which graph represents $y = sqrt3{x + 6}-3$?

Explanation:

Step1: Analyze the parent - function

The parent - function of $y = \sqrt[3]{x+6}-3$ is $y=\sqrt[3]{x}$. The general form of a cube - root function transformation is $y = a\sqrt[3]{x - h}+k$, where $(h,k)$ is the point of inflection.

Step2: Identify the transformation

For the function $y=\sqrt[3]{x + 6}-3$, we have $h=-6$ and $k = - 3$. This means the graph of the parent - function $y=\sqrt[3]{x}$ is shifted 6 units to the left and 3 units down.

Step3: Check key points

The parent - function $y=\sqrt[3]{x}$ has a point $(0,0)$. After the transformation, the corresponding point for $y=\sqrt[3]{x + 6}-3$ is obtained by substituting $x=-6$ into the function: $y=\sqrt[3]{-6 + 6}-3=-3$. So the point $(-6,-3)$ should be on the graph.

Answer:

The graph that has a point at $(-6,-3)$ and has the general shape of a cube - root function (smooth curve with a point of inflection) is the correct graph. Without seeing all the options, we can't exactly pick the right one from a list, but we know it should have the described transformation characteristics.