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Slope fields allow us to visualize the solution curves $y = f(x)$ of a first-order ordinary differential equation $y' = F(x,y)$. Specifically, at each point $(x,y)$ on the plane, we can draw or visualize a short segment with slope $dy/dx = F(x,y)$. Why is this useful? Well, if $y = f(x)$ is a solution to $y' = F(x,y)$, then the segments drawn are all tangent to the graph of the curve $y = f(x)$. By the way, it turns out that no two solution curves intersect: do you see why?