Graphing Antiderivatives

The antiderivative of velocity gives position. Differentiating the position function yields velocity, which represents the slope of the position graph. This inverse relationship means that integrating velocity over time accumulates the changes in position, showing how a car's location evolves as its speed varies throughout its journey.

When velocity is positive and increasing, the position graph is increasing and concave up. The positive velocity means the slope is positive, so position rises. The increasing velocity indicates the slope itself is growing steeper, creating the upward curvature characteristic of concave-up behavior.

The concavity of an antiderivative graph depends on whether the original function is increasing or decreasing. When velocity increases, position is concave up; when velocity decreases, position is concave down. This relationship reflects how second derivatives and the shape of a graph determine whether the curve bends upward or downward.

As velocity approaches zero, the slope of the position graph becomes zero, and position remains nearly constant. The antiderivative's rate of change diminishes, producing a flattening curve. This represents a moment where the car is nearly stationary before potentially changing direction or accelerating again.

Analyze trends in the original function to infer the antiderivative's shape. Observe where the function is positive or negative, and how sharply it changes. Positive values create upward slopes; negative values create downward slopes. Larger magnitudes produce steeper transitions, while constant functions yield straight-line antiderivatives with uniform slopes.

A constant positive velocity produces a straight-line position graph with a steady, positive slope. Since velocity is unchanging, the slope remains uniform throughout that interval. This linear behavior reflects zero acceleration and represents motion at a constant speed without any curvature.

Negative velocity means the position is decreasing, so the slope is negative. Decreasing velocity indicates the slope is becoming less negative, or increasing in value. This combination produces a concave-down curve that slopes downward but flattens as the velocity approaches zero, representing deceleration in the negative direction.