To determine the scale factor and the center of dilation from the graph, we follow these steps:

1. Find the Scale Factor

The scale factor compares the distances from the center of dilation to a point on the preimage $L, M, N, O$ and the corresponding image points $L', M', N', O'$.

• First, identify the coordinates of points:
  • $L = (2, 3)$ and $L' = (4, 6)$
  • $M = (3, 5)$ and $M' = (6, 10)$
  • $N = (5, 4)$ and $N' = (10, 8)$
  • $O = (4, 2)$ and $O' = (8, 4)$

Now, calculate the distances between corresponding points:

$\text{Scale factor} = \frac{\text{distance to image point}}{\text{distance to preimage point}}$

By comparing the coordinates:

• From $L$ to $L'$:
  $\text{Scale factor} = \frac{4}{2} = 2$
• From $M$ to $M'$:
  $\text{Scale factor} = \frac{6}{3} = 2$

Thus, the scale factor is 2.

2. Find the Center of Dilation

The center of dilation is the point from which all the dilations are performed. To locate it:

1. Draw lines connecting each preimage point $L, M, N, O$ to its corresponding image point $L', M', N', O'$.
2. Extend these lines. The point where they intersect is the center of dilation.

From observation, the center of dilation is at the origin $(0, 0)$.

Final Answer:

• Scale factor: $2$
• Center of dilation: $(0, 0)$

Do you need further clarification or help with any part of the solution? Let me know!

Related Questions:

1. How do you determine the center of dilation algebraically?
2. Can a scale factor be negative, and what does it represent?
3. What happens to a figure when the scale factor is $\frac{1}{2}$?
4. How does dilation affect angles and side ratios in a figure?
5. If the scale factor were 3, what would happen to the coordinates of $LMNO$?

Tip:

To confirm the center of dilation, always test multiple points and verify where the connecting lines intersect.