https://file.aiquickdraw.com/mathbot/file/17254908472172ddyxedg.jpg

The image shows the graph of the function \( y = (2x)^3 - 2 \).

This function represents a cubic curve, and the graph follows a general cubic shape, but it's been transformed due to the coefficient of \( 2x \) and the vertical shift of \( -2 \). Let's break down the equation and its graph:

### Analysis of the Function \( y = (2x)^3 - 2 \):

1. **Base Function**: The base function is \( y = x^3 \), which is a standard cubic curve. This function passes through the origin and has an S-shaped curve with points of inflection at \( x = 0 \).
   
2. **Horizontal Stretch**: The term \( 2x \) indicates a horizontal compression by a factor of \( \frac{1}{2} \). This means that the graph of the cubic function is "squeezed" towards the y-axis, making it steeper.

3. **Vertical Shift**: The \( -2 \) at the end shifts the entire graph down by 2 units. Thus, the graph no longer passes through the origin but is shifted downward.

### Key Points of the Graph:

- The inflection point has shifted down from \( (0, 0) \) to \( (0, -2) \).
- The graph steepens compared to the regular \( y = x^3 \) function due to the horizontal compression by the factor \( 2x \).
- The curve exhibits symmetry about the origin, as is typical of cubic functions, but is shifted vertically.

Let me know if you'd like further details on the transformations, or if you have any other specific questions!

### Relative Questions:
1. How would the graph change if we replaced \( 2x \) with \( 3x \)?
2. What happens to the graph if we remove the vertical shift \( -2 \)?
3. Can you describe the end behavior of the function \( y = (2x)^3 - 2 \)?
4. How do horizontal stretches and compressions affect the overall shape of cubic functions?
5. What would the graph look like if the cubic term was \( (x/2)^3 \) instead?

### Tip:
To analyze transformations of a graph, always focus on horizontal and vertical shifts, stretches, compressions, and reflections in that order!

The user uploaded an image showing the graph of the function y = (2x)^3 - 2.

Explore the graph of the function y = (2x)^3 - 2, focusing on transformations such as horizontal compression and vertical shifts. This problem is suitable for high school students in Grades 10-12.

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