Y = A cos(B(x - C)) + D

Definition

The expression $y = A \\cos(B(x - C)) + D$ represents a general form of the cosine function, which can be used to model various periodic phenomena. The parameters $A$, $B$, $C$, and $D$ allow for adjustments to the amplitude, frequency, phase shift, and vertical shift of the cosine curve, respectively.

5 Must Know Facts For Your Next Test

1. The parameter $A$ in the expression $y = A \\cos(B(x - C)) + D$ determines the amplitude of the cosine function, which is the distance between the midline and the maximum or minimum value of the curve.
2. The parameter $B$ in the expression $y = A \\cos(B(x - C)) + D$ determines the frequency of the cosine function, which is the number of complete cycles per unit of the independent variable $x$.
3. The parameter $C$ in the expression $y = A \\cos(B(x - C)) + D$ determines the phase shift of the cosine function, which is the horizontal shift of the curve.
4. The parameter $D$ in the expression $y = A \\cos(B(x - C)) + D$ determines the vertical shift of the cosine function, which is the distance between the midline and the $x$-axis.
5. The general form of the cosine function, $y = A \\cos(B(x - C)) + D$, can be used to model a wide variety of periodic phenomena, such as sound waves, electrical signals, and mechanical vibrations.

Review Questions

• Explain how the parameter $A$ in the expression $y = A \\cos(B(x - C)) + D$ affects the graph of the cosine function.
  • The parameter $A$ in the expression $y = A \\cos(B(x - C)) + D$ determines the amplitude of the cosine function. Specifically, $A$ represents the distance between the midline (the horizontal line at $y = D$) and the maximum or minimum value of the curve. As the value of $A$ increases, the amplitude of the cosine function increases, resulting in a taller graph. Conversely, as the value of $A$ decreases, the amplitude of the cosine function decreases, resulting in a shorter graph.
• Describe how the parameter $B$ in the expression $y = A \\cos(B(x - C)) + D$ affects the graph of the cosine function.
  • The parameter $B$ in the expression $y = A \\cos(B(x - C)) + D$ determines the frequency of the cosine function. Specifically, $B$ represents the number of complete cycles of the cosine function per unit of the independent variable $x$. As the value of $B$ increases, the frequency of the cosine function increases, resulting in more cycles per unit of $x$ and a more compressed graph. Conversely, as the value of $B$ decreases, the frequency of the cosine function decreases, resulting in fewer cycles per unit of $x$ and a more stretched-out graph.
• Analyze how the parameter $C$ in the expression $y = A \\cos(B(x - C)) + D$ affects the graph of the cosine function.
  • The parameter $C$ in the expression $y = A \\cos(B(x - C)) + D$ determines the phase shift of the cosine function. Specifically, $C$ represents the horizontal shift of the curve along the $x$-axis. As the value of $C$ increases, the cosine function is shifted to the right, and as the value of $C$ decreases, the cosine function is shifted to the left. This phase shift can be used to model periodic phenomena that are not aligned with the $x$-axis, such as sound waves or electrical signals that are out of phase with a reference signal.

Related terms

Amplitude:

The distance between the midline and the maximum or minimum value of the cosine function, represented by the parameter $A$.

Frequency:

The number of complete cycles of the cosine function per unit of the independent variable $x$, represented by the parameter $B$.

Phase Shift:

The horizontal shift of the cosine function, represented by the parameter $C$.