Answer:
![y=2x+8](https://tex.z-dn.net/?f=y%3D2x%2B8)
Step-by-step explanation:
Given:
The original equation of the line 'l' is given as:
![y=2x+2](https://tex.z-dn.net/?f=y%3D2x%2B2)
Scale factor of dilation is, ![k=4](https://tex.z-dn.net/?f=k%3D4)
Now, when a line is dilated by some scale factor 'k' centered at the origin, then the parallelism of the line is reserved and only the y intercept changes.
The rule of dilation for a point
with a scale factor 'k' is given as:
![(x,y)\rightarrow (kx,ky)](https://tex.z-dn.net/?f=%28x%2Cy%29%5Crightarrow%20%28kx%2Cky%29)
The standard form of a line is of the form
, where, 'm' is the slope and 'b' is the y-intercept of the line. On comparing the line 'l' with the standard form, we get:
![Slope,m=2\\y-intercept,b=2](https://tex.z-dn.net/?f=Slope%2Cm%3D2%5C%5Cy-intercept%2Cb%3D2)
The y-intercept is at the point (0, 2). Now, applying dilation rules on this point, we get:
![(0,2)\rightarrow (4\times 0,4\times 2)=(0,8)](https://tex.z-dn.net/?f=%280%2C2%29%5Crightarrow%20%284%5Ctimes%200%2C4%5Ctimes%202%29%3D%280%2C8%29)
So, the new y-intercept after dilation is at (0, 8) or the new value of 'b' is 8.
Therefore, the dilated line equation is given as:
![y=mx+b\\y=2x+8](https://tex.z-dn.net/?f=y%3Dmx%2Bb%5C%5Cy%3D2x%2B8)