Given the function $y=\sin (x-2)+3$, what are the values of the vertical shift $k$ and the phase shift $h$?

• A. $k=2$, $h=3$
• B. $k=-3$, $h=-2$
• C. $k=3$, $h=2$
• D. $k=0$, $h=2$

Answer: (C)

Verified step by step guidance

1. Identify the general form of the sine function with transformations: \(y = \sin(x - h) + k\), where \(h\) represents the phase shift and \(k\) represents the vertical shift.
2. Compare the given function \(y = \sin(x - 2) + 3\) to the general form to determine the values of \(h\) and \(k\).
3. Notice that inside the sine function, the expression is \((x - 2)\), which means the phase shift \(h\) is \$2$ units to the right.
4. Observe the constant added outside the sine function is \(+3\), indicating the vertical shift \(k\) is \$3$ units upward.
5. Conclude that the phase shift \(h = 2\) and the vertical shift \(k = 3\) based on the comparison.