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

Jun 13, 2018

See below

#### Explanation:

The second question answer the first, I'd say. Let's see the effect of the transformations: given a parent function $f \left(x\right)$, 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 $f \left(x\right) \setminus \to k f \left(x\right)$. 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 $f \left(x\right) \setminus \to f \left(k x\right)$. 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 $f \left(x\right) \setminus \to f \left(x\right) + k$. 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 $f \left(x\right) \setminus \to f \left(x + k\right)$. 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

$\sqrt{x} \setminus \to \sqrt{x - 3} \setminus \to \sqrt{x - 3} + 2$

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]}