Sine and cosine graph transformations quiz

📚 Quick Study Guide

• 📐 Amplitude: The amplitude of a sine or cosine function is half the distance between the maximum and minimum values. It's represented by $|A|$ in the general forms $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$.
• ↔️ Period: The period is the length of one complete cycle of the function. For $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$, the period is given by $\frac{2\pi}{|B|}$.
• phase shift: Phase Shift: The phase shift is the horizontal shift of the function. It's determined by $\frac{C}{B}$ in the general forms $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$. A positive value shifts the graph to the right, and a negative value shifts it to the left.
• ⬆️ Vertical Shift: The vertical shift is the upward or downward shift of the function. It's represented by $D$ in the general forms $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$.
• 📝 General Forms: The general forms of sine and cosine functions are: $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$, where:
  • $A$ is the amplitude
  • $B$ affects the period
  • $C$ affects the phase shift
  • $D$ affects the vertical shift

Practice Quiz

1. What is the amplitude of the function $y = 3\sin(2x) + 1$?
   1. 1
   2. 2
   3. 3
   4. 4


2. What is the period of the function $y = \cos(4x)$?
   1. $\pi$
   2. $\frac{\pi}{2}$
   3. $2\pi$
   4. $4\pi$


3. What is the phase shift of the function $y = \sin(x - \frac{\pi}{2})$?
   1. $\frac{\pi}{2}$ to the left
   2. $\frac{\pi}{2}$ to the right
   3. $\pi$ to the left
   4. $\pi$ to the right


4. What is the vertical shift of the function $y = 2\cos(x) - 3$?
   1. 3 units up
   2. 3 units down
   3. 2 units up
   4. 2 units down


5. Which transformation affects the maximum and minimum values of the function?
   1. Period
   2. Phase Shift
   3. Amplitude
   4. Vertical Shift


6. If the period of a sine function is $\pi$, what is the value of $B$ in $y = \sin(Bx)$?
   1. 1
   2. 2
   3. 0.5
   4. $\pi$


7. What is the range of the function $y = \sin(x) + 2$?
   1. [-1, 1]
   2. [1, 3]
   3. [-3, -1]
   4. [-2, 2]