Transcript Polynomial & Rational Inequalities

Analyzing Polynomial & Rational
Functions & Their Graphs
1)
2)
3)
4)
Steps in Analysis of Graphs of Poly-Rat Functions
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.
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.
Record behavior specific to intervals of roots/holes.
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 Example
Solve & graph:
(x + 3)
> (x + 3) .
(x2 - 2x + 1)
(x – 1)
Step 1:
(x + 3) –
(x – 1)2
(x + 3)(x – 1)
(x – 1)(x – 1)
> 0.
(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’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
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)
_
(-x – 1)2
No symmetry.
  f(-x) _
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 < .
Poly-Rat Inequalities Example cont’d
f(x) = (x + 3)(2 – x)/(x – 1)2.
Test evaluations of f(x) to get sign in intervals
   x  3  3  x  11  x  2 2  x  
-4
f(-4) = -6/25
Below
x-axis
(-4, -6/25)
0
f(0) = 6
3/2
f(3/2) = 9
3
f(3) = -3/2
Above
x -axis
Above
x -axis
Below
x-axis
(0, 6)
(3/2, 9)
(3, -3/2)
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, 1); (1, 2]; [2, ).
(-, -3];
neg
pos  pos 0
0
[
)( ]
-3 -2 -1 0 1 2
neg
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 2nd degree, there
is a horizontal asymptote at y = – 1/1 as |x| increases.
Poly-Rat Inequalities Example cont’d
___
Solve & graph:
(x + 3)(2 – x) > 0
(x – 1)2
963        
-5 -4 -3 -2 -1 0 
1 2 3 4 5
Test
Hole
Intercepts:
values:
& Asymptotes:
(-4,
(-3,
Sketch
-6/25),
0),Sketch
(1,
of 0),
(3/2,
(0,
f(x)
ofx6),
>f(x)
=9),
01,graph
graph
y(3,(2,
= -3/2)
-20)
in red