Download Example #2

Algebra 1
Unit 13
Chapter 12.1-12.9
Guided Notes
NAME __________________________
Period _____________
Teacher __________________
Date: ________________
12.1 Inverse Variation
Notes
Direct Variation: When y __________as x ___________.
Written as ___________ where k is a nonzero constant.
Inverse Variation: y varies inversely as x (y __________ as x ___________ or vice versa)
if there is some nonzero constant k where it can be written _____________
Constant of variation = ______
Example 1:
The owner of Superfast Computer Company has calculated that the time t in hours
that it takes to build a particular model of computer varies inversely with the number
of people p working on the computer. The equation pt = 12 can be used to represent
the people building a computer. Complete a table and draw a graph of the relation.
p
t
2
4
6
8
10
12
t
p
Example #2
Graph an inverse variation in which y varies inversely as x and y = 15 when x = 6
Use inverse variation equation:
xy=k
Substitute and solve for k
So equation becomes
xy= ___
Find values to graph:
x
y
-9
-6
-3
-2
0
2
3
6
9
The graphs of a direct variation equations are in the form y=kx and are linear.
The graphs of inverse variation equations are not linear in the form 8*3 and are curves
that approach a value.
In example #1, the p and t cannot be zero so the graph will never touch the p or t axis.
In example #2, neither x nor y can be 0 as it would be undefined. The graph approaches
these axis lines.
Example #3
Graph an inverse variation in which y varies inversely as x and y = 1 when x = 4
Use inverse variation equation:
xy=k
Substitute and solve for k
So equation becomes
xy= ___
Find values to graph:
x
y
-4
-2
-1
0
1
2
4
If (x1, y1) and (x2, y2) are solutions of an inverse variation, then x1y1=k and x2y2=k.
This means that ___________ = ___________. This is the product rule for inverse
variations and can be used to form proportions to solve inverse variation problems.
x1y1 = x2y2
product rule
x1y1 = x2y2
x2y1 x2y1
divide each side by x2y1
x1 = y2
x2
y1
simplify
Example #4
If y varies inversely as x and y = 4 when x = 7, find x when y = 14.
Let x1 = 7, y1= 4, and y2 = 14. Solve for x2
x1y1 = x2y2
Example #5
If y varies inversely as x and y = 8 when x = 6, find y when x = 4.
Let x1 = 6, y1= 8, and x2 = 4. Solve for y2
x1y1 = x2y2
Example #6
Anna is designing a mobile to suspend from a gallery ceiling in art class. A chain is
attached eight inches from the end of a bar that is 20 inches long. On the shorter end
of the bar is a sculpture weighing 36 kilograms. She plans to place another piece of
artwork on the other end of the bar. How much should the second piece of art weigh if
she wants the bar to be balanced?
Date: ________________
12.2 Rational Expressions
Notes
Rational expressions are algebraic expressions that can be written as a
__________. The numerator and denominator are polynomials. Some
examples are:
Denominators of a fraction can not be ________.
Excluded Values – values of a variable that result in a ____________
of __________ and must be excluded from the ________ of the
variable.
Example #1
State the excluded value of each of the following:
3b-2
b+7
5
a-4
12x
3x-1
5a2+2
a2-a-12
We simplify rational expressions the same way we simplify fractions,
eliminate any common factors of the numerator and denominator.
Example #2
Simplify
-7a2b3
21a5b
-7a2b3 = (7a2b)(-b2)
21a5b
(7a2b) (3a3)
=
Example #3
Simplify
(-b2)
(3a3)
35xy2
14y2z
Example #4
Simplify x2 – 2x – 15
x2 – x - 12
The GCF of the numerator and
denominator is 7a2b
Divide out the GCF
Example #5
Simplify 3x – 15
x2 – 7x + 10
Example #6
Simplify x2 – 9
x2 + 6x – 27
Date: ________________
12.3 Multiplying Rational Expressions
Notes
Recall that to multiply rational numbers expressed as fractions, you multiply numerators
and multiply denominators, then simplify the result. We do the same for multiplying
rational expressions.
Example #1
Find 5ab3 * 16c3
8c2
15a2b
5ab3 * 16c3 = 80ab3c3
8c2
15a2b
120a2bc2
Example #2
Find 12xy2 * 27m3p
45mp2 40x3y
= 40abc2(2b2c)
40abc2(3a)
the GCF of the numerator and
denominator is 40abc2.
= 2b2c
3a
simplify
Example #3
Find
x-5
x
*
x2
x2-2x-15
Example #4
Find
a2 + 7a +10 *
a+1
3a + 3
a+2
Example #5
Find
b+3
4b -12
b2 –4b + 3
b2 – 7b - 30
*
Date: ________________
Section 12-4 – Dividing Rational Expression
Notes
Recall that to divide rational numbers expressed as fractions you multiply by the reciprocal
of the divisor. Keep Change Flip!
