What does phase shift describe in a sine or cosine graph?

What does phase shift describe in a sine or cosine graph?

• A. Vertical scaling
• B. Horizontal shift (left or right)
• C. Amplitude
• D. Midline shift

Answer: (B)

Phase shift describes a horizontal shift of the sine or cosine graph along the x-axis. When the inside of the sine or cosine is written as sin(Bx − C) or cos(Bx − C), the amount of horizontal shift is C/B: a positive value moves the graph to the right, a negative value to the left. If you write it as sin(B(x − h)) or cos(B(x − h)), the shift is h units to the right when h is positive and to the left when h is negative. This shift only changes where the cycle starts; it does not change how tall the waves are or how often they repeat. The amplitude is determined by A (the peak heights), and the period is determined by B (the cycle length), specifically 2π divided by B. Moving the graph up or down—the midline—comes from a vertical shift D, not from phase shift. For example, sin(2x) and sin(2x − π/2) describe the same shape, but the second is shifted to the right by π/4, since sin(2x − π/2) = sin(2(x − π/4)). The same idea applies to cosine with its own horizontal shifts.

Phase shift describes a horizontal shift of the sine or cosine graph along the x-axis. When the inside of the sine or cosine is written as sin(Bx − C) or cos(Bx − C), the amount of horizontal shift is C/B: a positive value moves the graph to the right, a negative value to the left. If you write it as sin(B(x − h)) or cos(B(x − h)), the shift is h units to the right when h is positive and to the left when h is negative.

This shift only changes where the cycle starts; it does not change how tall the waves are or how often they repeat. The amplitude is determined by A (the peak heights), and the period is determined by B (the cycle length), specifically 2π divided by B. Moving the graph up or down—the midline—comes from a vertical shift D, not from phase shift.

For example, sin(2x) and sin(2x − π/2) describe the same shape, but the second is shifted to the right by π/4, since sin(2x − π/2) = sin(2(x − π/4)). The same idea applies to cosine with its own horizontal shifts.