Section 1-2 © Sandra Nite Math 131 Lecture Notes Section 1.2 – Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description of a real-world situation. Often the model is a function rule or equation of some type. Modeling Process Real-world Formulate Test/Check problem Real-world Mathematical predictions model Interpret Mathematical Solve conclusions Linear Models Characteristics: • The graph is a line. • In the form y = f(x) = mx + b, m is the slope of the line, and b is the y-intercept. • The rate of change (slope) is constant. • When the independent variable ( x) in a table of values is sequential (same differences), the dependent variable has successive differences that are the same. • The linear parent function is f(x) = x, with D = ℜ = (-∞, ∞) and R = ℜ = (-∞, ∞). • The direct variation function is a linear function with b = 0 (goes through the origin). • In a direct variation function, it can be said that f(x) varies directly with x, or f(x) is directly proportional to x. Example: See pp. 26-28 of the text. 1 Section 1-2 © Sandra Nite Polynomial Functions = n + n−1 +⋅⋅⋅+ 2 + + A function P is called a polynomial if P(x) an x an−1 x a2 x a1 x a0 where n is a nonnegative integer and a0 , a1 , a2 ,..., an are constants called the coefficients of the polynomial. If the leading coefficient an ≠ 0, then the degree of the polynomial is n. Characteristics: • The domain is the set of all real numbers D = (-∞, ∞). • If the degree is odd, the range R = (-∞, ∞). • The graph crosses the x-axis at most n times and has at most n – 1 vertices. • The quadratic parent function is f(x) = x2, with D = (-∞, ∞) and R = [0, ∞). • The general equation for a quadratic function is f(x) = ax 2 + bx + c. • The cubic parent function is f(x) = x3, with D = (-∞, ∞) and R = (-∞, ∞). • The end behavior for polynomials functions is shown below: 2 Section 1-2 © Sandra Nite Power Functions A function of the form f(x) = ax b, where a and b are nonzero constants, is called a power function . Characteristics: • The domain D = (-∞, ∞). • f(x) varies directly with the bth power of x, or f(x) is directly proportional to the bth power of x • Polynomial functions that are monomials (one term) are power functions. • When b is an even positive integer, f is an even function. • When b is an odd positive integer, f is an odd function. • When b is a fraction, f is a root function or radical function . • When a = 1, and b = -1, f is the reciprocal function . • When b = -1, f is a rational function or inverse variation function . Rational Functions P(x) A rational function is a ratio of two polynomials: f (x) = where P and Q are Q(x) polynomials. The domain is the set of all x-values such that Q(x) ≠ 0. Characteristics: 1 • The parent rational function is f (x) = , with D = (-∞, 0) ∪ (0, ∞) and x R = (-∞, 0) ∪ (0, ∞) and asymptotes are both axes. • For most real-world applications, the domain is (0, ∞). • The function has vertical asymptotes where Q(x) = 0 and P(x) ≠ 0. • The function has a removable discontinuity (hole) where Q(x) = 0 and P(x) = 0. Algebraic Functions A function f that is constructed using algebraic operations, such as addition subtraction, multiplication, division, and taking roots, of polynomials is an algebraic function . Characteristics: • Algebraic functions have a variety of shapes (see p. 33). • Algebraic functions have a variety of domains and ranges. Example: f (x) = x 2 +1 3 Section 1-2 © Sandra Nite Trigonometric Functions Trigonometric functions can be defined as functions of an angle in terms of the ratio of two of the sides of a right triangle containing the angle. Characteristics: • Trig functions are periodic and sometimes called circular functions. • When no units are given for the angle measure, it is understood to be radian measure. • The sine and cosine functions have domain D = (-∞, ∞) and range R = [-1, 1] and a period of 2 π. opp adj opp • Using ratios of the sides of a right triangle, sin = , cos = , and tan = . hyp hyp adj π • The tangent function has domain D = { x| x ∈ ℜ , x ≠ + kπ , k ∈ I} 2 1 1 1 • In terms of sin, cos, and tan, cot = , sec = , and csc = . tan cos sin Exponential Functions A function of the form f(x) = ab x, where a ≠ 0 and b > 0, is called an exponential function . Characteristics: • The parent exponential function with base e is f(x) = ex with D = (-∞, ∞) and R = (0, ∞). • f(x) varies exponentially with x, or f(x) is an exponential function of x. • The graph crosses the y-axis at f(0) = a and has the x-axis as an asymptote. Logarithmic Functions A function of the form f(x) = log abx, where a ≠ 0 and b > 0, is called a logarithmic function . Characteristics: • The parent logarithmic function with base e is f(x) = log ex or f(x) = ln x with D = (0, ∞) and R = (-∞, ∞). • The graph crosses the x-axis at f(a) = 0 and has the y-axis as an asymptote. • Logarithmic functions are inverses of exponential functions. 4 .