University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2023
Views:

410

Pages:

2
Author:

Niklas

IV. Polynomials GRAPHING (Ch2-2,2-3, and 2-4) I. GRAPHING AND ANALYZING HIGHER DEGREE POLYNOMIAL FUNCTIONS a. shared qualities among all polynomial functions with degree n: domain: (-∞,∞) continuous on (-∞,∞) differentiable on (-∞,∞) ↳ n-roots n-1 critical points b. multiplicity: number of times a root repeats: ex. double root, triple root, etc. odd multiplicity: graph will pass through the repeated root. even multiplicity: graph will rebound at the repeated root. c. end behavior by degree and leading coefficient: for all polynomial functions of the form anxn+an−1xn−1+...+ax+a0 the end behavior is odd n even n as x→∞, as x→∞, y→∞ y→∞ y→∞ as x→−∞, as x→−∞, y′→−∞ y→∞ as x→∞ y→−∞ x→∞ y′→−∞ as x→−∞, as x→−∞, y′→∞ y′→−∞ x3−2x+7 2. −x6−4x4+5x2+6 3 x2+7x+2 as x→∞, as x→∞, y→−∞ as x→∞, y→∞ x→−∞, x→−∞, y→∞ y→∞ as x→0, y→∞ x→−∞, y→∞ a. Graph polynomial functions: Let P(x) = x^4 - 2x^3 - 4x^2 + 2x + 3 y-int: P(0) = 3 x-int: Let P(x) = 0 + - + - + 2 2 0 2 2 0 2 2 1 3 -3 1 -1 5 -3 0 2 0 2 2 1 0 0 4 3 6 3 1 2 1 0 (x - 1)(x - 3)(x + 1) = 0 (x - 1)(x - 3)(x + 1) = 0 end behavior: as x -> ∞, y -> ∞ as x -> -∞, y -> ∞ number of cp: 3 domain: (-∞, ∞) range: [-9, ∞) *P(2) = -9, so this is a rough estimate of the absolute minimum (in reality, as seen through calculus, the absolute min is closer to -9.915)