Multiple Choice

1. What is the period of the function $y = \tan(x)$?

   $\pi$

   $4\pi$

   $2\pi$

   $\pi/2$

2. Which of the following describes the vertical asymptotes of the function $y = \tan(x)$?

   $x = \pi/2 + n\pi$

   $x = n\pi$

   $x = 2n\pi$

   $x = \pi/2 + 2n\pi$

3. What are the x-intercepts of the function $y = \tan(x)$?

   $x = n\pi$

   $x = \pi/2 + n\pi$

   $x = \pi/2 + 2n\pi$

   $x = \pi/4 + n\pi$

4. How does the graph of $y = \tan(x)$ behave as $x$ approaches its vertical asymptotes?

   It oscillates between $-1$ and $1$.

   It becomes a horizontal line.

   It approaches positive or negative infinity.

   It approaches $0$.

5. What is the period of the function $y = \tan(2x)$?

   $\pi$

   $\pi/2$

   $\pi/4$

   $2\pi$

6. Where do the vertical asymptotes of $y = \tan(2x)$ occur?

   $x = \pi/2 + n\pi$

   $x = n\pi$

   $x = \pi/4 + n\pi/2$

   $x = \pi/4 + n\pi$

7. What are the x-intercepts of the function $y = \tan(3x)$?

   $x = \pi/3 + n\pi$

   $x = n\pi/3$

   $x = \pi/6 + n\pi/3$

   $x = n\pi$

8. What is the period of the function $y = \tan(x/2)$?

   $4\pi$

   $2\pi$

   $\pi/2$

   $\pi$

9. Identify the period of the function $y = -3\tan(4x+ \pi/2)$.

   $4\pi$

   $\pi/2$

   $\pi/4$

   $\pi$

10. Which of the following is a vertical asymptote for the function $y = \tan(x - \pi/4)$?

    $x = \pi/2$

    $x = \pi$

    $x = \pi/4$

    $x = 3\pi/4$

11. For the function $y = \tan(x)$, what is the interval between two consecutive vertical asymptotes?

    $\pi$

    $\pi/2$

    $2\pi$

    $3\pi/2$

12. If the period of $y = \tan(bx)$ is $\pi/3$, what is the value of $b$? (Assume $b > 0$)

    $b = 3$

    $b = 1/3$

    $b = \pi/3$

    $b = 3\pi$

13. What is the effect of the '$-2$' in the function $y = \tan(x) - 2$ on the graph?

    Shifts the graph $2$ units left.

    Stretches the graph vertically by a factor of $2$.

    Shifts the graph $2$ units up.

    Shifts the graph $2$ units down.

14. Which transformation affects the period of the tangent function $y = \tan(x)$?

    Multiplying the entire function by a constant, e.g., $y = a\tan(x)$.

    Adding a constant to $x$, e.g., $y = \tan(x+c)$.

    Multiplying $x$ by a constant, e.g., $y = \tan(bx)$.

    Adding a constant to the entire function, e.g., $y = \tan(x)+d$.

15. The graph of $y = \tan(x)$ passes through the origin $(0,0)$. Which point does the graph of $y = \tan(x-\pi/2)$ pass through?

    $(0, \pi/2)$

    $(\pi/2, 0)$

    $(0, 0)$

    $(\pi/2, 1)$

16. Which of the following functions has vertical asymptotes at $x = \pi/6 + n\pi/3$?

    $y = \tan(x/2)$

    $y = \tan(x - \pi/6)$

    $y = \tan(3x)$

    $y = \tan(x/3)$

17. Describe the primary effect of the '$3$' in $y = 3\tan(x)$.

    Stretches the graph vertically.

    Changes the period to $3\pi$.

    Shifts the graph $3$ units up.

    Shifts the graph $3$ units left.

18. What is the period of the function $y = \tan(\frac{1}{3}x + \pi)$?

    $2\pi$

    $\pi$

    $\pi/3$

    $3\pi$

19. Which statement is true about the x-intercepts of $y = \tan(x+\pi)$?

    They are shifted $\pi$ units to the right compared to $y = \tan(x)$.

    They are different from $y = \tan(x)$.

    They are shifted $\pi$ units up compared to $y = \tan(x)$.

    They are the same as $y = \tan(x)$.

20. A tangent function has a period of $2\pi$. Which of the following could be its equation?

    $y = \tan(4x)$

    $y = \tan(x/2)$

    $y = \tan(x/4)$

    $y = \tan(2x)$