this graph shows both a reflection and a translation. what function is being graphed?\n$y =…

Explanation:

Step1: Analyze basic cube - root function

The parent function of a cube - root function is $y = \sqrt[3]{x}$. Its graph passes through the origin $(0,0)$.

Step2: Consider vertical translation

The general form of a vertical translation of a function $y = f(x)$ is $y=f(x)+k$, where $k$ is the amount of vertical shift. If $k=- 1$, the graph of the function is shifted down by 1 unit. All the given functions have $-1$ at the end, so they all have a vertical shift of 1 unit down.

Step3: Consider horizontal translation

The general form of a horizontal translation of a function $y = f(x)$ is $y = f(x - h)$. If the graph is shifted to the right by $h$ units, $h>0$. If shifted to the left by $h$ units, $h < 0$. The graph seems to be shifted to the right. For the function $y=\sqrt[3]{x - h}-1$, when $h = 2$, we have $y=\sqrt[3]{x - 2}-1$. But we also need a reflection.

Step4: Consider reflection

A reflection of the function $y = f(x)$ about the $x$ - axis is given by $y=-f(x)$. Combining the horizontal translation of 2 units to the right, vertical translation of 1 unit down and reflection about the $x$ - axis, the function is $y=-\sqrt[3]{x - 2}-1$.

Answer:

$y =-\sqrt[3]{x - 2}-1$