Name:______________
Block:______________
Algebra2 Unit 7
Rational Functions
Unit 7: Rational Functions
Day
Day
Day
Day
Day
Day
Day
Day
1:
2:
3:
4:
5:
6:
7:
8:
Multiplying and Dividing
Adding and Subtracting (exit ticket on multiply and divide)
Review and Assess
Solving Rational Equations
Graphing (exit ticket on solving)
Review and Practice
Test on Rational Operations and Equations
Inverse Variation
Day 1: Multiplying and Dividing RATIOnal Expressions
In these notes we will
Introduce, multiply and divide rational expressions
So we can eventually GRAPH them
Warm-up
Factor the following expressions completely.
1. 10 x  5 x 2
2. x 2  2 x  48
3. x 2  25
4. x2 – 49
5. 6x2 + 14x + 4
6. 2x2 + 9x – 6
7. x3 – 125
8. Simplify
16 x 4 y 2 z
2 x5 yz
9. Divide
3 3

8 2
When can we “cancel” (aka "divide out")? How do we look for “ones in disguise”?
Simplify the following.
3x  6
10.
3
13.
4  7x3
x
x6
11.
3
14.
x( x  1)  ( x  1)
( x  1)
12.
x 2  7 x3
x
15.
x  ( x  1)
( x  1)
What IS a RATIONAL FUNCTION?
p( x)
A function of the form f(x)=
q( x)
where q(x) ≠ 0.
In other words…….
Simplify Rational Expressions
Property
ac
ac
a


bc
bc
b
Method:
Factor and Divide out COMMON FACTORS (aka "cancel")
Include Domain Restrictions for Real Numbers (set any denominator  0)
Examples:
4 x  12
4( x  3)
4


x  2 x  15
( x  5)( x  3)
x 5
15
35
3


65
13  5
13
1.
x2  7 x
x2
2.
x 2  8 x  16
x 2  2 x  24
3.
x4
x 2  16
Multiplying Rational Expressions
2
Property
a c
ac
 
Simplify if possible.
b d
bd
Method
1. Multiply Numerator
2. Multiply Denominator
3. Factor and divide out COMMON FACTORS
5 x 2 y 3 27 x5
4.

3xy 4 15 x 4 y
5.
2 x 2  10 x x  3
 2
x 2  25
2x
6.
x4
 ( x 2  2 x  4)
x3  8
Dividing Rational Expressions
Property
a c
a d
ad
   
Simplify if possible.
b d
b c
bc
Multiply the first fraction by the reciprocal of the second fraction
7.
x 2  4 x  21 x 2  3x  70

5 x  15
x 2  100
8.
3x 2  13x  10
 (3x 2  2 x)
2
6x
9. Complex Fraction
x 2  6 x  27
2 x2  2 x
2
x  14 x  45
x2
Expression
Which factors
cancel?
Expression
1
x(x 5)
(x 5)
4
x 5
5
2
16x
4
5
(x 3)(x 5)
(x 3)
3
3(x 2)
(x 2)
6
4ab
4a
Which factors
cancel?
How do you get to Carneige Hall? PRACTICE, PRACTICE, PRACTICE!
For the following RATIONAL EXPRESSIONS, simplify, multiply or divide.
1.
2  x  1
 x  1 x  3
x4
4. 2
x  16
7.
3x5 y 2 6 xy 2

8 xy 9 x3 y
4x
x2  2x

10.
5 x  20 x 2  6 x  8
2.
40 x  20
10 x  30
3.
x2  2 x  3
5.
x2  x  6
8.
2 x 2  10 x x  3
 2
x 2  25
2x
6.
9.
2 x 2  3x  5
  2 x2  5x 
11.
6x
4x
x  x  2
2 x 2  10 x
3x 2  16 x  5
x5
  x 2  x  1
x3  1
Day 2: ADDING AND SUBTRACTING RATIONAL EXPRESSIONS
In these notes we will
ADD and SUBTRACT rational expressions (which means we must learn to FIND COMMON
DENOMINATORS for them)
So we can SOLVE PROBLEMS that have rational functions
Warm-Up – Add or subtract the following.
What’s the fuss about common denominators?
1.
3 10

