Solved Examples: Graphing Sine and Cosine Functions with Varying Amplitude and Period

📚 Quick Study Guide

• 📈 Amplitude: The amplitude of a sine or cosine function is the distance from the midline to the maximum or minimum value. 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. It's calculated as $\frac{2\pi}{|B|}$ for both sine and cosine functions.
• ↔️ Phase Shift: The phase shift is the horizontal shift of the function, calculated as $\frac{C}{B}$. 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, represented by $D$.
• ✏️ Graphing Steps:
  1. Identify $A$, $B$, $C$, and $D$.
  2. Determine the amplitude, period, phase shift, and vertical shift.
  3. Find key points by dividing the period into four equal intervals.
  4. Plot the points and draw the curve.

🧪 Practice Quiz

1. What is the amplitude of the function $y = 3\sin(2x)$?
   1. A) 1
   2. B) 2
   3. C) 3
   4. D) 6
2. What is the period of the function $y = \cos(4x)$?
   1. A) $\pi$
   2. B) $\frac{\pi}{2}$
   3. C) $2\pi$
   4. D) $4\pi$
3. What is the phase shift of the function $y = \sin(x - \frac{\pi}{2})$?
   1. A) $\frac{\pi}{2}$ to the left
   2. B) $\frac{\pi}{2}$ to the right
   3. C) $\pi$ to the left
   4. D) $\pi$ to the right
4. What is the vertical shift of the function $y = 2\cos(x) + 1$?
   1. A) 1 unit down
   2. B) 2 units up
   3. C) 1 unit up
   4. D) 2 units down
5. The function $y = -2\sin(x)$ has a reflection about the:
   1. A) y-axis
   2. B) x-axis
   3. C) origin
   4. D) None of the above
6. What is the period of the function $y = 5\sin(\frac{1}{2}x)$?
   1. A) $\pi$
   2. B) $2\pi$
   3. C) $4\pi$
   4. D) $8\pi$
7. What is the amplitude of the function $y = -4\cos(3x)$?
   1. A) -4
   2. B) 3
   3. C) 4
   4. D) -3