Which expression exhibits a sine function with both a phase shift and a vertical shift?

• A. y = 2 sin(3x - 1) + 4
• B. y = 2 sin(3x) + 4
• C. y = 2 sin(3x - 1) - 4
• D. y = 2 cos(3x - 1) + 4

Answer: (A)

A sine function with both a phase shift and a vertical shift is best understood by writing it in the form y = A sin(Bx − C) + D. Here A is the amplitude, B affects the period, C gives the horizontal shift, and D shifts the graph vertically.

The expression y = 2 sin(3x − 1) + 4 fits this pattern. inside the sine, 3x − 1 means there is a horizontal (phase) shift of 1/3 to the right. outside the sine, the +4 raises the graph by 4 units, giving a vertical shift upward. It remains a sine function, not cosine.

The other options fail one part of the requirement. One uses sin(3x) with no horizontal shift, so it has no phase shift. Another uses cosine, not sine, so it isn’t a sine function. Therefore, this expression is the correct example of a sine function that has both a phase shift and a vertical shift.