7 7
2.
1 3

4 8
3.
5 2

3 7
4.
4 8

5 5
5.
8 6

3 5
6.
9 3

2 8
7. Describe how you found common denominators.
Now let’s apply this to RATIONAL EXPRESSIONS.
Adding and Subtracting with LIKE Denominators – Life is GOOD!
1.
12 2


5x 5x
2.
3x
1


2x  5 2x  5
3.
2 x2
2
 2

2
x 1 x 1
4.
4x 2x  7


x3 x3
5.
x2
x2


x2 x2
Properties
Addition:
Adding and Subtracting with UNLIKE Denominators
a b
ad bc
ad  bc
 


c d
cd dc
cd
– ONE MORE STEP!
1. Write each expression in factored form
2. Construct a common denominator
(one of EACH factor to the highest degree)
3. Multiply to create the common denominator
4. Add or Subtract numerators
5. Simplify if possible. (Factor and divide out common factors))
1.
x
4

x2 x5
LCD
Factored form:
2.
4x 2x  7


x  3 x2  9
Factored form:
LCD
Subtraction:
a b
ad bc
ad  bc
 


c d
cd dc
cd
3.
2 x2
2
 2

2
x  1 3x  3
LCD
Factored form:
4.
x
2 x  1
 2
3 x  15
x  4x  5
LCD
Factored form:
5.
7
x
 2
2
9x
3x  3x
Factored form:
LCD
Your turn!
1.
5x
4x  9
 2
x  8 x  5 x  24
LCD
Factored form:
More Practice! 
1.
3.
x 2 - 9 4x -12
−
x -3
x -3
5
x+2
+
x + 3 x -3
2.
16
x 2 + 8x
+
x+4
x+4
4.
x7 x2
x5 x4
Day 4: SOLVING EQUATIONS WITH RATIONAL FUNCTIONS
In these notes we will
GET RID OF DENOMINATORS and solve rational equations
So that we can FIND WHERE 2 RATIONAL EXPRESSIONS INTERSECT
Steps:
1) Get rid of all denominators.
A) If you have one rational expression on each side: cross-multiply.
B) If you have more than one rational expression on each side then find LCD of
all rational expressions and multiply all terms by LCD.
2) Solve for x.
3) Check for extraneous solutions! (denominator cannot equal 0)
Solve by cross multiplying – only if you see FRACTION = FRACTION
Examples:
1.
7
11

x  2 2 x  10
3
1
=
x + 4x x + 4
2
2.
3.
1
x

2 x  5 11x  8
Solve by using the LCD. Multiply each term by the LCD to eliminate denominators.
Check the denominator for an extraneous solution!!
Remember, the denominator cannot equal 0.
Examples:
4.
5.
6.
3
1 12


x
2
x
5x
5
4
x 1
x 1
6
8x2
4x
 2

x 3
x 9
x3
Find LCD = _____________________
Find LCD = _____________________
Find LCD, factor all denominators
LCD = _____________________
Day 5: Graphing Rational Functions
In these notes we will define and investigate horizontal and vertical asymptotes
So that we can graph rational functions
Warm-Up Investigation – Getting ready for asymptotes!
1. What value must be omitted from the domain of the this function: f(x) =
1
______
x
Use your calculator to sketch the graph.
Change your x and y min-max to 5 and 5 to
get a better graph.
Describe what happens when x = 0.
2. What value must be omitted from the domain of this function:
f(x) =
1
______
x3
Use your calculator to sketch the graph.
Be sure to put ( ) around x  3.
Set your x min-max to 2, 8. Keep y set
to 5, 5.
Describe what happens at x = 3!!!
Compare the graph to the graph from #1.
Fill in the following tables. What happens to the function when the x value grows without bound?
3.
f (x) 
1
x
x
y
10
1000
100000
10
1000
100000
4.
f (x) 
3x
x2
x
y
10
1000
100000
10
1000
100000
Why do we "restrict a domain"? Why do we declare some solutions "extraneous"?
Find the domain of f ( x) 
1
x2
Find the domain of f ( x) 
1
x  4x  5
2
Having a DENOMINATOR that is a polynomial and a function that is a RATIO introduces new
twists which gives us the…..
ASYMPTOTE - ________________________________________________________
The graph of a rational function is called a HYPERBOLA. This graph is made of symmetrical
parts called branches.
Investigation: Match the function to its graph. Remember that numerators and denominators
should have parentheses around them. Discuss how the function definitions are different. How do
these differences affect their graphs?
y

_____1. f ( x) 
1
x


B.

y

A.
x











1
_____2. f ( x) 
x2






x

















y




1
3
_____3. f ( x) 
x2

C.


x









VERTICAL ASYMPTOTE 1.
2.
HORIZONTAL ASYMPTOTE 1.
2.



