Given:
Perpendicular line:
[tex]$m(r)=-\frac{2}{3} r-3[/tex]
[tex]f(-2)=-9[/tex]
To find:
The equation of the linear function f.
Solution:
[tex]$m(r)=-\frac{2}{3} r-3[/tex]
Slope of line m is = [tex]-\frac{2}{3}[/tex]
If two lines are perpendicular then the slope of one equation is negative reciprocal of the other.
Slope of line f = [tex]\frac{3}{2}[/tex]
[tex]f(-2)=-9[/tex]
This means f has point (-2, -9).
Using point-slope formula:
[tex]y-y_1=m(x-x_1)[/tex]
Here [tex]x_1=-2, y_1=-9[/tex] and [tex]m=\frac{3}{2}[/tex]
[tex]$y-(-9)=\frac{3}{2} (x-(-2))[/tex]
[tex]$y+9=\frac{3}{2} (x+2)[/tex]
[tex]$y+9=\frac{3}{2} x+\frac{3}{2} \cdot 2[/tex]
[tex]$y+9=\frac{3}{2} x+3[/tex]
Subtract 9 from both sides.
[tex]$y+9-9=\frac{3}{2} x+3-9[/tex]
[tex]$y=\frac{3}{2} x-6[/tex]
Substitute x = r and y = f.
[tex]$f(r)=\frac{3}{2} r-6[/tex]
The linear equation is [tex]f(r)=\frac{3}{2} r-6[/tex].
