Let's break down each trigonometric function to find the period and then explain how to graph them.

Part a) $y = \tan(4x)$

Period:

The general form of the tangent function is $y = \tan(kx)$, where the period $T$ is given by:

$T = \frac{\pi}{|k|}$

For $y = \tan(4x)$, $k = 4$. Therefore, the period is:

$T = \frac{\pi}{4}$

Graph:

The graph of $y = \tan(4x)$ is a vertical stretch of the standard tangent graph, but the period is compressed to $\frac{\pi}{4}$, meaning the function repeats every $\frac{\pi}{4}$. The vertical asymptotes occur at $x = \frac{\pi}{8} + \frac{\pi}{4}n$, where $n \in \mathbb{Z}$.

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Part b) $y = \tan\left(2\left(x + \frac{\pi}{2}\right)\right)$

Period:

Again, the general form is $y = \tan(kx)$, where $k = 2$. Thus, the period is:

$T = \frac{\pi}{|2|} = \frac{\pi}{2}$

The $x + \frac{\pi}{2}$ inside the function represents a horizontal shift to the left by $\frac{\pi}{2}$, but this does not affect the period itself.

Graph:

The graph of $y = \tan\left(2\left(x + \frac{\pi}{2}\right)\right)$ is shifted to the left by $\frac{\pi}{2}$ units compared to $y = \tan(2x)$. The vertical asymptotes will now occur at $x = -\frac{\pi}{4} + \frac{\pi}{2}n$.

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Part c) $y = 3\sec\left(2\left(x - \frac{\pi}{2}\right)\right)$

Period:

The secant function is the reciprocal of the cosine function, and the general form of the cosine function is $y = \sec(kx)$, where the period $T$ is:

$T = \frac{2\pi}{|k|}$

Here, $k = 2$, so the period is:

$T = \frac{2\pi}{2} = \pi$

The $x - \frac{\pi}{2}$ represents a horizontal shift to the right by $\frac{\pi}{2}$, but this shift does not affect the period.

Graph:

The graph of $y = 3\sec\left(2\left(x - \frac{\pi}{2}\right)\right)$ is stretched vertically by a factor of 3 and shifted to the right by $\frac{\pi}{2}$. The vertical asymptotes for the secant function will occur at the same places as the zeros of the cosine function $x = \frac{\pi}{4} + \frac{\pi}{2}n$.

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Would you like me to generate detailed graphs for these functions or further explanations?

Five Related Questions:

1. How does the amplitude of $y = 3 \sec(2x)$ compare to $y = \sec(2x)$?
2. What is the period of $y = \tan\left(5x - \frac{\pi}{3}\right)$?
3. How do horizontal shifts affect the vertical asymptotes of tangent and secant graphs?
4. What is the difference between the period of a sine function and a tangent function?
5. How do vertical stretches and compressions impact the shape of trigonometric graphs?

Tip:

When graphing trigonometric functions, always check for shifts and period changes first, as they significantly affect the shape and position of the graph.