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.