frac{3\pi}{4},0)) and ((\frac{\pi}{4},0))

Since no options are provided to choose from, we have described the key - features of the graph of (y = 2\tan(x+\frac{3\pi}{4})) which can be used to identify the correct graph.

Explanation:

Step1: Find the period

The period formula for (y = A\tan(Bx - C)+D) is (T=\frac{\pi}{|B|}). Here (B = 1), so (T=\pi).

Step2: Determine the phase - shift

The phase - shift formula is (\frac{C}{B}). For (y = 2\tan(x+\frac{3\pi}{4})), (C =-\frac{3\pi}{4}) and (B = 1), so the phase - shift is (-\frac{3\pi}{4}) (left - shift).

Step3: Identify vertical stretch

The coefficient (A = 2) stretches the graph of (y=\tan(x)) vertically by a factor of 2.

Step4: Find vertical asymptotes

Set (x+\frac{3\pi}{4}=\frac{\pi}{2}+k\pi) and solve for (x) to get (x=-\frac{\pi}{4}+k\pi,k\in\mathbb{Z}).

Step5: Evaluate at key points

Evaluate the function at (x =-\frac{3\pi}{4}) and (x=\frac{\pi}{4}) to get (y = 0) at these points.