Question

dilate line f by a scale factor of 2 with the center of dilation at the origin to create line f. where are points a and b located after dilation, and how are lines f and f related? the locations of a and b are a(0,4) and b(4,0); lines f and f are parallel. the locations of a and b are a(0,2) and b(2,0); lines f and f are the same line. the locations of a and b are a(0,2) and b(4,0); lines f and f intersect at point a. the locations of a and b are a(0,4) and b(2,0); lines f and f intersect at point b

Explanation:

Step1: Find coordinates of A and B

Assume point \(A=(0,2)\) and \(B=(2,0)\) (from the line in the graph).

Step2: Apply dilation formula

The dilation formula with center at the origin \((x,y)\to(kx,ky)\), where \(k = 2\).
For point \(A=(0,2)\), \(A'=(2\times0,2\times2)=(0,4)\).
For point \(B=(2,0)\), \(B'=(2\times2,2\times0)=(4,0)\).

Step3: Analyze the relationship between lines \(f\) and \(f'\)

Since dilation with center at the origin is a similarity transformation, and the line passes through the origin (because when \(x = 0,y=0\) satisfies the line equation \(y=-x + 2\) before dilation, and after dilation \(y=-x+4\) also passes through the origin in a sense of linear transformation through the center of dilation). But actually, for a line passing through the center of dilation, after dilation, the lines \(f\) and \(f'\) are the same line.

Answer:

The locations of \(A'\) and \(B'\) are \(A'(0,4)\) and \(B'(4,0)\); lines \(f\) and \(f'\) are the same line.