8.1 Model Inverse and Joint Variation
Goal  Use inverse variation and joint variation models.
VOCABULARY
Direct variation
The equation y = ax represents direct variation between x and y.
Constant of variation
The nonzero constant a in the direct variation equation y= ax
Inverse variation
a
Two variables x and y show inverse variation if they are related as follows: y  , a  0.
x
Constant of variation
The nonzero constant a in a variation equation
Joint variation
When a quantity varies directly with the product of two or more other quantities
DIRECT VARIATION
Equation The equation y = __ax__ represents direct variation between x and y, and y is said to __vary
directly__ with x. The nonzero constant a is called the constant of _variation_.
INVERSE VARIATION
a
Two variables x and y show inverse variation if they are related as follows: y = x , a  0
_____
The constant a is the constant of variation, and y is said to __vary inversely__ with x.
Example 1
Classify direct and inverse variation
Tell whether x and y show direct variation, inverse variation, or neither.
Given Equation
Rewritten Equation
Type of Variation
y
x
9
_____
_____
b. xy = 3
_____
_____
a.
Checkpoint Tell whether x and y show direct variation, inverse variation, or neither.
1. y = x + 2
2. yx = 5
y
3. 2.6  x
Example 2
Write an inverse variation equation
The variables x and y vary inversely, and y = 3 when x = 6. Write an equation that relates x and y.
Find y when x = 9.
Checkpoint Complete the following exercise.
4. The variables x and y vary inversely, and y = 4.4 when x = 5. Write an equation that relates x and y.
Find y when x = 8.
Example 3
Check data for inverse variation
Determine whether m and n show inverse variation. If they do, write a model that gives n as a function of m.
Find n when m = 45.
m
5
10
15
20
25
n
45
22.5
15
11.25
9
Checkpoint Do the data below show inverse variation? If so, write a model that gives y as a function
of x.
x
2
4
6
8
10
y
18
9
6
4.5
3.6
JOINT VARIATION
Joint variation occurs when a quantity varies directly with the product of _two or more_ other quantities. In the
equation below, a is a nonzero constant.
z = __axy__
z varies jointly with x and y.
Example 4
Write a joint variation equation
The variable z varies jointly with x and y. Also, z = 84 when x = 4 and y = 3. Write an equation that relates x,
y, and z. Find z when x = 5 and y = 2.
Example 5
Compare different types of variation
Write an equation for the relationship.
Relationship
a. m varies jointly with n, p, and q.
Equation
m = _____
b. r varies inversely with s.
r = _____
c. x varies inversely with the cube of y.
x = _____
d. k varies jointly with x and y and inversely
with m.
k = _____
e. t varies directly with u and inversely with
w.
t = _____
Checkpoint Complete the following exercises.
6. The variable z varies jointly with x and y. Also, z = 44 when x = 4 and y = 1. Write an equation that
relates x, y, and z. Find z when x = 6 and y = 3.
7. Write an equation for the relationship: x varies jointly with y and z and inversely with the square of t.
8.2 Graph Simple Rational Functions
Goal  Graph rational functions.
VOCABULARY
Rational function
px 
A function of the form f x  
where p(x) and g(x) are polynomials and g(x)  0
qx 
PARENT FUNCTION FOR SIMPLE RATIONAL FUNCTIONS




1
The graph of the parent function f(x) =
is a hyperbola , which consists of two symmetrical parts
x
called branches .
The domain and range are all nonzero real numbers .
The asymptotes are x = _0_, and y = _0_ .
a
Any function of the form g(x) =
(a = 0) has the same _asymptotes_ , _domain_, and _range_ as the
x
1
function f(x) = .
x
Example 1
Graph a rational function of the form y =
Graph the function y =.
4
x
Checkpoint Graph the function.
1. y =
3
x
a
x
GRAPHING TRANSLATIONS OF SIMPLE RATIONAL FUNCTIONS
To graph a rational function of the form y =
a
+ k, follow these steps:
xh
Step 1 Draw the asymptotes x = _h_ and y =__k__.
Step 2 Plot points to the left and to the right of the vertical asymptote .
Step 3 Draw the two branches of the hyperbola so that they pass through the plotted points and approach the
asymptotes.
Example 2
a
k
Graph a rational function y = y 
xh
6
 2. State the domain and range.
Graph y =
x3
Example 3
ax  b
c d
4x  2
x
Graph y 
State the domain and range.
x 1
Graph a rational function of the form y 
Checkpoint Graph the function. State the domain and range.
2.
y
3
1
x2
8.3 Graph General Rational Functions
Goal  Graph rational functions with higher-degree polynomials.
GRAPHS OF RATIONAL FUNCTIONS
Let p(x) and q(x) be polynomials with no common factors other than ±1.
f(x) =
p( x)

q( x)
a m x m  a m1x m1      a 1x  a 0
b n x n  b n x n1      b x  b
1
1
0
1. The x-intercepts of the graph of f are the real zeros of _p(x)_.
2. The graph of f has a vertical asymptote at each real zero of _q(x)_.
3. The graph of f has at most one horizontal asymptote, determined by the degrees m and n of p(x) and q(x).

If m < n, the line _y = 0_ is a horizontal asymptote.

