To find the image of the point $(0,4)$ after a dilation by a scale factor of 3 centered at the origin, use the dilation formula:

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

where $k = 3$, $x = 0$, and $y = 4$:

$x' = 3 \cdot 0 = 0$ $y' = 3 \cdot 4 = 12$

So, the image of $(0,4)$ after dilation is $(0,12)$.

Would you like further explanation?

Here are five related questions to deepen understanding:

1. How does dilation affect the coordinates of a point when centered at a different point than the origin?
2. What happens if the scale factor is a fraction (e.g., $\frac{1}{2}$)?
3. How does dilation relate to similarity transformations in geometry?
4. What is the effect of a negative scale factor in dilation?
5. How would you apply dilation to a shape rather than just a single point?

Tip:

When dilating points, always multiply both the x- and y-coordinates by the scale factor to find the new coordinates.