Answer
$y=-2x$
Work Step by Step
The slope of the line can be found as two perpendicular lines' slopes' product equals to $-1$.
Therefore the product of the slope of the line $y=\frac{1}{2}x+4$ and the one we are trying to find equals to $-1$.
As $m_1=\frac{1}{2}=0.5$, then
$0.5\times m_2=-1$
$m_2=-2$, this is the slope of the line we are trying to find.
If we have the slope and a point on the graph $(1,-2)$ we can use the point-slope formula to calculate the equation of the line:
$y-y_1=m(x-x_1)$
which translates to $y-(-2)=-2(x-1)$
The equation of the line is therefore: $y+2=-2x+2$
or $y=-2x$
