Graph the following function by moving the green and blue dots (if necessary).
y, equals, cube root of, x, minus, 5, end cube root, minus, 5
y= 
3
  
x−5
​
 −5

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?

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#### 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.

Graph the following function by moving the green and blue dots (if necessary): y = cube root of (x - 5) - 5

Learn how to graph the function y = √[3]{x - 5} - 5 with detailed steps on transformations, including horizontal and vertical shifts. This problem is ideal for students in Grades 9-12 studying algebra and function transformations.

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