Line has the equation y=2x+2. Write the equation of the image of after a dilation with a scale factor of 4, centered at the origin.Write your answer in slope-intercept form.

Answer:[tex]y=2x+8[/tex]Step-by-step explanation:Given:The original equation of the line 'l' is given as:[tex]y=2x+2[/tex]Scale factor of dilation is, [tex]k=4[/tex]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 [tex](x,y)[/tex] with a scale factor 'k' is given as:[tex](x,y)\rightarrow (kx,ky)[/tex]The standard form of a line is of the form [tex]y=mx+b[/tex], 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:[tex]Slope,m=2\\y-intercept,b=2[/tex]The y-intercept is at the point (0, 2). Now, applying dilation rules on this point, we get:[tex](0,2)\rightarrow (4\times 0,4\times 2)=(0,8)[/tex]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:[tex]y=mx+b\\y=2x+8[/tex]