Curve Sketching - Polynomials We're going to do the Curve Sketching Algorithm. f(x) = x5- 5x3+ 4x f' (x ) = 5x - 15x7 + 4 f" (x) = 20x - 30x A: Domain: B: Intercepts: y-int (0,0) x-int (op) (-1, 0), (1,0), (-2,0), (2,0) let u = x 2 Let x= 0 : let y= 0 : f ( 2 ) y= 05 - 510)3+ 410) =0 0 = 265 5x3+4x = 0= 2 (29-5x2+4) -70 = 2 ( 2+1) (2-1) ( 2+2) ( 20-2) C: Asymptotes: Vertical Asymptote: N / A (if there are any vertical asymptotes, you need Horizontal Asymptote: N /A to check the behaviour on both sides of each) Slant Asymptote: N / A - No asymptotes ! Polynomials are continuous for all RE IR D: Intervals of Increase/Decrease: and y EIR Critical numbers are values that f'(x) = 0 or f(x) Does Not Exist (DNE) Critical Numbers: sub u = x2 542 - 154 + 9 =0 ('(x) = 525 -15x2 +4 = 0 then use the quadratic formula 15# 1 145 (1 20 ) 10 15+ 1 145 = + 15 + 1 145 15 - JOYS IS - JIGS 10 lo X = 1. 64, - 1.64, 0.54, - 0.54 E: Local Extrema: Classify any local extrema and find their y-values here ( 1 1.64 ) = -3.63 - local min + 10.54 ) = 1.42 local max + 1-0.54 ) = - 1.42 - local min + 1-1.64) = 363 - local max Page 1 of 3F: Intervals of Concavity: Critical numbers are values that f"(x) = 0 or f(x) Does Not Exist (DNE) Critical Numbers: 1" (x ) = 20x3 - 30x =0 =7 10x (2x2-3) = 0 = X=0, - J. JE (" ( 20 ) G: Points of Inflection: Find the y-values for any points of inflection. + (0 ) = 0 +1- 13 ~ 153 + (13 ) = - 1.53 H: Sketch 12 p ) 6 1 P) 60 PD ( 1 0 ) ( 2 0 ) de.2. State the domain and determine the asymptotes, if any, of the following functions. a. f(x) = 8x +9 2 x - 16 2 b. f (x) = 5x - 3x + 1 x+2 3. Follow the Curve Sketching Algorithm to sketch the graph of f (x) = x - x