Question

1. line $y - 2x = 1$ is transformed by a dilation with a scale factor of 4 and centered at the origin.

which statement about the lines image is true?
a it has slope 2 and y - intercept $(0,4)$
b. it has slope $-2$ and y - intercept $(0,4)$
c. it has slope $1/2$ and y - intercept $(4,0)$
d. it has slope $-1/2$ and y - intercept $(4,0)$

Explanation:

Step1: Rewrite the original line equation

Rewrite \(y - 2x=1\) in slope - intercept form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the \(y\) - intercept).
We get \(y=2x + 1\), so the slope \(m = 2\) and the \(y\) - intercept is \((0,1)\).

Step2: Apply the dilation rule

A dilation centered at the origin with a scale factor \(k\) transforms a point \((x,y)\) to \((kx,ky)\).
For a line \(y=mx + b\), after dilation with scale factor \(k = 4\), the equation of the image line is \(y=mx + kb\).
Substitute \(m = 2\) and \(k = 4,b = 1\) into the equation. The equation of the image line is \(y=2x+4\).

Answer:

A. it has slope 2 and y - intercept \((0,4)\)