In so far as the statements of geometry speak about reality, they are not certain,
 and in so far as they are certain, they do not speak about reality.

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 in the relationship between
the two variables. The relationship between the two variables does not change over a range of conditions.

Two Types of Curves

In economics we use two types of curves to represent the changing relationships or changing tradeoffs: concave curves and convex curves. A graph is concave if it is bowed upward with the opening facing the origin or the horizontal axis. It is convex if it is bowed downward with the opening facing up or away from the origin. In economic modeling, especially at the introductory level, we assume that all curves are "well-defined", and "well-behaved", which essentially means that they are all smoothly-curved, no corners, no flat spots, no dips, no pointed peaks. Thus, if you have any number of separate points they should all be on a smooth curve.

(NOTE: The terms concave and convex are not used uniformly across the disciplines of physics, mathematics, and economics. The reason for this is that historically the terms were used first to describe the shape of lenses, which actually have 2 surfaces. Lenses have both a convex surface and a concave surface. Thus, in mathematics it is common to refer to a curved surface as either convex up or down, or even concave up or down. However, the good news is that in economics we are internally consistent.)

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.