The Graph of a Rational Function:
1)
Removable Discontinuity (Hole in the graph) – occurs when p(x) and q(x) have
a common factor and we can divide it out
2)
Non-removable Discontinuity (Vertical Asymptote) –when the denominator equals zero
3)
Horizontal Asymptote – the value that the function approaches as x increases or
decreases without bound (what happens as x goes to +∞? What happens as x goes to -∞?)
a)
If the degree of the numerator < the degree of the denominator,
y = 0 is the horizontal asymptote.
b)
If the degree of the numerator = the degree of the denominator,
lead coefficient of the numerator
y=
is the horizontal asymptote.
lead coefficient of the denominator
c)
If the degree of the numerator > the degree of the denominator,
there is NO horizontal asymptote (there is something else!).
4)
x-intercept – zero(s) of the numerator
5)
y-intercept – the value of f(0)
6)
Slant Asymptote occurs when the degree of the numerator is exactly one more than
the degree of its denominator (will study in next class!)
Graphing a Rational Function of the Form
y
a
x
Notice: There is NO variable in the numerator. Technically, we have 0x in the numerator.
1. Identify and draw the asymptotes (as dotted lines)
Vertical
x=0
Key Question:
Horizontal y = 0
Key Question:
2. Plot 2 points on the left and right side of the vertical asymptotes (use easy numbers).
3. Draw the branches of the hyperbola through the plotted points.
1. Graph
y
4
x
2. Graph
y
6
x
V.A. ____________
V.A.____________
H.A. ____________
H.A. ____________
 y
 y







    


x





    









Domain:
Domain:
Range:
Range:
Graphing a Rational Function of the Form
y
x





a
k
xh
Discuss: What has changed?
1. Identify and draw the asymptotes (as dotted lines)
Vertical
x=
Key Question:
Horizontal y =
Key Question:
2. Plot 2 points on the left and right side of the vertical asymptotes (use easy numbers).
3. Draw the branches of the hyperbola through the plotted points.
1. Graph
y
1
3
x2
2. Graph
V.A. _________ H.A. _________
y
V.A._________ H.A.__________
 y
 y







    


x





    







Domain:
Range:
Range:
y


Domain:
3. Graph
3
2
x 1
5
2
x 1
4. Graph
V.A. _________ H.A. _________
y




2
1
x3
 y









V.A._________ H.A.__________
 y
    
x

x





    









Domain:
Domain:
Range:
Range:
x





Can you identify asymptotes?
1. Which are asymptotes for the function
x = -3
y = -3
y=3
y = -3
y=3
x=3
y=3
y
y
Horizontal
y=
Key Question:
y=0
no asymptotes
no asymptotes
1
 3?
x3
y=0
y=2
y
ax  b
cx  d
1. Identify and draw the asymptotes (as dotted lines)
Key Question:
no asymptotes
1
2?
x3
Discuss: What has changed?
x=
y=0
y=2
Graphing a Rational Function of the Form
Vertical
no asymptotes
y  x  3?
y=0
5. Which are asymptotes for the function
y=0
1
?
x3
x=0
4. Which are asymptotes for the function
x=3
y
x=0
3. Which are asymptotes for the function
x=3
3
?
x
x=0
2. Which are asymptotes for the function
x=3
y
no asymptotes
2. Plot 2 points on the left and right side of the vertical asymptotes (use easy numbers).
3. Draw the branches of the hyperbola through the plotted points.
1. Graph
y
3x  6
x2
2. Graph
V.A. _________ H.A. _________
y
V.A._________ H.A.__________
 y
 y







    


x





    







Domain:
Range:
Range:
y


Domain:
3. Graph
2x 1
4x  2
2 x  4
x4
4. Graph
V.A. _________ H.A. _________
y




3x
x2
 y









V.A._________ H.A.__________
 y
    
x

x





    









