Question

graph the following function: $y = 1+\frac{1}{2}\tan(\frac{2pi}{5}x + 3pi)$
step 2 of 2: determine how the general shape of the graph, chosen in the previous step, would be shifted, stretched, and reflected for the given function. graph the results on the axes provided.
answer
x - axis reflection
reflect graph across x - axis
shift graph vertically
up down noneunits:
shift graph horizontally (phase shift)
left right noneunits:
stretch/compress graph vertically
yes no factor:
stretch/compress graph horizontally (period)
yes no
to find the period divide p by:

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 $\frac{2}{5}$