The function provided is:

$y = 4 \tan(x - \pi/2)$

Let's break down the components:

1. Amplitude: The tangent function does not have an amplitude because it extends to infinity.

2. Period: The period of the tangent function is given by: $\text{Period} = \frac{\pi}{B}$ Here, $B = 1$, so the period remains $\pi$.

3. Phase Shift: The phase shift is determined by solving $x - \pi/2 = 0$: $\text{Phase Shift} = \pi/2 \, \text{to the right}.$

4. Vertical Stretch: The factor $4$ means the tangent function is stretched vertically by a factor of 4.

5. Vertical Asymptotes: Tangent functions have vertical asymptotes at the edges of their period. Starting from the phase shift $x = \pi/2$: $\text{Asymptote 1: } x = \pi/2$ $\text{Asymptote 2: } x = \pi/2 + \pi = 3\pi/2$

Final Parameters:

• Period: $\pi$
• Phase Shift: $\pi/2$
• Asymptote 1: $x = \pi/2$
• Asymptote 2: $x = 3\pi/2$

Select the graph type that resembles a stretched tangent graph with these asymptotes.

Do you want me to explain how to interpret these results further or calculate other properties?

Related Questions:

1. What is the general formula for the period of trigonometric functions?
2. How does the vertical stretch factor affect the graph of the tangent function?
3. How can we determine asymptotes for other trigonometric functions?
4. What is the importance of phase shift in graphing periodic functions?
5. How can we use transformations to sketch a trigonometric graph without plotting points?

Tip: Always identify the key properties (period, phase shift, asymptotes) first when graphing trigonometric functions.