Domain:
Domain:
Range:
Range:
x





How about these?
x2
x2  4
x2  4
1. Graph y 
x2
2. Graph
V.A. _________ H.A. _________
V.A._________ H.A.__________
y
 y
 y







    


x





    









Domain:
Domain:
Range:
Range:
x





Compare and contract the two functions above. What makes their graphs
so different?
Day 8: INVERSE VARIATION
In these notes we will
define and investigate inverse variation
So that we can apply the process to real-world problems.
Direct Variation
y  ax
Inverse Variation
a
y
x
Joint Variation
y  axz
How to Solve a VARIATION Problem
1. Determine which formula to use.
2. Substitute values. Solve for a (the constant of variation)
3. Find an equation that relates to the variables by substituting a
into the formula from Step 1
Understanding VARIATION
Direct Variation and Cookies
Inverse Variation and Cookies
You are bringing cookies for your class.
You only have 2 cookies to eat (the
You want each person to have 2 cookies
constant of variation). You are a very
(the constant of variation). The number of generous person. If someone asks you to
cookies you bring will depend on the
share, you do. If more people ask you to
number of people!!
share, you do. The amount of cookie you
will have left is inversely proportional to
y = 2x where x = # of people
the number of people sharing the cookie.
and
y = # of cookies you will
2 total: you each get 1 cookie
1
bring
4 total: you each get
cookie
2
More people? More cookies!!
2
y=
where x = # of people
x
Y will always be 2 times x!!
and y = the amount of cookie for
each person
More people? Less cookie per person!!
The product of x and y will always be 2!
Inverse Variation
1. The variables x and y vary inversely, and y = 15 when x =
1
. Write an equation that relates x and y.
3
Then find y when x = -10.
Find Equation
Find y when x = -10
2. The number of songs that can be stored on an MP3 player varies inversely with the average size of a
song. The MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB).
a. Write an equation that gives the number n of songs that will fit on the MP3 player as a function
of the average song size s (in megabytes).
b. Create a table of values for the following size of songs:
Average Size of Song (MB)
2
2.5
3
Number of Songs
5
Check For Inverse Variation - Does the data have the same Constant of Variation?
y
Determine if the table represents inverse variation.
x
y
32
40
50
60
1
2
10
8
6
12
Calculate x  y for each data pair.
32  12
a
x
2
3
1
= _____
2
50  8 = _____
40  10 = _____
60  6
2
= _____
3
Joint Variation – join together
Joint Variation occurs when a quantity varies directly with the product of two or more other quantities. In
the equations below, a is a nonzero constant.
Example:
z = axy
z varies jointly with x and y
p = aqrs
p varies jointly with q, r, and s
3. The variable z varies jointly with x and y. Also, z = 60 when x = -4 and y = 5. Find z when x = 7 and
y = 2.
Find Equation
Find z when x = 7 and y = 2
Other types of Variation
y varies inversely with x.
y
a
x
y varies inversely with the square of x.
y
a
x2
z varies directly with y and inversely with x.
z
ay
x
Example:
In kick boxing, it is found that the force, f, needed to break a board, varies inversely
with the length, l, of the board. If it takes 5 lbs of pressure to break a board 2 feet
long, how many pounds of pressure will it take to break a board that is 6 feet long?
1. Set up the formula.
2. Find the missing constant, k,
using the first set of data given.
3. Using the formula and constant, k,
find the missing value in the problem.
Practice:
1. Variable M varies directly with p. If M = 75 when p = 10, find M when p = 16.
2. R varies inversely with variable T. If R is 168 when T = 24, find R when T = 30.
3. Variable Y varies jointly with P and Q. If Y = 144 when P = 12 and Q = 8,
find Y when P = 15 and Q = 25.
4. The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume of a
gas is 200cc when the pressure is 1.6 liters per square centimeter, find the volume (to the nearest
tenth) when the pressure is 2.8 liters per sq centimeter.
5. In science, one theory of life expectancy states that the lifespan of mammals varies inversely to
the number of heartbeats per minute of the animal. If a gerbil's heart beats 360 times per minute
and lives an average of 3.5 years, what would be the life expectancy of a human with an average
of 72 beats per minute? Does this theory appear to hold for humans?