It seems you're dealing with a graphing problem involving the function:

$y = \sqrt[3]{x - 5} - 5$

Let’s break down how to graph this function by transforming it step-by-step:

Step 1: Understand the Base Function

The base function is the cube root function, $y = \sqrt[3]{x}$. This function has a characteristic shape:

• The graph of $y = \sqrt[3]{x}$ passes through the origin (0,0).
• It increases from left to right, and is symmetric about the origin.

Step 2: Transformations

Now, let’s apply the transformations to the base function.

1. Horizontal Shift:

   • The expression inside the cube root is $x - 5$.
   • This results in a horizontal shift to the right by 5 units.
   • So, the graph now starts from the point $(5, 0)$.

2. Vertical Shift:

   • There is also a -5 outside the cube root, meaning the entire graph shifts down by 5 units.
   • The starting point now moves from $(5, 0)$ to $(5, -5)$.

Step 3: Plotting Key Points

To plot the graph, consider these points:

• Starting Point: The shifted version starts at $(5, -5)$.
• Additional Points: Calculate a few values to understand the behavior:
  • When $x = 6$, $y = \sqrt[3]{6 - 5} - 5 = \sqrt[3]{1} - 5 = -4$.
  • When $x = 13$, $y = \sqrt[3]{13 - 5} - 5 = \sqrt[3]{8} - 5 = -3$.

Graph Description

• The graph will have an inflection point at $(5, -5)$.
• To the right of $x = 5$, the graph gradually increases.
• To the left, the values of $y$ continue to decrease, but at a decreasing rate.

Moving the Dots (Green and Blue)

• Green Dot: Move it to the point $(5, -5)$ to mark the new "origin" after the transformation.
• Blue Dot: Use it to mark any other point that helps illustrate the increasing nature, for example, $(6, -4)$.

Would you like more detailed instructions on how to perform these transformations or see a graph of this function?

---

Questions for Further Exploration:

1. What are the differences between the graphs of cube root functions and square root functions?
2. How does a horizontal shift affect the domain of a function?
3. What happens if we have a negative coefficient in front of the cube root?
4. Can you determine the range of this transformed function?
5. How would the graph change if the transformation was $y = \sqrt[3]{x + 5} - 5$?

Tip: When graphing transformed functions, it’s helpful to first sketch the base function and then apply each transformation one by one.