Calculus AB and Calculus BC

CHAPTER 1 Functions


C1. Polynomial Functions.

A polynomial function is of the form

f (x) = a0 xn + a1 xn − 1 + a2 xn − 2 + · · · + an − 1 x + an,

where n is a positive integer or zero, and the ak’s, the coefficients, are constants. If a0 ≠ 0, the degree of the polynomial is n.

A linear function, f (x) = mx + b, is of the first degree; its graph is a straight line with slope m, the constant rate of change of f (x) (or y) with respect to x, and b is the line’s y-intercept.

A quadratic function, f (x) = ax2 + bx + c, has degree 2; its graph is a parabola that opens up if a > 0, down if a < 0, and whose axis is the line Image

A cubic, f (x) = a0 x3 + a1 x2 + a2 x + a3, has degree 3; calculus enables us to sketch its graph easily; and so on. The domain of every polynomial is the set of all reals.

C2. Rational Functions.

A rational function is of the form


where P(x) and Q(x) are polynomials. The domain of f is the set of all reals for which Q(x) ≠ 0.