To find the image of the point $(-3, -8)$ after a dilation by a scale factor of 4, centered at the origin, we use the dilation formula. The dilation formula is:

$(x', y') = (k \cdot x, k \cdot y)$

where:

• $(x, y)$ is the original point,
• $k$ is the scale factor, and
• $(x', y')$ is the image of the point after dilation.

Here:

• $k = 4$,
• the original point is $(-3, -8)$.

Now, apply the scale factor to the coordinates:

$x' = 4 \cdot (-3) = -12$ $y' = 4 \cdot (-8) = -32$

So, the image of the point $(-3, -8)$ after dilation by a scale factor of 4 centered at the origin is:

$(-12, -32)$

Would you like further explanation or any clarification?

Here are some related questions you could explore:

1. How would the image change if the scale factor was a fraction (e.g., 1/2)?
2. What happens if the dilation is centered at a point other than the origin?
3. How does dilation affect the distances between points on a coordinate plane?
4. Can dilation preserve angles in a geometric figure? If so, why?
5. How does a negative scale factor affect the image of a point?

Tip: When performing dilation, always remember to multiply the coordinates by the scale factor. If the center of dilation is not at the origin, you need to adjust the points accordingly.