Sinusoidal function

Definition

A sinusoidal function is a type of periodic function that describes smooth, repetitive oscillations, typically modeled using the sine or cosine functions. These functions are fundamental in trigonometry and have applications in various fields such as physics, engineering, and signal processing.

5 Must Know Facts For Your Next Test

1. The general form of a sinusoidal function can be written as $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$.
2. The amplitude $A$ represents the peak value of the wave from its central axis.
3. The period $T$ of the function is given by $T = \frac{2\pi}{B}$.
4. The phase shift is determined by the value of $C$, which shifts the graph horizontally.
5. The vertical shift is represented by $D$, which moves the graph up or down.

Review Questions

• What parameters affect the amplitude and period of a sinusoidal function?
• How do you determine the phase shift and vertical shift in a sinusoidal function?
• Write the equation for a sine wave with amplitude 3, period $\pi$, phase shift $\frac{\pi}{4}$ to the left, and vertical shift up by 2 units.

Related terms

Amplitude: The maximum distance a point on a wave reaches from its central axis.

Period: The length of one complete cycle of a periodic function. For sinusoidal functions, it is given by $T = \frac{2\pi}{B}$.

Phase Shift: The horizontal displacement of a periodic function. It shifts the graph left or right based on the value of $C$ in the general form.