Multiple Choice

1. What is the period of the function $y = 3\sin(2x)$?

   $2\pi$

   $\dfrac{\pi}{2}$

   $\pi$

   $3\pi$

2. What is the y-intercept of the function $y = 2\cos(x) + 1$?

   $3$

   $1$

   $2$

   $0$

3. The graph of $y = \sin(x)$ is dilated by a factor of 2 from the x-axis. What is the equation of the transformed graph?

   $y = \sin\left(\dfrac{x}{2}\right)$

   $y = \sin(2x)$

   $y = 2\sin(x)$

   $y = \dfrac{1}{2}\sin(x)$

4. The graph of $y = \cos(x)$ is dilated by a factor of $\tfrac{1}{3}$ from the y-axis. What is the equation of the transformed graph?

   $y = -\cos(3x)$

   $y = \cos\left(\dfrac{x}{3}\right)$

   $y = 3\cos(x)$

   $y = \cos(3x)$

5. What are the coordinates of the first positive x-intercept of the function $y = 2\sin\left(\tfrac{x}{2}\right)$?

   $\left(\tfrac{3\pi}{2}, 0\right)$

   $\left(\pi, 0\right)$

   $\left(2\pi, 0\right)$

   $\left(\tfrac{\pi}{2}, 0\right)$

6. What is the period of the function $y = 3\sin(2x)$?

   $\dfrac{\pi}{2}$

   $2\pi$

   $3\pi$

   $\pi$

7. What is the y-intercept of the function $y = 2\cos(x) + 1$?

   $1$

   $0$

   $2$

   $3$

8. The graph of $y = \sin(x)$ is dilated by a factor of 2 from the x-axis. What is the equation of the new graph?

   $y = \sin(2x)$

   $y = \dfrac{1}{2}\sin(x)$

   $y = \sin\left(\dfrac{x}{2}\right)$

   $y = 2\sin(x)$

9. The graph of $y = \cos(x)$ is dilated by a factor of $\tfrac{1}{3}$ from the y-axis. What is the equation of the new graph?

   $y = -\cos(3x)$

   $y = \cos\left(\dfrac{x}{3}\right)$

   $y = \cos(3x)$

   $y = 3\cos(x)$

10. What are the coordinates of one of the main points on the midline of the function $y = 4\cos(x) + 2$?

    $(0, 0)$

    $(2\pi, 2)$

    $\left(\tfrac{\pi}{2}, 2\right)$

    $(\pi, 6)$

11. What is the sine function represented by the given graph? (Note: The graph is not provided here, so this question cannot be accurately answered without the graph.)

    
    $y = 2\sin(x - \tfrac{\pi}{2})$

    $y = 2\sin(x + \pi)$

    $y = 2\sin(x - \tfrac{\pi}{2})$

    $y = \sin\left(x - \tfrac{\pi}{2}\right)$

12. What is the sine function represented by the given graph? (Note: The graph is not provided here, so this question cannot be accurately answered without the graph.)

    
    $y = 2\sin\left(x - \tfrac{\pi}{2}\right) + 1$

    $y = \sin\left(x - \tfrac{\pi}{2}\right) + 1$

    $y = 2\sin(x) + 1$

    $y = 2\sin\left(x + \tfrac{\pi}{2}\right) + 1$

13. What is the sine function represented by the given graph (amplitude $2$, period $\pi$, phase shift $\tfrac{\pi}{2}$ to the right)?

    
    $y = 2\sin\left(x - \tfrac{\pi}{2}\right)$

    $y = \sin\left(x - \tfrac{\pi}{2}\right)$

    $y = 2\sin\left(2(x - \tfrac{\pi}{2})\right)$

    $y = \sin(2x)$

14. What is the period of the function $y = 3\tan(2x)$?

    $\pi$

    $\dfrac{\pi}{2}$

    $3\pi$

    $2\pi$

15. What is the y-intercept of the function $y = \tan(x) + 1$?

    $1$

    $-1$

    $0$

    $\pi$

16. The graph of $y = 2\tan(x)$ is a vertical dilation of $y = \tan(x)$ by a factor of:

    $3$

    $-2$

    $2$

    $\dfrac{1}{2}$

17. The function $y = \tan\left(\tfrac{x}{3}\right)$ represents a horizontal dilation of $y = \tan(x)$ by a factor of:

    $2$

    $3$

    $\dfrac{1}{3}$

    $-3$

18. What are the x-intercepts of the function $y = \tan(x)$ in the interval $[0, 2\pi)$?

    $x = 0, \pi$

    $x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}$

    $x = 0, \pi, 2\pi$

    $x = \pi$

19. Find the general solution of the equation $\sin x = \tfrac{1}{2}$.

    $x = \tfrac{\pi}{6} + 2n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{3} + 2n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{5\pi}{6} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{6} + 2n\pi$ or $x = \tfrac{5\pi}{6} + 2n\pi$, where $n \in \mathbb{Z}$

20. Solve the trigonometric equation $\cos(2x) = -\tfrac{\sqrt{3}}{2}$ for $x$.

    $x = \tfrac{5\pi}{12} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{6} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{5\pi}{12} + n\pi$ or $x = \tfrac{7\pi}{12} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{5\pi}{12} + 2n\pi$ or $x = \tfrac{7\pi}{12} + 2n\pi$, where $n \in \mathbb{Z}$

21. What is the general solution to $\tan(3x) = 1$?

    $x = \tfrac{\pi}{12} + \tfrac{n\pi}{3}$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{3} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{12} + \tfrac{2n\pi}{3}$, where $n \in \mathbb{Z}$

    $x = \tfrac{\pi}{4} + n\pi$, where $n \in \mathbb{Z}$

22. Find the general solution for $x$ in the equation $2\sin x + \sqrt{3} = 0$.

    $x = \tfrac{7\pi}{6} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{4\pi}{3} + 2n\pi$ or $x = \tfrac{5\pi}{3} + 2n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{4\pi}{3} + n\pi$, where $n \in \mathbb{Z}$

    $x = \tfrac{5\pi}{6} + n\pi$, where $n \in \mathbb{Z}$