To graph the function \( y = x^{\frac{2}{3}} - 1 \), you can follow these steps:

### 1. **Understand the Function:**
   - The function \( y = x^{\frac{2}{3}} - 1 \) consists of two parts:
     - \( x^{\frac{2}{3}} \): This represents the cubic root of \( x^2 \), which is always non-negative.
     - The \(-1\) shifts the graph of \( x^{\frac{2}{3}} \) downward by 1 unit.

### 2. **Identify Key Points:**
   - **At \( x = 0 \):** 
     \[
     y = 0^{\frac{2}{3}} - 1 = -1
     \]
     So, the point \((0, -1)\) is on the graph.
   - **At \( x = 1 \):** 
     \[
     y = 1^{\frac{2}{3}} - 1 = 0
     \]
     So, the point \((1, 0)\) is on the graph.
   - **At \( x = -1 \):** 
     \[
     y = (-1)^{\frac{2}{3}} - 1 = 1 - 1 = 0
     \]
     So, the point \((-1, 0)\) is also on the graph.

### 3. **Behavior of the Function:**
   - **For \( x > 0 \):** 
     The function \( y = x^{\frac{2}{3}} - 1 \) increases slowly since the exponent \( \frac{2}{3} \) grows more slowly than a linear function.
   - **For \( x < 0 \):** 
     The function behaves similarly as for \( x > 0 \), because \( x^{\frac{2}{3}} \) is still positive for negative values of \( x \).

### 4. **Draw the Graph:**
   - **Plot the key points** identified above: \((0, -1)\), \((1, 0)\), and \((-1, 0)\).
   - **Sketch the curve**:
     - The curve is symmetric with respect to the y-axis since \( y = x^{\frac{2}{3}} - 1 \) does not change if \( x \) is replaced by \( -x \).
     - Near \( x = 0 \), the function behaves like a steep curve and flattens as \( |x| \) increases.
   - **Vertical shifts**: The graph of \( x^{\frac{2}{3}} \) is shifted downward by 1 unit.

### 5. **Final Shape of the Graph:**
   - The graph passes through \((0, -1)\) and has a gentle upward curve towards the right and left, flattening out as \( x \) moves away from 0. The graph is symmetric around the y-axis.

If you'd like, I can create this graph for you. Would you like me to do that?

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Here are some questions related to this graph:

1. How does the graph of \( y = x^{\frac{2}{3}} - 1 \) compare to \( y = x^{2} - 1 \)?
2. What is the range of the function \( y = x^{\frac{2}{3}} - 1 \)?
3. How do transformations affect the graph of basic functions like \( y = x^{\frac{2}{3}} \)?
4. What happens to the graph if we replace \( x^{\frac{2}{3}} \) with \( x^{\frac{4}{3}} \)?
5. What are the real roots of \( y = x^{\frac{2}{3}} - 1 \)?

**Tip:** Always consider the effect of horizontal and vertical transformations when graphing functions.

Learn how to graph the function y = x^(2/3) - 1 with detailed steps and key points. Understand the behavior of the function, identify key points like (0, -1), (1, 0), and (-1, 0), and analyze the impact of transformations. Suitable for high school students.

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