https://file.aiquickdraw.com/mathbot/file/1730911736253f4z65f9h.jpg

To find the image of \(\triangle ABC\) after a dilation with a scale factor of 2 centered at the origin, we'll multiply the coordinates of each vertex by 2.

### Step-by-Step Solution

1. **Identify the Coordinates of Each Vertex:**
   - Point \( A \) is at \((-4, 0)\).
   - Point \( B \) is at \((2, 0)\).
   - Point \( C \) is at \((-2, 3)\).

2. **Apply the Dilation Transformation:**
   - The formula for dilation centered at the origin with a scale factor of \(k\) is \((x, y) \rightarrow (kx, ky)\).
   - Here, \( k = 2 \).

3. **Calculate the Dilated Coordinates:**
   - For \( A \): \((-4 \cdot 2, 0 \cdot 2) = (-8, 0)\).
   - For \( B \): \((2 \cdot 2, 0 \cdot 2) = (4, 0)\).
   - For \( C \): \((-2 \cdot 2, 3 \cdot 2) = (-4, 6)\).

4. **Plot the Image Points:**
   - After dilation, the vertices of \(\triangle ABC\) (let's label it \(\triangle A'B'C'\)) are:
     - \( A' = (-8, 0) \)
     - \( B' = (4, 0) \)
     - \( C' = (-4, 6) \)

### Conclusion

The new triangle \(\triangle A'B'C'\) will have its vertices at \((-8, 0)\), \((4, 0)\), and \((-4, 6)\).

Would you like me to provide further details on how to plot this or answer any additional questions?

### Related Questions:

1. What happens to a shape’s orientation after a dilation?
2. How do different scale factors affect the size of the dilated image?
3. What are the differences between dilation and translation?
4. Can a dilation result in a smaller figure? How?
5. How would the coordinates change if the center of dilation were not the origin?

### Tip:
Remember, in a dilation transformation, all points move directly away from or toward the center by the scale factor, maintaining the shape’s proportions.

Graph the image of △ABC after a dilation with a scale factor of 2, centered at the origin.

Learn how to graph the image of △ABC after a dilation with a scale factor of 2, centered at the origin. This solution covers key geometric concepts of dilation transformations, using the dilation formula to calculate new coordinates, and plotting the image points.

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