Answer
$ f(x)=-\frac{1}{2}x+1$.
Work Step by Step
Because we are given a point on the line, we will first determine the point-slope form of the line's equation.
$y-y_1=m(x-x_1)$.... (1)
where $(x_1,y_1)=(-6,4)$.
We are given that our line is perpendicular on a line passing through the points $(2,0)$ and $(0,-4)$ because $2$ is its $x$-intercept and $-4$ is its $y$-intercept. Let $m_1$ be the slope of this line.
Two lines are perpendicular if their slopes are negative reciprocal to each other, therefore $m$ and $m_1$ check the equation:
$m\cdot m_1=-1$......... (2).
Determine the slope $m_1$.
$\Rightarrow m_1=\frac{Change \; in \; y}{Change \; in \; x}$
$\Rightarrow m_1=\frac{-4-0}{0-2}$
Simplify.
$\Rightarrow m_1=\frac{-4}{-2}$
$\Rightarrow m_1=2$.
Plug in $m_1=2$ in equation (2) to find $m$:
$\Rightarrow m\cdot 2=−1$
Divide both sides by $2$:
$\Rightarrow m=−\frac{1}{2}$
Plug in $(x_1,y_1)=(-6,4)$ and $m=-\frac{1}{2}$ in equation (1):
$\Rightarrow y-4=-\frac{1}{2}(x-(-6))$........(3).
The standard equation of a line in a slope- intercept form is
$\Rightarrow y=mx+c$.
Bring equation (3) to the slope-intercept form. Use the distribution property and add $4$ to each side:
$y-4+4=-\frac{1}{2}x-3+4$
Simplify.
$\Rightarrow y=-\frac{1}{2}x+1$
$\Rightarrow f(x)=-\frac{1}{2}x+1$.
