RATIONAL
FUNCTIONS
In the form:
Y=
1
k
xh
In the form: y=
p( x)
q ( x)
and NO common factors
Shape of graph: Hyperbola
In the form: y=
p( x)
q ( x)
and common factors
xintercepts: p(x)=0
zeros of numerator
Vertical asymptotes at x=h
(zeros of the denomin
Int Alg 5.3 Properties of Logs
Change of Base Thm: logax =
Ex) log232=
log 32
ln 32
or
log 2
ln 2
log x
log a
or
ln x
ln a
This is very useful!
Try these on your calc. The answer is 5.
Simplify these with the change of base thm.
1) log750
2) log22000
3) l
Transformations of the graphs of exponential functions: f(x) = ax
Horizontal: g(x) = ax+/ c
Vertical: g(x) = ax +/ c
Refl over x: g(x) = ax
Refl over y: g(x) = a(x)
Natural Base e
e=2.718281828.(neverends,neverrepeats)
So.f(x)=ex means2.71828x
Evaluat
Int Alg 5.5 Day 1
Formulas to know for day 1:
Continuous growth: A = Pert
Compound growth: A = P(1+ r )nt
n
Half Life: y = a()x/t where x = amt of time and t = length of halflife
1) How long will it take an investment to double at 7%
compounded annually?
Int Alg 5.2 Logs
When looking for an exponent: USE LOGS!
y = 2x and y = log2 x are inverse functions (reflections over y=x). You
use inverses to solve equations, which is why I say when you are solving an
equation for an exponent you need to take the log
Int Alg 5.5 Day 2 WOW!
y=abx is the exponential model we will use.
The book uses y=aebx .
1) The population of zhs in 2000 was 1565. In 2011 it is 1912.
Find an exponential model for this growth. Let 2000 be year 0.
Algebraically
Calculator
2) R = log I
I
Adding/Subtracting with unlike denominators
1
3
2
8
EX 3: Simplify and determine the values where x is undefined.
A
x2
x 3
3x 4
+
B
x
8
+ 2
x 2
x 4
C
2x 2
x2
2x
x 4
30 x 5
x 3
9
5
20140416 09:34:35
1/2
Chapter 8 Notes.pdf (5/12)
EX 4: Simplify the compl
Algebra 2 H
83 Adding and Subtracting Rational Expressions
complex fraction

Adding/Subtracting fractions with like denominators
Identify any values for which the denominator is undefined
Keep the denominator in the answer the same
2
5
Add/Subtract the
8.28.3 List of Rules
When adding or subtracting you need a common denominator.
Add/Subtract the top, the common denominator stays.
To get a common denominator: multiply the two/three denominators
together factoring can help.
When multiplying, multipl
Algebra 2 H
82 Multiplying and Dividing Rational Expressions
CAUTION: As you simplify a
rational expression, take note of
rational expression quotient of two polynomials
values that must be excluded.
The excluded values are those
that make the rational
Section 82/83 QUIZ REVIEW
Holt Alg 2H
Name_
Simplify each expression. Note any values where the variables are undefined.
1
4 x 2 6x
2x
2
2
3
1
1
x 2 x 2
4
5
3 3 1
4 8 2
6
1
a 1
3
1
a 1
8
7
1
u
1
v 1 u 2 v 2
a 1
1
1
a
3x 2 10 x 3
2x 2 5x 3
12 a 2 10
5
84 Graphing Rational Functions Activity
1. Opener Refrigerator Problem
In this exercise, you will work with a group or partner to determine a function based on real
world information. You will then interpret your function using its graph.
2. Sorting Gra
Algebra 2 H
85 Solving Rational Equations and Inequalities
Day 2: Solving Rational Inequalities
Solve for x. Make sure you consider all cases and check your answer.
3x 18
x
x 6
3x 18
3
11
20140425 12:17:23
1/2
Chapter 8 Notes.pdf (11/12)
8
x 5
15
x 3
Algebra 2 H
84 Rational Functions
Rational functions are all discontinuous but they can have many different shapes. Some
arehyperbola, somehave asymptotesandsomehaveholes.First,welllookat an
example of each one.
F(x)=
1
x 5
G(x) =
3
x2 4
x 1
H(x) =
x2 1
Algebra 2 H
85 Solving Rational Equations and Inequalities
Day 1: solving rational equations
rational equation
extraneous solution
rational inequality
To solve a rational equation, multiply each term of the equation by the LCD
(_). This should remove an