Question

for the figure below, do a dilation centered at the origin with a scale factor of \\( \frac { 1 } { 4 } \\).
then, give the endpoints for both the original figure and the final figure.
endpoints of original figure:
left: right:
endpoints of final figure:
left: right:

Explanation:

Step1: Find original endpoints

From the graph, the left - hand endpoint (the upper - left point) of the original figure has coordinates \((1,16)\) and the right - hand endpoint (the lower - right point) has coordinates \((16,1)\).

Step2: Apply the dilation formula

The formula for dilation centered at the origin \((x,y)\to(kx,ky)\), where \(k = \frac{1}{4}\).
For the point \((1,16)\):
\(x_1=\frac{1}{4}\times1=\frac{1}{4}\), \(y_1=\frac{1}{4}\times16 = 4\)
For the point \((16,1)\):
\(x_2=\frac{1}{4}\times16 = 4\), \(y_2=\frac{1}{4}\times1=\frac{1}{4}\)

Answer:

Endpoints of original figure: Left: \((1,16)\), Right: \((16,1)\)
Endpoints of final figure: Left: \((\frac{1}{4},4)\), Right: \((4,\frac{1}{4})\)