To determine the image of the point $(5, -4)$ after a dilation by a scale factor of $44$ centered at the origin, we apply the following rule:

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

where:

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

Given:

• $(x, y) = (5, -4)$,
• $k = 44$.

We calculate:

1. $x' = 44 \cdot 5 = 220$,
2. $y' = 44 \cdot (-4) = -176$.

Thus, the image of $(5, -4)$ is:

$(220, -176)$

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Would you like a detailed explanation of the dilation process or an example with a different scale factor?

Here are 5 related questions to explore further:

1. What happens to a point when dilated with a scale factor of $k = 1$?
2. How does the origin affect dilation transformations?
3. What are the coordinates of $(5, -4)$ after a dilation with a scale factor of $\frac{1}{2}$?
4. How do you determine the scale factor given the original and transformed points?
5. How does dilation differ from other transformations like reflection or translation?

Tip: When scaling points, remember that a scale factor $k > 1$ enlarges the figure, while $0 < k < 1$ reduces it.