Answer
The linear function in slope-intercept form is $ y=\frac{x}{4}+\frac{3}{2}$.
Work Step by Step
Let us consider the coordinates $\left( -2,1 \right)$ and $\left( 6,3 \right)$ as $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ respectively.
And find the equation of the function by taking $ f\left( x \right)$ as $ y $:
$\begin{align}
& \frac{y-{{y}_{1}}}{x-{{x}_{1}}}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
& \frac{y-1}{x-\left( -2 \right)}=\frac{3-1}{6-\left( -2 \right)} \\
& \frac{y-1}{x+2}=\frac{2}{8} \\
& \frac{y-1}{x+2}=\frac{1}{4}
\end{align}$
Then, simplify it further, to get
$\begin{align}
& 4\left( y-1 \right)=x+2 \\
& 4y-4=x+2 \\
& 4y=x+2+4 \\
& y=\frac{x+6}{4}
\end{align}$
And breaking the terms into individual terms,
$\begin{align}
& y=\frac{x}{4}+\frac{6}{4} \\
& y=\frac{x}{4}+\frac{3}{2}
\end{align}$
Thus, the slope-intercept form is $ y=\frac{x}{4}+\frac{3}{2}$.
