The fundamental difference between using linear graphs as models and using curves as models is that curves provide more information because their slope changes at every point.

Lines have unique slopes that provide two pieces of information. First, they show the trade-off, or rate of (ex)change, between the independent and dependent variables. Second, they illustrate whether the relationship or trade-off is negative (" inverse") or positive ("direct").

Since the slope of a line is constant, we know there is no change the relationship between the two variables does not change over a range of conditions.

Concave and convex curves also illustrate the trade-offs and the direction (or sign) in the relationship between the variables. However, since the slope changes at every point on concave and convex curves, the relationship or interaction between the variables does change continuously. The slopes of curves illustrate the "acceleration or deceleration" of the interaction between the variables. That is, the changing slope along a curve tells us whether the existing conditions are

i. increasing at an increasing rate

ii. increasing at a decreasing rate

iii. decreasing at an rate decreasing

iv. decreasing at a increasing rate

The slope of the concave curve above along Tangent Line A is positive, with the vertical variable changing far more rapidly than the horizontal variable. The value of the slope approaches Positive Infinity, +ƒ.

As we move from left to right, along the curve, the slope remains Positive, but becomes continuously smaller, appoaching 0, zero as the Tangent Lines B, C, and D become flatter. Along this arc of the curve the horizontal variable begins to accelerate and change faster than the vertical variable which decelerates in contrast.

Along Tangent Line E the slope of the curve is 0, zero, at the peak, or maximum point. Here the vertical variable does not change at all, as the horizontal variable changes.

Continuing to move along the curve to the right the Tangent Lines F, G, H, and I become increasingly negative, with the horizontal variable first rising rapidly as the vertical variable falls slowly. Then as the vertical variable begins to fall more rapidly, the horizontal variable increases more slowly. At Tangent Line I the value of the slope is approaching Negative, -ƒ.