The graph of linear function `g` passes through the points `\left(-5,-1\right)`and `\left(5,\ 5\right),`as shown. What are the slope and y-intercept of the graph `g`?

Since the graph passes through the points $(-5,-1)$ and $(5,5),$ the slope is

\[\frac{5 - (-1)}{5 - (-5)} = \boxed{1}.\]To find the $y$-intercept, we can substitute the point $(5,5)$ into $y = 1x + b$ to get $5 = 1 \cdot 5 + b,$ so $b = \boxed{0}.$