Answer
The slope-intercept form of line is\[f\left( x \right)=-\frac{1}{4}x-6\]
Work Step by Step
Consider the equation $4x-y-6=0$.
Isolate y terms on one side.
$\begin{align}
& 4x-y-6=0 \\
& -y=-4x+6
\end{align}$
Divide both sides of the equation by $-1$ to remove the negative sign.
$\begin{align}
& -\frac{y}{-1}=\frac{-4x+6}{-1} \\
& y=4x-6
\end{align}$
Hence, the slope ${{m}_{1}}$ of the line $4x-y-6=0$ is ${{m}_{1}}=4$ and the y-intercept is $-6$.
Now, let the slope of the line $f$ be ${{m}_{2}}$.
So,
$\begin{align}
& {{m}_{1}}\cdot {{m}_{2}}=-1 \\
& 4\cdot {{m}_{2}}=-1 \\
& {{m}_{2}}=-1\left( \frac{1}{4} \right) \\
& {{m}_{2}}=-\frac{1}{4}
\end{align}$
The y-intercept is equal -- that is, $\left( 0,-6 \right)$.
Now, the equation of $f$ having point $\left( 0,-6 \right)$ and slope $-\frac{1}{4}$ is:
$\begin{align}
& f\left( x \right)=mx+b \\
& =-\frac{1}{4}\left( x \right)+\left( -6 \right) \\
& =-\frac{1}{4}x-6
\end{align}$
Hence, the equation of the line in slope intercept form of the line which is perpendicular to the line $4x-y-6=0$ and has the same y intercept is $f\left( x \right)=-\frac{1}{4}x-6$.
