To graph the transformation of the given function \( Y = \sqrt[3]{8x} \), let’s break down the steps for understanding how the transformation affects the graph.

### Original Function: \( Y = \sqrt[3]{x} \)
The cube root function \( Y = \sqrt[3]{x} \) has a characteristic S-shape. It passes through the origin, is symmetric around the origin, and continues to both positive and negative infinity, though at a slower rate than a quadratic function.

### Transformed Function: \( Y = \sqrt[3]{8x} \)

The transformation here is inside the cube root function, which affects the \( x \)-axis. The transformation \( 8x \) compresses the graph horizontally by a factor of \( \frac{1}{8} \). This means the graph will stretch or shrink along the \( x \)-axis.

- **Horizontal compression**: The factor inside the cube root (the 8) compresses the graph horizontally. Instead of reaching 1 at \( x = 1 \), now the graph reaches 1 at \( x = \frac{1}{8} \), meaning the graph stretches faster as we move along the x-axis.

### Key points for the graph:
1. **At \( x = 0 \), \( Y = 0 \)** (the graph still passes through the origin).
2. **At \( x = \frac{1}{8} \), \( Y = 1 \)** (it reaches 1 faster compared to \( \sqrt[3]{x} \)).
3. **At \( x = -\frac{1}{8} \), \( Y = -1 \)** (similarly, negative values are reached faster).

The transformation results in a graph that is horizontally compressed compared to the original cube root function. The graph is still symmetric about the origin.

Would you like a visual plot of this graph for clarity?

### Relative Questions:
1. How does a horizontal stretch differ from a vertical stretch in transformations?
2. What would happen to the graph if we had \( \sqrt[3]{x - 2} \) instead?
3. How do transformations inside the function differ from those outside the function?
4. How does the graph of a cube root function differ from that of a square root function?
5. What effect would multiplying by a factor outside the cube root (e.g., \( Y = 2\sqrt[3]{x} \)) have on the graph?

### Tip:
When transforming graphs, operations inside the function affect the x-axis (horizontal), and operations outside affect the y-axis (vertical).

Explore the graph transformation of the cube root function Y = √[3]{8x}. Learn how horizontal compression by a factor of 1/8 affects the graph, and understand the changes compared to the original cube root function Y = √[3]{x}. Suitable for students in Grades 9-11.

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