Simplifying Rational Expressions and Stating Domain Restrictions
Investigate and explain characteristics of rational functions including domain, range, zeros, points of
discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry,
asymptotes, and end behavior (MM4A1a)
Vocabulary:
 A rational expression is an expression that can be written as a ratio of two polynomials.
 It is in simplest form if the numerator and denominator have no factors in common other
than 1.
 The domain of an algebraic expression is the set of real numbers that the variable is
permitted to have. A rational expression is undefined when the denominator is zero.
Steps to simplify rational expressions.
1. Factor the numerator and denominator.
2. Find the restrictions for the denominator.
3. Simplify if possible.
Simplify the following. State any domain restrictions.
8
x
a.
b.
2x
3x 2  9x
d.
g.
x 2  5x  4
x 2  16
27x 3  8
6x 2  x  2
e.
h.
x3 x
x 2  5x  6
x 3  x 2  4x  4
3x 3  x 2  10x
c.
f.
8
x 2  4x  12
x2 1
x2  x 2
Factoring Review
• GCF (Greatest Common Factor)
ALWAYS LOOK FOR A GCF FIRST FOR EVERY PROBLEM!!!
Numbers - find the largest number that divides equally into ALL
the coefficients.
Example: 20x2 + 12x + 8
Variables – if all terms have the same variable then factor out the
variable with the lowest power.
Example: 20x2y3 + 12xy2 + 8y4
Binomials – if all terms are multiplied by the same binomial then
factor out the binomial.
Example: 3x(2x – 5) – 4(2x – 5)
• Grouping
use when there are 4 terms
- make 2 groups that are 2 terms each
- factor out the variable and number GCF’s of each group
- factor out the binomial GCF
Example: 6x3 – 15x2 – 8x + 20
• Difference of Two Squares: a2 – b2 = (a + b)(a – b)
use when there are 2 terms that are each perfect squares and one term is
negative
Example: 225x8 – 100x2
• Perfect Square Trinomial:
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
use when there are 3 terms in which the leading coefficient and the constant
are perfect squares, and the middle term is 2 times the product of the square
root of the first and the square root of the last
Example: 36x2 -84x + 49
• Other Trinomials
use when there are 3 terms and perfect square trinomial does not work
- factor out GCF, if necessary
- multiply the leading coefficient and the constant together
- list all the factors of the product of the leading coefficient times the constant
- find the two factors that add up to the middle term
- rewrite the middle term as a sum of the two factors
- factor the 4 terms by grouping
Examples:
x2 – 9x + 18
6x2 – 7x – 20
• Sum and Difference of Two Cubes
(a3 + b3) = (a + b)(a2 - ab + b2)
(a3 – b3) = (a – b)(a2 + ab + b2)
“short parentheses, long parentheses, same, opposite, always a
root, cube root, square, multiply, square”
Examples:
x3 + 27
343x3 – 8
plus; cube
Horizontal and Slant Asymptotes of Rational Functions
Investigate and explain characteristics of rational functions including domain, range, zeros, points of discontinuity, intervals of
increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior (MM4A1a)
Vocabulary
 Horizontal Asymptote: If f (x )  a as x   or x   , then the horizontal line y = a is a
horizontal asymptote.
 Slant (Oblique) Asymptote: An oblique line (neither horizontal or vertical) which the graph of a
function approaches as the independent variable goes to positive or negative infinity. Often occurs in
rational functions when the degree of the numerator is one higher than the degree of the denominator.
How to find the horizontal asymptote (HA) or Slant Asymptote (SA)
Case 1: The degree of the numerator (N) < (less than) the degree of the denominator (D). There will be
a horizontal asymptote at
y = 0.
x2 1
Example: f (x )  3
x  2x
Case 2:
Has HA y = 0.
The degree of the numerator (N) = (equal to) the degree of the denominator (D). There
will be a horizontal asymptote y 
4x 2  1
Example: f (x ) 
5x 2  2x
Case 3:
degree
==
23
degree
leading coefficient of the N
leading coefficient of the D
degree
degree == 22
Has HA y 
4
.
5
The degree of the numerator (N) > (greater than) the degree of the denominator (D).
There will be a slant asymptote if N = D + 1. Use long division to find the equation of the
SA. Ignore any remainder.
2
degree = 2
2x  1 
degree = 1
2x 2  x  1
x 1
Example: f (x ) 
x 1
x 1
2x 2  x  1
SA: y = 2x – 1
(2x 2  2x )
x  1
( x  1)
2
Holes and Vertical Asymptotes of Rational Functions
Investigate and explain characteristics of rational functions including domain, range, zeros, points of discontinuity, intervals of
increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior (MM4A1a)
Vocabulary
 If f (x )   as x  a from the left or right, then x = a is a vertical asymptote.
 f(x) contains a hole at  a , b  if f (x )  b as x  a from the left and right, but f (a )  b .
How to find holes and vertical asymptotes
Holes
Factor the numerator and denominator. If they have common factors, cross them out. Set the factor
equal to zero and solve for x. Find the
y-coordinate of the hole by plugging the value into the remaining equation.
f x  
Example
f (4) 
2
x  6x  40
2
x  16

x
 10   x  4 
x  4 x  4

 x  10 
x  4
f x  
4 4
x 4
 4  10   14  7  1.75
 4  4 8 4
Vertical asymptotes (VA)
Using the simplified equation set the denominator equal to zero and solve for x.
Example
x 40
x
2
 6x  40  x  10   x  4 

2
x  4 x  4
x  16
 x  10 

x  4
x 40
You try
Find the coordinate of any holes in the graph and the equation of all vertical asymptotes.
1.
f (x ) 
3x  1
x2 9
2.
f (x ) 
x2 1
x 2  3x  2
4 4
x  4
Domain, X-intercepts, & Y-intercepts of Rational Functions
Investigate and explain characteristics of rational functions including domain, range, zeros, points of discontinuity, intervals of
increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior (MM4A1a)
Vocabulary
 The domain of a rational function consists of all real numbers x except those for which the denominator
is zero.
 The x-intercepts of a rational function are the points where the graph crosses the x-axis. The xintercepts are where f(x) = 0 and are listed as points (# , 0).
 The y-intercept of a rational function is the point where the graph crosses the y-axis. The y-intercept is
where x = 0 and is listed as the point (0 , #).
How to find the domain, x-intercepts, and y-intecepts
Domain
Factor the numerator and denominator. Do NOT cancel yet. The domain is all real numbers not equal to the
zeros of the denominator.
x 2  6x  40  x  10  (x  4) Domain: all reals  4, -4
f x  

2
 x  4  (x  4)
x  16
Example
x 40
x 40
x-intercepts
Using the reduced factored equation, set the numerator equal to zero and solve for x.
x  10  0
10 10
 x  10 
x 2  6x  40  x  10   x  4 


Example f  x  
2
x  4
x  4 x  4
x  16
x  10
(-10,0)
y-intercept
Using the reduced factored equation, plug 0 in the numerator and the denominator and simplify.
0  10 10 5
 x  10 
x 2  6x  40  x  10   x  4 

 =2.5
f
x





Example
04
4 2
2
x  4
x  4 x  4
x  16
(0,2.5)
You try
State the domain, x-intercepts, and y-intercept of each function.
x 2  5x  4
1.
f(x) =
2.
x 2  16
3.
f(x) =
x2 1
x2  x 2
4.
f(x) =
f(x) =
x3 x
x 2  5x  6
x 3  x 2  4x  4
3x 3  x 2  10x