Final Answer:
The graph of the slope field for the differential equation dy/dx = x^2 resembles a family of curves with increasing slopes as x values increase. The curves curve upwards more steeply as x becomes larger in magnitude.
Step-by-step explanation:
The slope field is a graphical representation of the slope of a solution curve at various points in the x-y plane for a given differential equation. For dy/dx = x^2, each point (x, y) on the slope field represents a slope, which is equal to x^2.
The slope field for dy/dx = x^2 depicts lines or line segments with slopes determined by the value of x at each point. Since x^2 increases as x becomes larger (either positively or negatively), the slopes of the curves in the slope field increase in magnitude as x moves away from the origin. This causes the curves to bend more steeply upwards as x increases or decreases.
The solutions to the differential equation dy/dx = x^2 can be represented by curves that follow the direction of the lines in the slope field. These curves are parabolas because the antiderivative of x^2 is (1/3)x^3 plus an arbitrary constant. These parabolic solutions match the overall behavior depicted in the slope field, showing the increasing nature of the curves as x values increase, forming a family of curves spreading away from the origin.