Descartes’ rule of signs predicts how many positive and negative real zeros a polynomial can have. Count the sign changes in f(x), written in descending order: the number of positive real zeros equals that count or is less by an even number. Count the sign changes in f(−x) the same way to get the possible number of negative real zeros.
The rate of change of a function tells how fast its output changes as the input changes. Over an interval [a, b], the average rate of change is [f(b) − f(a)]/(b − a). Comparing these rates on back-to-back intervals shows the shape of the graph: rates that increase make it bend up (concave up), and rates that decrease make it bend down (concave down).
The difference quotient of a function f is [f(x + h) − f(x)]/h, with h ≠ 0. It gives the slope of the secant line between the points at x and x + h, which is the average rate of change over that interval. To simplify it, find f(x + h), subtract f(x), and cancel the factor of h.
An even function satisfies f(−x) = f(x) for every x in its domain, so its graph is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x), so its graph is symmetric about the origin. To test a function, replace x with −x, simplify, and compare the result with f(x) and −f(x).