Answer
a. Rate of change = $-\frac{1}{2}$
b. $f(x) = -\frac{1}{2} + 2$
Work Step by Step
a. The rate of change is how much $y$-increases for every increase in the $x$ value. Use the slope formula:
$\text{Slope}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{2}}$.
Pick two points from the graph and plug them into the formula, for example $(4,0)$ and $(0,2)$ to get the slope, which is also the rate of change:
$\text{Slope}=\dfrac{2-0}{0-4}=-\dfrac{2}{4}=-\dfrac{1}{2}$.
b. The rate of change is $a$ and the $y$-intercept is $b$. The $y$-intercept occurs when $x = 0$, which means that, based off the graph given, $b= 2$.
So the overall equation for the function is
$f(x) = -\dfrac{1}{2}x + 2$