To determine the image of the point [tex]\((5, -4)\)[/tex] after a dilation by a scale factor of 4 centered at the origin, you need to follow these steps:
1. Identify the coordinates of the original point:
- The original point given is [tex]\((5, -4)\)[/tex].
2. Determine the scale factor for the dilation:
- The scale factor provided is 4.
3. Apply the dilation transformation:
- When a point [tex]\((x, y)\)[/tex] is dilated by a scale factor [tex]\(k\)[/tex] from the origin, the new point [tex]\((x', y')\)[/tex] is given by:
[tex]\[
x' = k \cdot x
\][/tex]
[tex]\[
y' = k \cdot y
\][/tex]
4. Substitute the coordinates of the original point and the scale factor into the transformation equations:
- For the [tex]\(x\)[/tex]-coordinate:
[tex]\[
x' = 4 \cdot 5 = 20
\][/tex]
- For the [tex]\(y\)[/tex]-coordinate:
[tex]\[
y' = 4 \cdot (-4) = -16
\][/tex]
5. State the coordinates of the new point:
- The new coordinates after the dilation are [tex]\((20, -16)\)[/tex].
Therefore, the image of the point [tex]\((5, -4)\)[/tex] after a dilation by a scale factor of 4 centered at the origin is [tex]\((20, -16)\)[/tex].
