Trigonometric functions and graphs Knowledge-Based Practice

Questions

1. 1.

   A sinusoidal function of the form $f(x) = a\cos(bx) + d$, where $a > 0$ and $b > 0$, has a maximum value of 7, a minimum value of $-1$, and a period of $\frac{2\pi}{3}$. What is the value of $f\left(\frac{\pi}{6}\right)$?

   • A) 3
   • B) 1
   • C) 5
   • D) $-1$
2. 2.

   What is the sum of all $x$-intercepts of the function $f(x) = 2\sin(2x) - \sqrt{3}$ on the interval $[0, \pi]$?

   • A) $\frac{\pi}{2}$
   • B) $\frac{2\pi}{3}$
   • C) $\pi$
   • D) $\frac{5\pi}{6}$
3. 3.

   What are the coordinates of the first local maximum of the function $f(x) = 4\sin\left(2x - \frac{\pi}{3}\right) - 1$ for $x > 0$?

   • A) $\left(\frac{\pi}{3}, 3\right)$
   • B) $\left(\frac{5\pi}{12}, 3\right)$
   • C) $\left(\frac{7\pi}{12}, 3\right)$
   • D) $\left(\frac{5\pi}{12}, 4\right)$
4. 4.

   Which of the following gives the equations of all vertical asymptotes of the function $f(x) = \tan\left(2x - \frac{\pi}{3}\right)$, where $k \in \mathbb{Z}$?

   • A) $x = \frac{\pi}{6} + \frac{k\pi}{2}$
   • B) $x = \frac{\pi}{12} + \frac{k\pi}{2}$
   • C) $x = \frac{5\pi}{12} + k\pi$
   • D) $x = \frac{5\pi}{12} + \frac{k\pi}{2}$
5. 5.

   What are the period and the range of the function $f(x) = -3\cos\left(2x - \frac{\pi}{4}\right) + 1$?

   • A) Period is $\pi$, and range is $[-2, 4]$
   • B) Period is $2\pi$, and range is $[-2, 4]$
   • C) Period is $\pi$, and range is $[-4, 2]$
   • D) Period is $\frac{\pi}{2}$, and range is $[-3, 3]$

Answers and step-by-step solutions appear after you finish the test.