Chapter 5 Polynomial and Rational Functions 5.1 Quadratic Functions and Models 5.2 Polynomial Functions and Models 5.3 Rational Functions and Models A linear or exponential or logistic model either increases or decreases but not both. Life, on the other hand gives us many instances in which something at first increases then decreases or vice-versa. For situations like these, we might turn to polynomial models.

Higher Degree Polynomials f(x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0 To find y intercept, determine f(0) = c. To find x intercepts, solve f(x) = 0 by factoring or SOLVE command. Leading term determines global behavior (as power function). [possibly] more turning points Identify turning points approximately [point and click] by graph. FACTORED FORM: f(x) = a(x-x 1 )(x-x 2 )…(x-x k ) for x 1, x 2 … x k zeroes of f. Graph is always a smooth curve

The speed of a car (in mph) after t seconds is given by: f(t) =.005t 3 – 0.21t t + 49 (3.46, 51.27) (24.44, 28.35) According to Maple t-intercept is rate of change at t = 15 is rate of change at t = 26 is 0.52 These calculations agree with the graph, since slope of curve is negative at t = 15 and positive at t = 26.

The cost of a day’s production of x pots is given by: C(x) =.01x 3 – 0.65x x + 20 fixed costs T/F cost increases with # of pots (turning points) marginal cost (cost of producing one more) e.g. C(8)-C(7)

Rational Functions and Models A rational function is a quotient or ratio of two polynomials. ??? A Closer Look!

x f(x) Undefined As x → -3 -, y → ∞ As x → -3 +, y → - ∞ We say the graph of f has a vertical asymptote at x = -3.

vertical asymptote at x = -3. Vertical Asymptote at x = k k is not in the domain of f the values of f increase (or decrease) without bound as x approaches k near x = k, the graph of f resembles a vertical line

x f(x) As x → - ∞, y → 2 + As x → ∞, y → 2 - We say the graph of f has a horizontal asymptote at y = 2.

horizontal asymptote at y = 2. The quotient of leading terms determines the global behavior of a rational function. In fact, for x large in magnitude

horizontal asymptote at y = 2 vertical asymptote at x = -3 y intercept: f(0) = -1/3 x intercept:2x – 1 = 0 x = 1/2 two branches

horizontal asymptote: f(x) ≈ -2/x 2 ≈ 0 for x large in magnitude HA at y = 0 vertical asymptotes: x 2 – 1 = (x-1)(x+1) = 0 VA POSSIBLE at x = 1, -1 y intercept: f(0) = 2 x intercept:-2 = 0 no solution three branches

horizontal asymptote: f(x) ≈ 3x 2 /x 2 = 3 for x large in magnitude HA at y = 3 vertical asymptotes: x 2 – 4 = (x-2)(x+2) = 0 VA POSSIBLE at x = 2, -2 y intercept: f(0) = 0 x intercept:3x 2 = 0 x = 0 three branches

HW Page 255 #33-50 TURN IN: #33,36, 40( with Maple graph), 41(with Maple graph)