Test Questions on Cycles and Periodicity in Functions

📚 Quick Study Guide

• 🔍 Periodic Function: A function $f(x)$ is periodic if there exists a non-zero constant $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain. $T$ is the period of the function.
• ⏱️ Period: The smallest positive value of $T$ for which $f(x + T) = f(x)$ is the fundamental period.
• 📈 Amplitude: For functions like $f(x) = A\sin(Bx)$ or $f(x) = A\cos(Bx)$, the amplitude is $|A|$.
• 🔄 Frequency: The number of cycles completed per unit of time or space. It's the reciprocal of the period, i.e., $f = \frac{1}{T}$.
• 📐 Transformations: Understanding how transformations like shifts, stretches, and compressions affect the period and amplitude of trigonometric functions.
• 💡 Key Trigonometric Periods:
  • $\sin(x)$ and $\cos(x)$ have a period of $2\pi$.
  • $\tan(x)$ has a period of $\pi$.
• 📝 Period Changes: For $f(x) = \sin(Bx)$ or $f(x) = \cos(Bx)$, the period is $\frac{2\pi}{|B|}$. For $f(x) = \tan(Bx)$, the period is $\frac{\pi}{|B|}$.

🧪 Practice Quiz

1. What is the period of the function $f(x) = \sin(2x)$?
   1. $\pi$
   2. $2\pi$
   3. $\frac{\pi}{2}$
   4. $4\pi$
2. The function $f(x) = \cos(x) + \cos(2x)$ is periodic. What is its period?
   1. $\pi$
   2. $2\pi$
   3. $3\pi$
   4. $4\pi$
3. What is the amplitude of the function $f(x) = 3\sin(x)$?
   1. 1
   2. 2
   3. 3
   4. 4
4. If $f(x)$ has a period of $T$, what is the period of $f(x/2)$?
   1. $T/2$
   2. $2T$
   3. $T$
   4. $4T$
5. The period of $f(x) = \tan(3x)$ is:
   1. $\pi$
   2. $3\pi$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{6}$
6. Which of the following functions is NOT periodic?
   1. $f(x) = \sin(x)$
   2. $f(x) = x^2$
   3. $f(x) = \cos(x)$
   4. $f(x) = \tan(x)$
7. What is the period of the function $f(x) = 5\cos(4x)$?
   1. $5\pi$
   2. $4\pi$
   3. $\frac{\pi}{2}$
   4. $\frac{5\pi}{4}$