Answer
$(a)$ $\frac{f(x)-f(a)}{x-a}$
$(b)$
In general a linear function has a form $f(x)=mx+b$
In our case $m=c$ and $b=f(a)-ca)$
The slope is $c$
$y$-intercept is $f(a)-ca$
Work Step by Step
$(a)$ In general we have average rate of change formula: $c=\frac{f(x_2)-f(x_1)}{x_2-x_1}$
Then we simply input the $x$ and $a$ values:
$\frac{f(x)-f(a)}{x-a}$
$(b)$ We can simplify the equation:
$c=\frac{f(x)-f(a)}{x-a}$
$c(x-a)=f(x)-f(a)$
$f(x)=cx-ca+f(a)$
$f(x)=cx+(f(a)-ca)$
In general a linear function has a form $f(x)=mx+b$
In our case $m=c$ and $b=f(a)-ca)$
The slope is $c$
We have $y$-intercept when $x=0$
$f(0)=c\times 0(f(a)-ca)=f(a)-ca$