Example #1
Find
5x2
7
5x2
7
÷
÷
10x3
21
10x3 =
21
=
=
5x2 * 21
7
10x3
1
3
5x2 * 21
10x3
17
3
2x
Example #2
Find
n+1
n+3
÷
2n + 2
n+4
2x
Example #3
Find
6x4
5
÷
24x
75
Example #4
Find
÷
3m + 12
m+5
m+4
m–2
Example #5
Find
12x - 36
x-7
÷
(x - 3)
Example #6
Find
w2 - 11 w – 26
7
÷
w - 13
w+7
Example #7
In 1986, Jenna Yeager and Dick Rutan piloted an experimental aircraft named Voyager
around the world non-stop, without refueling. The trip took exactly 9 days and covered a
distance of 24,012 miles. What was the speed of the aircraft in miles per hour?
Date: ________________
Section 12-5 – Dividing Polynomials
Notes
DIVIDING POLYNOMIALS BY MONOMIALS: To divide a polynomial by a monomial,
divide each ___________ of the polynomial by the monomial.
Example #1
Find (
Example #2
Find
Example #3
Find
.
Example #4
Example #5
Use Long Division to find (
Use Long Division to find (
Date: ________________
Section 12-6 – Rational Expressions with Like Denominators
Notes
Adding Rational Expressions: To add fraction with ____________ denominators, simply
add the numerators, and write the sum over the common __________________. You can
add rational expressions with like _________________ in the same way.
Example #1 - Numbers in Denominator Find
Example #2 – Binomials in Denominator Find
Example #3 –
Find an expression for the perimeter of rectangle PQRS.
P
Q
S
R
Subtract Rational Expressions: To subtract rational expressions with like ____________,
subtract the numerator and write the difference over the ______________ denominator.
Recall that to subtract an expression, you ________________ its additive
____________________.
Example #4 – Subtract Rational Expressions –
Find
Example #5 – Inverse Denominators Find
Date: ________________
Section 12-7 – Rational Expressions with Unlike Denominators
Notes
Add Rational Expressions: The___________ _____________ ________________
(LCM) is the least number that is a common multiple of two or more numbers.
Example #1 – LCM of Monomials –
Find the LCM of
and 18
.
Example #2 – LCM of Polynomials Find the LCM of
Recall that to add fractions with _____________ denominators, you need to rename the
fractions using the least common ________________ (LCM) of the denominators, known
as the least common ______________(LCD).
KEY CONCEPT – ADD RATIONAL EXPRESSIONS
STEP 1 –
STEP 2 –
STEP 3 –
STEP 4 –
Example #3 – Monomial Denominators –
Find
Example #4 – Polynomial Denominators –
Find
Example #5 – Binomials in Denominators –
Find
Example #6 – Polynomials in Denominators Find
Date: ________________
Section 12-8 – Mixed Expressions and Complex Fractions
Notes
A number like
is a ________________ number because it contains the sum of an
_________, 2, and a fraction,
. An expression like 3 +
is called a _______________
_________________ because it contains the sum of a monomial, 3, and a rational
expression
Changing mixed expressions to rational expressions is similar to changing
mixed number to improper fractions.
Example #1 – Mixed Expression to Rational Expression –
Simplify
3+
KEY CONCEPT:
Any Complex Fraction ---------, where b
, can be expressed as -------.
SIMPLIFY COMPLEX FRACTIONS: If a fraction has one or more fractions in the
_______________ or _______________, it is called a _________________
_________________. You simplify an algebraic complex fraction the same way you
simplify a ________________ complex fraction.
Example #2 – Complex Fraction Involving Numbers –
If Katelyn has
pounds of cookie dough, and the average cookie requires
how many cookies can she make?
of dough,
Example #3 – Complex Fraction Involving Monomials Simplify
x2y2
a
2
xy
a2
Example #4 – Complex Fraction Involving Polynomials Simplify
a - 15
a–2
a+3
Example #5 – Complex Fraction Involving Polynomials Simplify
b2
b+3
b–4
Date: ________________
Section 12-9 – Solving Rational Equations
Notes
______________ _______________- are equations that contain rational expressions. You
can use ______________ products to solve rational equation, but only when both side so
the equation are ___________ fractions.
Example #1 – Use Cross Products Solve
Example #2 – Use the LCD –
Solve
n – 2 __ n – 3 = 1
n
n–6
n
Example #3 – Multiple Solutions Solve - 4
+ 3 = 1
a+1
a
Rational Equations can be used to solve __________ _____________.
Example #4 – Work Problems
Abbi has a lawn care service. One day, she asked her friend Jo-El to work with her.
Normally, it takes Abbi two hours to mow and trim Mrs. Hart’s Lawn. When Jo-El works
with her, the job took only 1 hour and 20 minutes. How long would it have taken Jo-El to
do the job himself?
Rational equations can also be used to solve ______________ _______________.
Example #5 – Rate Problem
Two trains leave from different locations 9.46 miles apart and travel toward each other.
The first train leaves at noon and arrives at its destination 24 minutes later. The second
train leaves at noon at arrives at its destination 28 minutes later. At what time do the two
trains pass each other?
EXTRANEOUS SOLUTIONS: Multiplying each side of an equation by the LCD of two
rational expressions can yield results that are not solutions to the original equation. Recall
that such solutions are called ____________ __________________.
Example #6 – No Solution
Solve
3x
+ 6x – 9 = 6
x–1
x-1
Example #7 – Extraneous Solution
Solve
2n
+ n+3
= 1
2
1–n
n -1
Example #8 –
Solve
3
x-1
=
x+2
x–1