The graph of (3(x+2)+1 / (x+2)^3)-6 is the same as the graph of 3x+1 / x^3 only it is shifted _____________.

start with 3x + 1/x^3

now replace x with x+2 and you will get

3(x+2) + 1/(x+2)^3 which will result in a horizonatal shift of 2 units to the LEFT

now if we subract 6 from that, resulting in a vertical shift of 6 units DOWN, we get your starting expression.

So the two graphs indeed have the same shape, the last one is "shifted" two to the left and 6 down to get the first.

Reverse the directions to get the shift from the first to the second

To determine the shift between the graphs of the two expressions, we need to understand how the second expression is derived from the first.

Starting with the expression 3x + 1/x^3, we can notice that by replacing x with x+2, we obtain 3(x+2) + 1/(x+2)^3. This substitution results in a horizontal shift of 2 units to the left.

Now, if we subtract 6 from the expression, we get (3(x+2) + 1/(x+2)^3) - 6, which leads to a vertical shift of 6 units down.

Therefore, the graph of (3(x+2) + 1/(x+2)^3) - 6 is the same as the graph of 3x + 1/x^3, but it is shifted 2 units to the left and 6 units down.

To reverse the directions and find the shift from the first expression to the second, we simply need to do the opposite operations.

So, starting with (3(x+2) + 1/(x+2)^3) - 6, if we add 6 to the expression, the graph will shift 6 units up. Next, if we replace x+2 with x, the graph will shift 2 units to the right.

Therefore, to obtain the graph of 3x + 1/x^3 from (3(x+2) + 1/(x+2)^3) - 6, we shift it 6 units up and 2 units to the right.