Polynomials and Rational Functions

WARM UP EXERCSE Roots, zeros, and x-intercepts. f (x) = x 2 ! 25 f (x) = x 2 + 25 f (x) = x 3 ! 25x f (x) = polynomial, f (a) = 0 ! f (x) = (x - a)g(x) 1 § 2-3 Polynomials and Rational Functions Students will learn about: •Polynomial functions –Behavior & graphs –Root approximation •Rational functions: –Behavior & graphs 2 1 Examples f (x) = (x ! 3)3 + 2 = f (x) = x 3 ! 4x Behavior as x gets big? How many intercepts? How many turning points? f ( x) = x 4 ! 6 x 2 f ( x) = x5 ! 5 x3 + 4 x + 1 f (x) = an x n + an!1 x n!1 + ... + a2 x 2 + a1 x1 + a0 3 Graphs of examples f(x)=3x^2+6x-1 y=x^4-6x^2 50 30 40 25 20 30 15 20 10 10 5 0 0 -6 -4 -2 0 2 4 -10 -4 -3 -2 -1 0 1 2 3 4 -5 -10 -15 Number of intercepts? Number of turning points? Behavior as x gets big? Behavior as x goes to negative infinity? How do your answers change if we shift these left or right? How do your answers change if we shift these up or down? 4 2 Graphs of examples y=x^3-27x 60 y=x^5-5x^3+4x+1 40 5 4 20 3 0 -20 2 0 10 20 30 40 50 60 1 -3 -40 -2 -1 0 -1 0 1 2 3 -2 -60 -3 Number of intercepts? Number of turning points? Behavior as x gets big? Behavior as x goes to negative infinity? How do your answers change if we shift these left or right? How do your answers change if we shift these up or down? 5 Examples x goes to infinity? f (x) = x 3 ! 2 h(x) = (x)2 (x ! 1) g(x) = (x ! 1)(x ! 2)(x ! 3) j(x) = (x ! 1)(x 2 + 1) x gets to negative infinity? f (x) = x 3 ! 2 h(x) = (x)2 (x ! 1) g(x) = (x ! 1)(x ! 2)(x ! 3) j(x) = (x ! 1)(x 2 + 1) How many intercepts? f (x) = x 3 ! 2 h(x) = (x)2 (x ! 1) g(x) = (x ! 1)(x ! 2)(x ! 3) j(x) = (x ! 1)(x 2 + 1) How many turning points? f (x) = x 3 ! 2 h(x) = (x)2 (x ! 1) g(x) = (x ! 1)(x ! 2)(x ! 3) j(x) = (x ! 1)(x 2 + 1) 6 3 In General f (x) = an x n + an!1 x n!1 + ... + a2 x 2 + a1 x1 + a0 Degree n EVEN: Behavior as x goes to infinity? Behavior as x gets to negative infinity? How many intercepts? Between 0 and n How many turning points? Between 1 and n-1 Degree n Odd: Behavior as x goes to infinity? Behavior as x gets to negative infinity? How many intercepts? How many turning points? 7 Rational Function Examples Graph the following: f (x) = 1 x Domain Range lim f (x) = x!" lim f (x) = x!"# lim f (x) = x + !2 lim f (x) = x " !2 8 4 Rational Function Examples Graph the following: g(x) = 1 +3= x Domain Range lim f (x) = x!" lim f (x) = x!"# Remark: Limits to infinity in general: lim x!" ax + b ax b = lim + = x!" cx + b cx + b cx + b How about as x approaches 0? g(1 / 10) = g(1 / 100) = g(!1 / 10) = g(!1 / 100) = lim f (x) = as x approaches 0 from the right f (x) approaches _______ x + !0 lim f (x) = as x approaches 0 from the left f (x) approaches _______ " 9 x !0 Rational Function Examples Graph the following: f (x) = 1 x!2 Domain Range lim f (x) = x!" lim f (x) = x!"# How about as x approaches 2? f (1.9) = f (1.99) f (2.1) = f (2.01) lim f (x) = x + !2 lim f (x) = x " !2 10 5 Rational Function Examples Graph the following: g(x) = 1 +3= 2(x ! 1) Domain Range lim f (x) = x!" lim f (x) = x!"# lim f (x) = + x ! lim f (x) = " 11 x ! Rational Functions • Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x). Example: Let P(x) = x + 5 and Q(x) = x – 2 then R(x)= x+5 x!2 Domain: Range: Zeros: x-intercepts: y-intercepts: 12 6 Graph of rational function 13 Rational Functions • Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x). We will focus on R(x)= ax + b cx + d Domain: Don’t want cx+d = 0. So… All real numbers except x=-d/c. The line x=-d/c is the vertical aspmptote Range: lim f (x) = a / c x!"# y=a/c is the horizontal asymptote lim f (x) = a / c Range: All real numbers except y=a/c. x!" Zeros: Want ax+b=0 so zero at x=-b/a (if a not zero) x-intercepts: (-b/a,0) y-intercepts: (0,b/d) lim f (x) = + x ! lim f (x) = " 14 x ! 7 3x + 5 f (x) = x +1 Example: Domain: Range: lim f (x) = x!"# lim f (x) = x!" Zeros: x-intercepts: y-intercepts: lim f (x) = + x ! lim f (x) = " 15 x ! 8