Question

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 1

Answer:

$y=2x+8$

Step-by-step explanation:

Given:

The original equation of the line 'l' is given as:

$y=2x+2$

Scale factor of dilation is, $k=4$

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 $(x,y)$ with a scale factor 'k' is given as:

$(x,y)\rightarrow (kx,ky)$

The standard form of a line is of the form $y=mx+b$, 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:

$Slope,m=2\\y-intercept,b=2$

The y-intercept is at the point (0, 2). Now, applying dilation rules on this point, we get:

$(0,2)\rightarrow (4\times 0,4\times 2)=(0,8)$

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:

$y=mx+b\\y=2x+8$