2. Analyzing Polynomial & Rational Functions & Their Graphs Steps in Analysis of Graphs of Poly-Rat Functions 1)Examine graph for the domain with attention to holes. (If f = p/q, “holes” are where q(x) = 0.) Here there will be vertical asymptotes. 2)Find any root(s) where f(x) = 0 or, if f = p/q, p(x) = 0. Note where f(x) = 0, p(x) = 0, or q(x) = 0, they may be factored. 3)Record behavior specific to intervals of roots/holes. 4)Determine any symmetry properties and any horizontal or oblique asymptotes.

3. Polynomial & Rational Inequalities Steps to Solution & Graph of Poly-Rat (In)equalities 1)Write as form: f(x) > 0, f(x) > 0, f(x) < 0, f(x) < 0, or f(x) = 0, with single quotient if f is rational. 2)Find any root(s) of f(x) = 0 &, if f = p/q, find “holes” where q(x) = 0. Factor f, p, q as possible. 3)Separate real number line into intervals per above. 4)For an x = x i in each interval find f(x i ). Note: sign[f(x i )] = sign[f(x)] for x i in interval i. Use this to sketch graph. Also, if f inequality was > or <, include in solution set roots of f from 2) above.

4. Poly-Rat Inequalities Example Solve & graph: (x + 3) > (x + 3). (x 2 - 2x + 1) (x – 1) Step 1: (x + 3) – (x + 3)(x – 1) > 0. (x – 1) 2 (x – 1)(x – 1) (x + 3)(2 – x) > 0 ___ (x – 1) 2 or f(x) = p(x)/q(x) > 0 with p(x) = (x + 3)(2 – x) and q(x) = (x – 1) 2.

5. Poly-Rat Inequalities Example cont’d Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Step 2: Note roots of f(x) = roots of p(x). They are at x = –3 and at x = 2. The point x = 1 is a zero of q(x) so there f(x) is undefined and x = 1 is a “hole” or not in the domain. Step 3: The intervals: (- , -3]; [-3, 1); (1, 2]; [2,  ). f(x) values: f(-4)= -6/25, f(0)= 6, f(3/2)= 9, f(3)= -3/2,

6. Poly-Rat Inequalities Example cont’d Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Step 2 & 3 Data Summery: x-intercepts: at x = –3 and at x = 2. y -intercept: at y = f(0) = 6. f(-x) = (-x + 3)(2 + x)   f(-x) _ _ (-x – 1) 2 No symmetry.

7. Poly-Rat Inequalities Example cont’d Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Step 2 & 3 Data Summery cont’d: Vertical asymptote: at x = 1. Hole: x = 1 Horizontal asymptote: at y = -1 Intervals: -  < x < -3, -3 < x < 1, 1 < x < 2, 2 < x < .

8. -4 f(-4) = -6/25 Below x-axis (-4, -6/25) 3/2 f(3/2 ) = 9 Above x -axis (3/2, 9) 3 f(3) = -3/2 Below x-axis (3, -3/2) Test evaluations of f(x) to get sign in intervals 0 f(0 ) = 6 Above x -axis (0, 6) Poly-Rat Inequalities Example cont’d f(x) = (x + 3)(2 – x)/(x – 1) 2.

9. -3 -2 -1 0 1 2 [ )( ] Poly-Rat Inequalities Example cont’d Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Step 4 Graphs: A) Solution set as intervals on the number line – (- , -3]; [-3, 1); (1, 2]; [2,  ). neg 0 pos  0 neg -3 -2 -1 0 1 2 [ )( ]

10. Poly-Rat Inequalities Example cont’d Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Step 4 Graphs: B) Solution set in graph sketch of f(x) versus x. First plot known points. Then sketch. Do not forget in sketching to include information about asymptotes. In this case, since x = 1 is a zero of multiplicity 2 in q(x), there is a vertical asymptote at x = 1. Also, since both p(x) and q(x) are of 2 nd degree, there is a horizontal asymptote at y = – 1/1 as |x| increases.

11. -5 -4 -3 -2 -1 0 1 2 3 4 5 3- 6- 9-      Solve & graph: (x + 3)(2 – x) > 0 ___ (x – 1) 2 Poly-Rat Inequalities Example cont’d Intercepts: (-3, 0), (0, 6), (2, 0) Hole & Asymptotes: (1, 0), x = 1, y = -2 Test values: (-4, -6/25), (3/2, 9), (3, -3/2) Sketch of f(x) graph Sketch of f(x) > 0 graph in red