If m = n, the line y = b

am
is a horizontal asymptote.
n
If m > n, the graph has _no horizontal asymptote_.
a
The end behavior is the same as y =
m mn
b
x
n
Example 1
Graph a rational function (m < n)
Graph y =
3
x
2
2
. State the domain and range.
.
8.4 Multiply and Divide Rational Expressions
Goal  Multiply and divide rational expressions.
VOCABULARY
Simplified form of a rational expression
A rational expression in which its numerator and denominator have no common factors (other than ±1)
SIMPLIFYING RATIONAL EXPRESSIONS
Let a, b, and c be nonzero real numbers or variable expressions. Then the following property applies.
ac

bc
Divide out common factor c.
a
b
Example 1
Simplify a rational expression
x 2  7 x  10
x2  4

MULTIPLYING RATIONAL EXPRESSIONS
Let a, b, c, and d be nonzero real numbers or variable expressions. The rule for multiplying rational expressions
is the same as the rule for multiplying numerical fractions: multiply _numerators_, multiply _denominators_,
and write the new fraction in simplified form.
a
b

c
d

ac
Simplify,
bd
Example 2
Multiply rational expressions
2x2  4 x
x 2  4 x  12

x2  9 x  18
2x
ac
bd
if possible.
Checkpoint Complete the following exercises.
1. Simplify the expression.
x 2  2 x  15
x2  4x  3
2. Multiply the expression.
6 x 2  18 x
x2  x  6

x2  x  2
x2  7x  8
Example 3
Multiply a rational expression by a polynomial
x 4
x3  1
2
 ( x  x  1)
DIVIDING RATIONAL EXPRESSIONS
To divide one rational expression by another, multiply the first expression by the reciprocal of the second
expression.
a

b
c

d
a

b
d

c
ad
Simplify bc if possible
Example 4
Divide rational
expressions
2
3
x7

8x  8x
x2  6x  7
Checkpoint Multiply or divide the expression.
3.
 2x2
x 3  27
2
 ( x  3 x  9)
x5
4.
9 x 2  18 x

2 x 2  11 x  5
2x2  5x  2
8.5 Add and Subtract Rational Expressions
Goal  Add and subtract rational expressions.
VOCABULARY
Complex fraction
A fraction that contains a fraction in its numerator or denominator
ADD (SUBTRACT) WITH LIKE DENOMINATORS
To add (or subtract) rational expressions with like denominators, simply add (or subtract) their numerators .
Then place the result over the common denominator. Let a, b, and c be polynomials with c  0
a b ab

Addition  
Subtraction: a  b  a b
b c
c
b c
c
Example 1
Add with like denominators
9
2



6x 6x
6x
6x
.
ADD (SUBTRACT) WITH UNLIKE DENOMINATORS
To add (or subtract) two rational expressions with unlike denominators, find the
least common denominator (LCD), which is the _least common multiple__ (LCM) of the denominators.
Rewrite each rational expression using the LCD, then add (or subtract) using the procedure for like
denominators. Let a, b, c, and d be polynomials with c  0 and d  0.
Addition
a b ad bc ad  bc
 


c d
cd cd
cd
Subtraction a  b  ad 
c d c
d
b
ad  b

cc
c c
d
d
Example 2
Add with unlike denominators
Add:
5

2
4x
x1
2x2  4x
Example 3
Subtract with unlike denominators
Subtract:
x 4
7
 2
3x  9 x  9
Checkpoint Perform the indicated operation and simplify.
1.
2.
8x
7

3 x 3 x
13
3x

2
3x
3.
4.
x 1
6
 2
x  6x  9 x  9
2
8
4x2

2 x
8 x 2  12 x
SIMPLIFYING COMPLEX FRACTIONS
A complex fraction is a fraction that contains a fraction in its numerator or denominator .
Method 1: If necessary, simplify the numerator and denominator by writing each as a single fraction. Then
divide the numerator by the denominator.
Method 2: Multiply the numerator and the denominator by the least common denominator (LCD) of every
fraction in the numerator and denominator. Then simplify.
Example 4
Simplify a complex fraction (Method 1)
8
Simplify:
5x

x 1
4

1
1
5
5
Your Notes
Example 5
Simplify a complex fraction (Method 2)
2x
x1
Simplify:
1
2

3x
x1
Checkpoint Simplify the complex fraction.
6x x

4
4
5.
8 3x

2
2
5
7
 2
6. 2x x
3
1

2
2x
x
8.6 Solve Rational Equations
Goal  Solve rational equations.
VOCABULARY
Cross multiplying
A method of solving a simple rational equation for which each side of the equation is a single rational
expression. Equal products are formed by multiplying the numerator of each expression by the denominator of
the other.
Example 1
Solve a rational equation by cross multiplying
20
5

2x  5 x  2
=
Checkpoint Solve the equation by cross multiplying.
1.
3
6

x  2 3x  8
2.
6
 12

x  2 x 1
Example 2
Solve a rational equation with one solution
8 11  14
 
x 3
x
Example 3
Solve a rational equation with two solutions

8
Solve: 3x 5  2 
x 3
( x  3)( x  4)
Checkpoint Solve the equation by using the LCD
3.
11 3 5
 
4 x 2x
4. 1 
4
9

x 2 x
Example 4
Check for extraneous solutions
Solve:
6x2
3x
4


2
x 1
x 4 x  4
6
Checkpoint Solve the equation by using the LCD. Check for extraneous solutions.
2
5. 8x  4x  2
x2  9 x  3 x  3