The problem asks for the coordinates of point $N'$ after the figure is dilated by a factor of 2, centered at the origin. Here's how to solve it step-by-step:

Step 1: Understanding dilation

A dilation centered at the origin with a factor of $k$ scales each point by multiplying its $x$- and $y$-coordinates by $k$. For this question: $k = 2$

Step 2: Original coordinates of point $N$

From the graph, the original coordinates of point $N$ are: $N = (-3, -1)$

Step 3: Apply the dilation formula

Using the dilation formula: $(x', y') = (k \cdot x, k \cdot y)$ Substitute $k = 2$ and $N = (-3, -1)$: $x' = 2 \cdot (-3) = -6$ $y' = 2 \cdot (-1) = -2$

Step 4: New coordinates of $N'$

The coordinates of $N'$ after the dilation are: $N' = (-6, -2)$

Final Answer:

The correct answer is: $\boxed{(-6, -2)}$

Let me know if you'd like additional explanations or have questions!

Related Questions:

1. What is the general formula for dilating a point centered at a specific origin?
2. How does the scale factor affect the size and position of the figure?
3. What happens to a point's coordinates if the scale factor is less than 1?
4. How would the coordinates of $N'$ change if the center of dilation were not the origin?
5. Can dilation change the orientation of a shape? Why or why not?

Tip:

Always check the center of dilation when applying a transformation—it determines how each point is scaled and shifted!