Which expression best represents the standard form for a transformed sine graph?

Which expression best represents the standard form for a transformed sine graph?

• A. y=a sin(bx) + d
• B. y=a sin(bx - c) + d
• C. y= a cos(bx - c) + d
• D. y= a sin(bx - c) - d

Answer: (B)

The main idea here is that a transformed sine graph is described by a single sine expression that shows both how tall it gets, how it’s stretched horizontally, and how it’s shifted both horizontally and vertically. The standard way to write this is y = a sin(bx - c) + d. Here, a controls the amplitude (how tall the waves are), b compresses or stretches the graph horizontally, the inside term bx - c creates a horizontal shift to the right by c/b, and the outside + d shifts the graph upward by d. This form captures all the common transformations in one neat expression. So the best choice is the one that uses both a horizontal shift inside the sine and a vertical shift outside, giving y = a sin(bx - c) + d. It matches the convention exactly: the phase shift is inside the sine with a minus sign, and the vertical shift is added at the end. The other forms don’t fit as well for this purpose. One omits the horizontal shift inside the sine, so it wouldn’t reflect a transformed graph with a phase shift. Another uses a cosine instead of sine, which would describe a different wave. The last option shows a downward vertical shift, which is just a sign choice for the vertical shift and isn’t the standard way to represent the vertical translation in the usual form.

The main idea here is that a transformed sine graph is described by a single sine expression that shows both how tall it gets, how it’s stretched horizontally, and how it’s shifted both horizontally and vertically. The standard way to write this is y = a sin(bx - c) + d. Here, a controls the amplitude (how tall the waves are), b compresses or stretches the graph horizontally, the inside term bx - c creates a horizontal shift to the right by c/b, and the outside + d shifts the graph upward by d. This form captures all the common transformations in one neat expression.

So the best choice is the one that uses both a horizontal shift inside the sine and a vertical shift outside, giving y = a sin(bx - c) + d. It matches the convention exactly: the phase shift is inside the sine with a minus sign, and the vertical shift is added at the end.

The other forms don’t fit as well for this purpose. One omits the horizontal shift inside the sine, so it wouldn’t reflect a transformed graph with a phase shift. Another uses a cosine instead of sine, which would describe a different wave. The last option shows a downward vertical shift, which is just a sign choice for the vertical shift and isn’t the standard way to represent the vertical translation in the usual form.