The graph of the cube root parent function $y=\sqrt[3]{x}$ is translated to form $f(x)$ shown on the graph.
Which equation represents $f(x)$ ?
$f(x)=\sqrt[3]{x+6}+1$
$f(x)=\sqrt[3]{x-6}+1$
$f(x)=\sqrt[3]{x+6}-1$
$f(x)=\sqrt[3]{x-6}-1$

Step 1 ：We can see that the cube root parent function $y=\sqrt[3]{x}$ is translated horizontally and vertically to form $f(x)$. Let's analyze each option.

Step 2 ：Option 1: $f(x)=\sqrt[3]{x+6}+1$. This represents a horizontal shift of 6 units to the left and a vertical shift of 1 unit up.

Step 3 ：Option 2: $f(x)=\sqrt[3]{x-6}+1$. This represents a horizontal shift of 6 units to the right and a vertical shift of 1 unit up.

Step 4 ：Option 3: $f(x)=\sqrt[3]{x+6}-1$. This represents a horizontal shift of 6 units to the left and a vertical shift of 1 unit down.

Step 5 ：Option 4: $f(x)=\sqrt[3]{x-6}-1$. This represents a horizontal shift of 6 units to the right and a vertical shift of 1 unit down.

Step 6 ：According to the given graph, the correct translation is a horizontal shift of 6 units to the right and a vertical shift of 1 unit up.

Step 7 ：Thus, the correct equation for $f(x)$ is \(\boxed{f(x)=\sqrt[3]{x-6}+1}\).