Answer
$f(x)=-\frac{1}{3}x+\frac{13}{3}$
Work Step by Step
Slope of the perpendicular line:-
$x-$intercept is $3$.
The point on line is $(3,0)$.
$y-$intercept is $-9$.
The point on line is $(0,-9)$.
Slope of the line passes through $(3,0)$ and $(0,-9)$ is
$\Rightarrow m_1=\frac{Change \; in \; y}{Change \; in \; x}$
$\Rightarrow m_1=\frac{-9-0}{0-3}$
Simplify.
$\Rightarrow m_1=\frac{-9}{-3}$
$\Rightarrow m_1=3$.
Two lines are perpendicular if their slopes are negative reciprocal to each other.
Hence, slope of the perpendicular line is
$m_2=−\frac{1}{m_1}$
$m_2=−\frac{1}{3}$
The standard equation of a line in a slope intercept form is
$\Rightarrow y=mx+c$
Where, $m=$ slope and $c=$ $y-$ intercept.
Substitute the value of slope.
$\Rightarrow y=(-\frac{1}{3})x+c$
We are given that $f$ passes through $(-5,6)$.
Plug $x=-5$ and $y=6$ into the line equation.
$\Rightarrow 6=(-\frac{1}{3})(-5)+c$
Simplify.
$\Rightarrow 6=\frac{5}{3}+c$
Subtract $\frac{5}{3}$ from both sides.
$\Rightarrow 6-\frac{5}{3}=\frac{5}{3}+c-\frac{5}{3}$
Simplify.
$\Rightarrow \frac{13}{3}=c$
Substitute the value of $c$ into the line equation.
$\Rightarrow y=(-\frac{1}{3})x+\frac{13}{3}$
Plug $y=f(x)$.
$\Rightarrow f(x)=-\frac{1}{3}x+\frac{13}{3}$
