We want to find the image of the following line after a dilation centered at the origin with a scale factor of 1.5.
y=4x-2
See that a dilation of a line is also line. We can write any line in the slope-intercept form. In the equation m is the slope of the line and b is the y-intercept.
y= mx+ b
We should find both of these values to write the equation of our line. First, see that dilation centered at a point outside the line does not change its slope. This means that the two lines have the same slope, 4.
y= 4x+b
Now, we are going to find the y-intercept of the original line. This will help us find the y-intercept of the image. To do so, we substitute 0 for x in the equation and solve for y.
We found that (0, - 2) is the point of intersection between the line and the y-axis. Now, let's remember the function rule for a dilation centered at the origin with a scale factor of 1.5.
We can use it to find the y-intercept of the image.
ccc
(x,y)& → & (1.5x, 1.5 y)
(0,- 2) & → & (0, - 3)
We found that the y-intercept of the dilated image is - 3. Let's substitute this value into the function rule and simplify.
y=4x+( - 3) ⇔ y=4x-3
The equation of the dilated image is y=4x-3.
