Trigonometric functions and graphs Knowledge-Based Download PDF

Questions

1. 1.

   A trigonometric function of the form $f(x) = A\cos(Bx) + D$, where $A > 0$ and $B > 0$, has an amplitude of 3, a midline at $y = -2$, and passes through the point $\left(\frac{\pi}{4}, -2\right)$. What is the smallest positive value of $B$?

   • A) 1
   • B) 2
   • C) 4
   • D) $\frac{1}{2}$
2. 2.

   What are all the $x$-intercepts of the graph of the function $f(x) = 2\cos\left(2x + \frac{\pi}{3}\right) - 1$ on the interval $[0, \pi]$?

   • A) $x = \frac{\pi}{6}$, $x = \frac{\pi}{2}$, and $x = \frac{5\pi}{6}$
   • B) $x = 0$, $x = \frac{2\pi}{3}$, and $x = \pi$
   • C) $x = 0$, $x = \frac{\pi}{3}$, and $x = \frac{2\pi}{3}$
   • D) $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$
3. 3.

   What are the coordinates of the absolute maximum point of the function $f(x) = 3\sin\left(2x - \frac{\pi}{3}\right) - 1$ on the interval $[0, \pi]$?

   • A) $\left(\frac{\pi}{12}, 2\right)$
   • B) $\left(\frac{5\pi}{12}, 2\right)$
   • C) $\left(\frac{5\pi}{12}, 4\right)$
   • D) $\left(\frac{11\pi}{12}, 2\right)$
4. 4.

   Which of the following equations represents all vertical asymptotes of the function $f(x) = \tan\left(2x - \frac{\pi}{4}\right)$, where $k \in \mathbb{Z}$?

   • A) $x = \frac{\pi}{8} + \frac{k\pi}{2}$
   • B) $x = \frac{3\pi}{4} + k\pi$
   • C) $x = \frac{\pi}{4} + k\pi$
   • D) $x = \frac{3\pi}{8} + \frac{k\pi}{2}$
5. 5.

   What are the range and the fundamental period of the function $f(x) = -2\cos\left(3x + \frac{\pi}{6}\right) + 5$?

   • A) Range: $[-7, -3]$, Period: $\frac{2\pi}{3}$
   • B) Range: $[3, 7]$, Period: $\frac{2\pi}{3}$
   • C) Range: $[3, 7]$, Period: $\frac{\pi}{3}$
   • D) Range: $[-2, 5]$, Period: $\frac{2\pi}{3}$