To solve this problem, we apply a dilation transformation to the point $(8, -4)$ using a scale factor of $\frac{1}{4}$, centered at the origin.

The formula for dilation centered at the origin is: $(x', y') = (k \cdot x, k \cdot y)$ where $k$ is the scale factor, and $(x, y)$ is the original point.

Given:

• $x = 8$, $y = -4$
• $k = \frac{1}{4}$

Calculation:

1. Multiply the $x$-coordinate by $\frac{1}{4}$: $x' = \frac{1}{4} \cdot 8 = 2$
2. Multiply the $y$-coordinate by $\frac{1}{4}$: $y' = \frac{1}{4} \cdot (-4) = -1$

Final Answer:

The image of $(8, -4)$ after dilation is: $(2, -1)$

Would you like me to explain the concept of dilation in more detail?

Related Questions:

1. What happens if the scale factor is negative in a dilation?
2. How does dilation affect the distance between points and the origin?
3. How can you determine the scale factor if given the original point and its image?
4. What is the image of $(6, -9)$ under a scale factor of $\frac{1}{3}$?
5. How do dilations differ from translations and reflections?

Tip: In a dilation centered at the origin, multiplying both coordinates by the scale factor always preserves the point's direction relative to the origin.