Answer:
[tex]y = \frac{1}{3} x + \frac{8}{3}[/tex]
Step-by-step explanation:
The equation of the line I is [tex]y = \frac{1}{3} x[/tex]
Scale factor = [tex]\frac{1}{2}[/tex], center = (0, 0)
To find the dilated line, we need follow the steps below:
Step 1: Draw the equation of the given line I
Step 2: Let's take a point from the line.
Let's take (2, 6) which is on the line I.
Let's find the dilated point by the scale factor [tex]\frac{1}{2}[/tex]
Multiply the point (2, 6) by [tex]\frac{1}{2}[/tex], we get
([tex]2.\frac{1}{2} , 6.\frac{1}{2} )[/tex] = (1, 3)
Step 3: Write the equation of the new line (Image I)
The dilated line has the same slope.
So slope (m) = [tex]\frac{1}{3}[/tex]
x = 1 and y = 3
Now let's find the slope intercept.
y = mx + b
Plug in x = 1, y = 3 and slope (m) = [tex]\frac{1}{3}[/tex]
3 = [tex]\frac{1}{3}[/tex].1 + b
3 = [tex]\frac{1}{3}[/tex] + b
b = 3 - [tex]\frac{1}{3}[/tex]
b = [tex]\frac{3.3 -1}{3} = \frac{9 -1}{3} = \frac{8}{3}[/tex]
Now let's find the equation of the image I.
y = mx + b
[tex]y = \frac{1}{3} x + \frac{8}{3}[/tex]
The required equation is [tex]y = \frac{1}{3} x + \frac{8}{3}[/tex] after the dilation of scale factor.
