Which expression exhibits a standard cosine graph with horizontal shift and vertical shift?

Which expression exhibits a standard cosine graph with horizontal shift and vertical shift?

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

Answer: (C)

A cosine graph with both horizontal and vertical shifts is described by y = a cos(b(x - h)) + k. This form shows a horizontal shift h and a vertical shift k. Writing it as y = a cos(bx - c) + d makes the horizontal shift explicit inside the cosine (shift to the right by c/b) and the vertical shift explicit at the end (up by d). The horizontal shift comes from the inside argument, while the vertical shift comes from the + d term. The pattern that uses cosine, includes a phase inside the cosine, and adds a vertical shift is the one with cos(bx - c) + d. The other forms either use sine, omit the horizontal shift, or place the vertical shift with a minus sign, which changes direction.

A cosine graph with both horizontal and vertical shifts is described by y = a cos(b(x - h)) + k. This form shows a horizontal shift h and a vertical shift k. Writing it as y = a cos(bx - c) + d makes the horizontal shift explicit inside the cosine (shift to the right by c/b) and the vertical shift explicit at the end (up by d). The horizontal shift comes from the inside argument, while the vertical shift comes from the + d term.

The pattern that uses cosine, includes a phase inside the cosine, and adds a vertical shift is the one with cos(bx - c) + d. The other forms either use sine, omit the horizontal shift, or place the vertical shift with a minus sign, which changes direction.