Line has the equation y= – 1/ 3 x–3. Write the equation of the image of after a dilation with a scale factor of 3, centered at the origin. Write your answer in slope-intercept form.

The the equation of the image of after a dilation with a scale factor of 3, centered at the origin will be [tex]y=\:-\:\frac{1}{3}x-9[/tex], which is shown as blue colored line in attached graph.

Step-by-step explanation:

Considering the equation of the line

[tex]y=\:-\:\frac{1}{3}x-3[/tex]

Comparing with the slope-intercept form [tex]y = mx + b[/tex], we can determine that the slope of line is [tex]m = \frac{-1}{3}[/tex], as shown in the attached figure. The red colored line indicated the this (original) line.

We have to write the equation of the image after a dilation with a scale factor of 3, centered at the origin.

Writing Steps:

First draw the line of the equation as shown in attached figure.

Then take a point from the line. Let suppose the point is (0, -3).

Then dilate the point (0, -3) with a scale factor of 3 by multiplying the coordinates of the point (0, -3) with 3 to determine the location of the dilated point. So,

(x, y) ⇒ (3x, 3y)

(0, -3) ⇒ (3×0, 3×-3) = (0, -9)

Now, lets write the equation of new line

As the dilated line has same slope. So,

[tex]m = \frac{-1}{3}[/tex]

As the dilated point is (0, -9), in which

x = 0, and y = -9

As the slope-intercept form is

[tex]y = mx + b[/tex] where [tex]m = \frac{-1}{3}[/tex]

So, putting the values in slope intercept form

[tex]y = mx + b[/tex]

[tex]-9=\:-\:\frac{1}{3}\left(0\right)+b[/tex]

[tex]\mathrm{Switch\:sides}[/tex]

[tex]-\frac{1}{3}\left(0\right)+b=-9[/tex]

[tex]\mathrm{Remove\:parentheses}:\quad \left(a\right)=a[/tex]

[tex]-\frac{1}{3}\cdot \:0+b=-9[/tex]

[tex]\mathrm{Apply\:rule}\:0\cdot \:a=0[/tex]

[tex]-0+b=-9[/tex]

[tex]b=-9[/tex]

So, the equation of image becomes

[tex]y=\:-\:\frac{1}{3}x+\left(-9\right)[/tex]

[tex]y=\:-\:\frac{1}{3}x-9[/tex]

So, the the equation of the image of after a dilation with a scale factor of 3, centered at the origin will be [tex]y=\:-\:\frac{1}{3}x-9[/tex], which is shown as blue colored line in attached graph.

Keywords: equation, dilation, scale factor of 3

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