Download Chapter 9: Rational Functions

CHAPTER 9:
RATIONAL FUNCTIONS
9.1 INVERSE VARIATION
Direct Variation
A function of the form y = kx
as x increases y increases.
(k is the constant of variation)
Example: If x and y vary directly, and x = 6
when y = 3, write an equation.
Inverse Variation
A function has the form
k
y
x
(k is a constant)
As one value of x and y increase, the other
decreases.
Modeling Inverse Variation
Suppose that x and y vary inversely, and x = 3 when
y = -5. Write the function that models the inverse
variation.
You Try! Modeling Inverse Variation
Suppose that x and y vary inversely, and x = -2
when y = -3. Write the function that models the
inverse variation.
Rational Functions
Identifying from a table
x
3
6
9
y
1
1/2
1/3
• Does this look like direct or inverse variation?
• Find the value of k:
• Write the equation:
You Try! Is the relationship between the variables in each table a direct
variation, an inverse variation, or neither? Write then functions.
1)
2)
3)
A Combined Variation has more than one
relationship.
EX:
kx is read as y varies directly with
y=
z
x (on top) and inversely with z (on the bottom).
Write the function that models each relationship.
1. Z varies jointly with x and y. (Hint jointly means
directly)
1. Z varies directly with x and inversely with the cube
of y
1. Z varies directly with x squared and inversely with y
Write a function
Z varies inversely with x and y.
Write a function when x = 2 and y = 4 and z = 2
9.2 Graphing Inverse Variation
9.3 Rational Functions
Investigation
Graph the following:
1. Y = 3/X
2. Y = -8/X
What do you notice?
Rational Functions
Rational functions in the form y = k/x is split
into two parts. Each part is called a Branch.
If k is POSITIVE the branches are in
Quadrants I and III
If k is NEGATIVE the branches are in
Quadrant II and IV
An Asymptote is a line that the graph
approaches but NEVER touches.
Asymptotes
From the form
k
y=
+c
x-b
• The Vertical Asymptote is x = b
• The Horizontal Asymptote is y = c
Identify the Asymptote from the following functions.
1.
3
y=
+1
x-2
2.
3
y=
-3
x-4
You Try! Identify the Asymptote from the following
functions.
1.
3
y=
x+3
2.
3
y=
-3
x+2
Translate:
1)
3
y
x
Up 3 units and Left 2 Units
1)
Down 5 units and Right 1 unit
1)
Right 4 units
1)
Such that it has a Vertical asymptote of x=3 and a
horizontal asymptote of y= -2
Rational Functions
A rational function can also be written in the
form
p(x)
f (x) =
q(x)
where p and q are polynomials.
Asymptotes
Vertical Asymptotes are always found
in the BOTTOM of a rational function.
Set the bottom equal to zero and solve.
Find the Vertical Asymptotes for the following.
1. y =
x -5
2.
(x + 3)(x - 4)
2
x + 2x +16
3. y =
(x - 3)
4
x + 2x + 4
y=
2
x - 25
2
x - 25
4. y = 2
x - 4x + 4
2
Asymptotes
What are the Asymptotes?
x(x - 4)
y=
(x - 4)
Graph it, what do you notice?!
There is a HOLE in the graph is when (x – a)
is a factor in both the numerator and the
denominator.
x(x - 4)
y=
(x - 4)
So on the graph, there is a HOLE at 4.
•A graph is Continuous if it has no jumps,
breaks or holes.
•A graph is Discontinuous if it does have
holes or jumps or breaks.
Find the places of Discontinuity:
x x6
y
2
3x 12
2
Find the places of Discontinuity:
x 1
y 2
x  2x  3
2
Find the places of Discontinuity:
2
x
y 2
x 9
Find the places of Discontinuity:
12x  24
y 2
x  2x
9.4
Simplifying, Multiplying and
Dividing
Simplest Form
• A rational expression is in SIMPLEST
FORM when its numerator and denominator
are polynomials that have no common
divisors.
• When simplifying we still need to
remember HOLES as points of discontinuity.
Simplify:
1.
(x - 5)(x + 3)(x + 4)
(x - 5)(x - 4)(x + 3)
Simplify:
1.
x  10x  25
2
x  9x  20
2
Simplify:
1.
3x - 6
2
x - 6x + 8
Simplify:
1.
2x - 3x - 2
2
x - 5x + 6
2
Multiplying Rational Expressions
1. Multiply the tops and the bottoms.
1. Simplify.
Example:
x 2 + 7x +12
x +3
· 2
x+5
x + 5x + 4
Example:
x - 4 x +1
· 2
2
x -1 x + 2x
2
Dividing Rational Expressions
Dividing by a fraction is the same thing as
multiplying by the reciprocal.
1. Flip the second fraction
2. Multiply the tops and the bottoms.
3. Simplify.
Dividing Example:
x4
5(x  4)

(3x  2)(x  2) (x  2)(7x  5)
Dividing Example
a 2  2a 15
a 1

2
a 16
3a 12
9.5 Adding and Subtracting
Rational Expressions
Simplify:
You Try! Simplify:
.
• A Common Denominator is also the
LEAST COMMON MULTIPLE.
• LCM is the smallest numbers that each
factor can be divided into evenly.
Finding LCM of 2 numbers
• 7:
• 21:
Find the LCM of each pair of numbers
1. 4, 5:
1. 3, 8:
2. 4, 12:
Unlike Denominators
1.
4 1
+
5 10
2.
5 2
12 9
For Variables LCM: Take the Largest Exponent!
Find the LCM:
1. x4 and x:
2. x3 and x2:
1. 3x5 and 9x8:
1.
2 5
+ 3
2
x
x
2.
3
1
5
2x 5x
For Expressions LCM: Include ALL factors
• 3(x + 2) and 5(x – 2)
• x(x + 4) and 3x2(x + 4)
For Expressions LCM: Include ALL factors
• (x2 + 2x - 8) and (x2 – 4)
• 7(x2 – 25) and 2(x2 + 7x + 10)
5
6
1.

x  3 x 1
2.
y
2

2
y  y  20 y  4
3.
x
3
 2
x2 x 4
4.
b - 2b - 8
6
2
b + b - 2 b -1
2
A complex fraction is a fraction that has a
fraction in its numerator or denominator.
To Simplify: Multiply the top and the bottom
by the COMMON DENOMINATOR
Simplify:
2
x
3
y
Simplifying Complex Fractions
2
1
3
4
9
Simplify:
2
6
x
1
y
9.6 SOLVING RATIONAL
EQUATIONS
To solve proportions you cross multiply:
a c
=
b d
Example:
3 x

8 48
Use cross multiplication to solve.
Check for Extraneous Solutions when
variables are in the bottom because we
cannot divide by zero!
1 2

2
x
x
Solve:
x2 x7

10
5
You Try ! Solve:
y
6

y3 y9
You Try! Solve:
5
15
= 2
2x - 2 x -1
Eliminate the Fraction
We can eliminate the fractions all together if
we multiply the whole equation by the LCM
of the denominators!
1
2 1


2x 5x 2
Sum or Difference Equations
Solve the equation for x.
x x
  10
3 2
Solve:
7
2
3
 
2
x  5x x 2x  10