Question

line l has the equation y = 1/2x - 1. write the equation of the image of l after a dilation with a scale factor of 4, centered at the origin. write your answer in slope - intercept form.

Explanation:

Step1: Recall dilation rule for lines

For a dilation with scale - factor \(k\) centered at the origin, if the original line is \(y = mx + b\), the new line is \(y=kmx + kb\). Here, \(k = 4\), \(m=\frac{1}{2}\), and \(b=-1\).

Step2: Calculate the new slope and y - intercept

The new slope \(m_{new}=k\times m\). Substituting \(k = 4\) and \(m=\frac{1}{2}\), we get \(m_{new}=4\times\frac{1}{2}=2\).
The new y - intercept \(b_{new}=k\times b\). Substituting \(k = 4\) and \(b = - 1\), we get \(b_{new}=4\times(-1)=-4\).

Step3: Write the new equation

The slope - intercept form of a line is \(y=mx + b\). So the equation of the dilated line is \(y = 2x-4\).

Answer:

\(y = 2x - 4\)