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.

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 = xi in each interval find f(xi). Note: sign[f(xi)] = sign[f(x)] for xi in interval i. Use this to sketch graph. Also, if f inequality was > or < , include in solution set roots of f from 2) above.

Poly-Rat Inequalities ExampleSolve & graph:

(x + 3) > (x + 3) .

(x2 - 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.

Poly-Rat Inequalities Example cont’dSolve & 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 ,

Poly-Rat Inequalities Example cont’dSolve & 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.

Poly-Rat Inequalities Example cont’dSolve & 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 < .

xxxx 2 21 13 3

-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’df(x) = (x + 3)(2 – x)/(x – 1)2.

-3 -2 -1 0 1 2

[ )( ]

Poly-Rat Inequalities Example cont’dSolve & 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 pos 0 neg

-3 -2 -1 0 1 2

[ )( ]

Poly-Rat Inequalities Example cont’dSolve & 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 2nd degree, there is a horizontal asymptote at y = – 1/1 as |x| increases.

-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