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) = ax^2 + 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
A cubic, f (x) = a 0 x^3 + a 1 x^2 + a 2 x + a 3 , 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.
D. TRIGONOMETRIC FUNCTIONS
The fundamental trigonometric identities, graphs, and reduction formulas are given in the Appendix.
D1. Periodicity and Amplitude.
The trigonometric functions are periodic. A function f is periodic if there is a positive number p such
that f (x + p) = f (x) for each x in the domain of f. The smallest such p is called the period of f. The
graph of f repeats every p units along the x-axis. The functions sin x, cos x, csc x, and sec x have
period 2π; tan x and cot x have period π.
The function f (x) = A sin bx has amplitude A and period g(x) = tan cx has period
EXAMPLE 10
Consider the function f (x) = cos (kx).
(a) For what value of k does f have period 2?
(b) What is the amplitude of f for this k?
SOLUTIONS:
(a) Function f has period since this must equal 2, we solve the equation getting k = π.
(b) It follows that the amplitude of f that equals has a value of
EXAMPLE 11
Consider the function
Find (a) the period and (b) the maximum value of f.
(c) What is the smallest positive x for which f is a maximum?
(d) Sketch the graph.