4.2 Square Root Functions

Notes

The parent function is $\displaystyle f(x) = \sqrt{x}$ with domain $x \geq 0$ and range $y \geq 0$. For $\displaystyle g(x) = a\sqrt{x - h} + k$:

• $h$ shifts right (left if $h$ is negative) and $k$ shifts up.
• A negative $a$ reflects the graph over the $x$-axis. $|a| > 1$ stretches it vertically.
• The starting point is $(h, k)$. The domain is $x \geq h$, and the range is $y \geq k$ if $a > 0$ or $y \leq k$ if $a < 0$.
• For $\displaystyle \sqrt{2x - 6}$, solve $2x - 6 \geq 0$ to get $x \geq 3$.