In the image, you're asked to apply a dilation to point $A$ with respect to the origin $(0, 0)$ using a scale factor of $\frac{1}{3}$.

Step-by-Step Solution:

1. Identify the coordinates of point $A$: Point $A$ is located at $(6, -6)$ on the coordinate plane.

2. Apply the dilation formula: When dilating a point about the origin by a scale factor of $k$, the formula for the new coordinates is: $A' = (k \cdot x, k \cdot y)$ In this case, the scale factor $k = \frac{1}{3}$.

3. Calculate the new coordinates:

   • For the $x$-coordinate: $\frac{1}{3} \times 6 = 2$
   • For the $y$-coordinate: $\frac{1}{3} \times (-6) = -2$

   So, the new coordinates of point $A'$ are $(2, -2)$.

4. Plot the image: The image of point $A$ after dilation will be located at $(2, -2)$ on the coordinate plane.

Would you like further details or assistance with this? Here are 5 related questions to deepen your understanding:

1. How do you determine the coordinates of a point after dilation when the origin is not the center?
2. What happens to the distances between points and the origin when the scale factor is less than 1?
3. How would a negative scale factor change the position of the point?
4. Can dilation result in the points being collinear with the origin?
5. What is the relationship between dilation and other transformations like rotation or translation?

Tip: Always double-check the coordinates before and after transformations to ensure accurate results.