graph the following function: $y = 1+\frac{1}{2}\tan(\frac{2pi}{5}x + 3pi)$\nstep 2 of 2: determine how the…

Explanation:

Step1: Analyze the general form of the tangent - function

The general form of a tangent function is $y = A\tan(Bx - C)+D$. For the given function $y = 1+\frac{1}{2}\tan(\frac{2\pi}{5}x + 3\pi)$, we can rewrite it as $y=\frac{1}{2}\tan(\frac{2\pi}{5}x+3\pi)+1$. Here, $A = \frac{1}{2}$, $B=\frac{2\pi}{5}$, $C=- 3\pi$, $D = 1$.

Step2: Check for x - axis reflection

Since $A=\frac{1}{2}>0$, there is no reflection across the x - axis.

Step3: Determine vertical shift

The value of $D = 1$. A positive value of $D$ in the formula $y = A\tan(Bx - C)+D$ means a vertical shift up by 1 unit.

Step4: Determine horizontal shift (phase - shift)

The phase - shift is given by $\frac{C}{B}$. For $y=\frac{1}{2}\tan(\frac{2\pi}{5}x+3\pi)$, we have $C=-3\pi$ and $B = \frac{2\pi}{5}$. The phase - shift is $\frac{-3\pi}{\frac{2\pi}{5}}=-\frac{15}{2}=-7.5$. A negative phase - shift means a shift to the left by $\frac{15}{2}=7.5$ units.

Step5: Determine vertical stretch/compression

The coefficient of the tangent function is $A=\frac{1}{2}$. So, there is a vertical compression by a factor of $\frac{1}{2}$.

Step6: Determine horizontal stretch/compression (period)

The period of the tangent function $y = A\tan(Bx - C)+D$ is $p=\frac{\pi}{|B|}$. For $B=\frac{2\pi}{5}$, the period $p=\frac{\pi}{\frac{2\pi}{5}}=\frac{5}{2}$. The standard period of the tangent function $y = \tan(x)$ is $\pi$. To find the period of the given function, we divide the standard period $\pi$ by $\frac{2}{5}$.

Answer:

x - Axis Reflection: No Shift Graph Vertically: Up, 1 unit Shift Graph Horizontally (Phase Shift): Left, 7.5 units Stretch/Compress Graph Vertically: Yes, Factor: $\frac{1}{2}$ Stretch/Compress Graph Horizontally (Period): Yes, To find the period divide $\pi$ by $\frac{2}{5}$