How do you graph `y=sqrt(x-3)+2` and how does it compare to the parent function?

The second question answer the first, I'd say. Let's see the effect of the transformations: given a parent function `f(x)`, we have four basic transformation: stretched and translation, both horizontal and vertical.

• Vertical stretch: we obtain it by multipling the whole function by some constant, so `f(x) \to kf(x)`. If `0<k<1` we have a vertical shrink, if `k>1` we have a vertical expansion. If `k` is negative the function is reflected about the `x` axis, and then stretched as above.
• Horizontal stretch: we obtain it by multiplying the variable by some constant, so `f(x) \to f(kx)`. If `0<k<1` we have a horizontal expansion, if `k>1` we have a horizontal shrink. If `k` is negative the function is reflected about the `y` axis, and then stretched as above.
• Vertical translation: we obtain it by adding some constant to the function, so `f(x)\to f(x)+k`. If `k` is positive the shift is upwards, otherwise it's downwards.
• Horizontal translation: we obtain it by adding some constant to the variable, so `f(x)\to f(x+k)`. If `k` is positive the shift is leftwards, otherwise it's rightwards.

In your case, you have both vertical and horizontal translation, so you have

`sqrt(x) \to sqrt(x-3) \to sqrt(x-3)+2`

The first transformation translates the function three units to the right, the second one two units higher. See the graph of the three steps to check:

Parent function: `y=sqrt(x)`
graph{sqrt(x) [-3.5, 10, -0.5, 5]}

First translation: `y = sqrt(x-3)`
graph{sqrt(x-3) [-3.5, 10, -0.5, 5]}

Second translation: `y = sqrt(x-3)+2`
graph{sqrt(x-3)+2 [-3.5, 10, -0.5, 5]}