The frequency of a sine graph refers to how often the sine wave repeats itself over a given interval, typically within a period of $2\pi$. It tells us the number of cycles (or oscillations) the wave completes within a unit of time or distance.

In the general form of the sine function: $y = A \sin(Bx + C) + D$

• $A$ is the amplitude (the height of the wave from the centerline).
• $B$ affects the frequency of the sine wave.
• $C$ is the phase shift (horizontal shift).
• $D$ is the vertical shift (moving the graph up or down).

How Frequency Works:

• Standard sine wave: $y = \sin(x)$

  • The period (length of one full cycle) is $2\pi$, and the wave completes 1 cycle over this interval.

• Effect of $B$: In $y = \sin(Bx)$, the value $B$ directly affects the frequency. The frequency is given by $|B|$, and the period is $\frac{2\pi}{|B|}$.

  • If $B = 2$, the sine wave completes 2 cycles in the interval from $0$ to $2\pi$, meaning the frequency is doubled.
  • If $B = \frac{1}{2}$, the sine wave completes half a cycle in the same interval, meaning the frequency is halved.

Key Points:

• Frequency is the number of oscillations per unit interval.
• It is inversely related to the period: $\text{frequency} = \frac{1}{\text{period}}$.
• As $B$ increases, the graph "compresses" horizontally, making the wave cycle faster (higher frequency).
• As $B$ decreases, the graph "stretches" horizontally, making the wave cycle slower (lower frequency).

Would you like more details or any further clarifications?

5 Related Questions:

1. How does amplitude affect the appearance of a sine graph?
2. What is the difference between frequency and period in sine functions?
3. How do phase shifts affect the position of sine and cosine graphs?
4. How does the sine graph compare to the cosine graph in terms of frequency and shifts?
5. Can you explain how vertical shifts modify the behavior of the sine graph?

Tip:

Remember, the higher the frequency, the more compressed the sine wave becomes, while lower frequencies stretch the wave over a larger interval.