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Unit 7—Rational Functions
Rational Expressions
• Quotient of 2 polynomials
3x 2 y
2x  5
2 x 2  3x  1
or
or
or x  9
3
15 xy
y 3
y 1
p
so
,q  0
q
Things to Consider
Graphing Rational Functions
1. Factor
2. Determine where discontinuities would occur in
the graph
3. Graph any asymptotes on the graph and pick points
on both sides to locate the branches of the
function
• If factors cancel, then you have a point of
discontinuity (hole)
• If factors remain in the denominator, you have a
vertical asymptote(s)
Horizontal Asymptotes
1. If the degree of the numerator is bigger than
the degree of the denominator, then there is
NO H.A. (top-heavy)
2. If the degree of numerator is smaller than
the degree of the denominator, then the HA
is at y=0 (bottom heavy)
3. If the degree of numerator is equal to degree
of the denominator, then the HA is equal to
the leading coefficients (equal weight)
Graphing Rational Functions
x 2  7 x  12
y 2
x  9 x  20
Graphing Rational Functions
3x  13x  10
y
x 5
2
Graphing Rational Functions
x  2x  3
y 2
x  x2
2
Graphing Rational Functions
x
y 2
x  2x  3
Graphing Rational Functions
x 9
y
x 3
2
Graphing Rational Functions
x 9
y 2
x 1
2
To simplify a rational expression
• Look for common factors
 27 x 3 y
ex.
9x4 y
( x  2)
ex.
( x  2)( x  1)
4x2  9
ex.
4 x 2  12 x  9
x3
ex.
x 3
To simplify a rational expression
• Look for common factors
x2
ex.
2 x
6  3x
ex.
4 x 2  5x  6
x5
ex.
25  x 2
Multiply Rational Expressions
• Factor, Reduce common factors first, then multiply
x2  9 x2  4x  4
ex. 2

x 4
x3
( x  3) 2
x4
ex. 2

x  7 x  12 x  3
Multiply Rational Expressions
• Factor, Reduce common factors first, then multiply
4ab3 b 2  16
ex.

2
4b  b
8a
x 3  27 x 2  25
ex. 2

x 9
x5
Dividing Rational Expressions
• Rewrite as Multiplication by reciprocal of 2nd
fraction, factor, Reduce common factors, then
multiply
3x  6
x 2  5x  6
ex.

12 x  24 3x 2  12
x2
3x
ex. 2
 2
x  2x  1 x 1
Dividing Rational Expressions
• Rewrite as Multiplication by reciprocal of 2nd
fraction, factor, Reduce common factors, then
multiply
3x 2  9 x x 2  9
ex.

x2
4x  8
4 x 3 16 x 2
ex. 4  2
3y
9y
Dividing Rational Expressions
x 3  25 x 2 x 2  2 x 2  5 x
ex. 2


2
x  6x  5
4x
7x  7
Dividing Rational Expressions
ax  ay  bx  by ax  ay  bx  by
ex.

ax  ay  bx  by ax  ay  bx  by
Dividing Rational Expressions
x  9 x  14
2
x  6x  5
2
x  8x  7
2
x  7 x  10
2
Dividing Rational Expressions
  x  1   x  1 
x
  
 
 
2
x 1   x 1   x  1  
Adding and Subtracting Rational
Expressions
• To add fractions, you must have a common
denominator
• To determine the LCD, list any common factor
that occurs in two or more of the
denominators only once in the LCD and then
include all other factors that are not common.
Adding and Subtracting Rational
Expressions
x 1
x
ex.

x4 x4
2x  5 x  4
ex. 2
 2
x 1 x 1
Adding and Subtracting Rational
Expressions
6x
4x
ex.

3x  1 2 x  5
5x  3
4x
ex.
 2
3 x x 9
Adding and Subtracting Rational
Expressions
4
3
ex. 2
 2
x  16 x  8 x  16
2x  4
x4
ex. 2

x  x x( x  1)( x  1)
Adding and Subtracting Rational
Expressions
1
5x
3
ex.
 2

2x x 1 x 1
7y
8
3
ex. 2


y  y 2 y y2
Adding and Subtracting Rational
Expressions
2 x  10
5 x
ex. 2

x  25 25  x 2
3y  2
7
ex. 2
 2
y  5 y  24 y  4 y  32
Complex Rational Expressions
1 1

x y
1
1
x
Complex Rational Expressions
x4
y2
2
x
1
1
x
Complex Rational Expressions
35
m
m  12
63
m
m2
Complex Rational Expressions
1 1

x y
1 1

x y
Complex Rational Expressions
x
x
3
x
x
6
Solving Rational Equations
1.
2.
3.
4.
Factor all denominators
Multiply both sides of equation by LCD
Solve
Eliminate any solution that would make the
denominator zero
5. Check remaining solutions
Solving Rational Equations
5
4
8

 2
x x  3 x  3x
Solving Rational Equations
x 1
4
2
5
x
Solving Rational Equations
x2
x

4
2
x  3 x  10 x  5
Solving Rational Equations
y 1
2

y 3 y 3
Solving Rational Equations
3
2

m 1 m  3
Solving Rational Equations
50 50
4


x x2 x
Solving Rational Equations
3
2y
5


2
y2 4 y
y2
Solving Rational Inequalities
1. State the excluded values
2. Solve the related equation
3. Use those values on a number line and test
the values
Solving Rational Inequalities
x2
x

2( x  3) x  3
Solving Rational Inequalities
4
1
c2
Graphing Rational Functions
Possible Graphs:
Direct and Inverse Variation
• Direct Variation can be expressed in the form
y=kx
• K is the constant of variation
• Equation of variation—equation representing
the relationship between the variable but
substitute the value of k
Ex. Y=2x (if k=2)
Direct and Inverse Variation
y varies directly w ith x. When y  12, x  3.
Find y when x  10
Direct and Inverse Variation
y varies directly w ith x. When y  25, x  5.
Find x when y  1
Direct and Inverse Variation
Q varies directly w ith square of p. When q  8, p  3.
Find q when p  4
Direct and Inverse Variation
1
L varies directly w ith cube of m. When L  , m  2.
2
Find L when m  3
Inverse Variation
• Expressed as y=k/x
Direct and Inverse Variation
2
y varies inversely with x. When y  3, x  .
5
3
Find y when x 
4
Direct and Inverse Variation
1
y varies inversely with the square of x. When y  2, x  .
2
Find y when x  4
Direct and Inverse Variation
y varies inversely with x. When y  6, x  -3.
Find x when y  8
Joint Variation
• When one quantity varies directly with the
product of two or more other quantities
• Combination Variation—when one quantity
varies directly with another quantity and
inversely with the other quantity
Joint or Combination Variation
y varies directly w ith x and the square of z. When y  4, x  2 and z  3.
Find y when x  8and z  6
Joint or Combination Variation
y varies directly w ith x and inversely with
the square of z. When y  20, x  4 and z  1.
Find y when x  5 and z  6