Question

the function $h(x)$ is a transformation of the square root parent function, $f(x)=\sqrt{x}$. what function is $h(x)$?
a. $h(x)=\sqrt{x+6}$
b. $h(x)=\sqrt{x}-6$
c. $h(x)=\sqrt{x-6}$
d. $h(x)=\sqrt{x}+6$

Explanation:

Step1: Identify parent function domain

The parent function $f(x)=\sqrt{x}$ has a domain of $x\geq0$, so its graph starts at $(0,0)$.

Step2: Identify $h(x)$ starting point

From the graph, $h(x)$ starts at $(-6,0)$. This means the domain of $h(x)$ is $x\geq-6$.

Step3: Match to transformation rule

A horizontal shift left by $a$ units transforms $f(x)$ to $f(x+a)$. To get a domain of $x\geq-6$, we replace $x$ with $x+6$ in the parent function.

Step4: Verify vertical position

The graph of $h(x)$ has the same vertical height as $f(x)$ at corresponding shifted points, so no vertical shift is present.

Answer:

A. $h(x) = \sqrt{x + 6}$