Download Algebra 2nd Semester Review

Math Exam Review Semester 2
By Kyle Skarr and Ryan McLaughlin
Solving First Power Equations in one
Variable

Example problem 4x=24-2x
How to solve
 4x=24-2x
 +2x +2x
6x=24
 /6 /6
 X=4

Solving First Power Equations in
one Variable continued

Equations
containing
fraction
coefficients
–
Example
equation
x 3 x
 
5 4 2
Least common
denominator is 20
4x  15 10x
6x  15  x  2.5
Solving First Power Equations in one
Variable continued

Equations with
variables in the
denominators–
Example
10 5

x 2x
Multiply by 2x because it
is the least common
denominator
10
5
2x   2x 
x
2x
20  5  25
Solving First Power Equations in one
Variable continued

Special cases– Example 5x  25  25  5x
–
Example
8x  16  8x
xx
16  0
All real
No solution
Properties


Addition Property of Equality
If a=b then a+c = b+c and c+a = c+b
Properties


Multiplication Property of Equality
If a,b,c are any real numbers and a=b then
ca=cb and ac=bc
Properties


Reflexive Property of Equality
If a is a real number then a=a
Properties


Symetric property of equality
a=b then b=a
Properties


Transitive property of equality
If a=b and b= c then a=c
Properties


Associative property of Addition
(a+b) + c = a + (b+c)
Properties


Associative property of multiplication
(ab)c = a(bc)
Properties



Commutative Property of Addition
a+b = b+a
ab=ba
Properties

Commutative property of multiplication
45  5 4
Properties


Distributive Property
a(b+c) = ab+ac
Properties


Prop. Of opposites or inverse property of
addition
5+(-5)=0
Properties


Property of reciprocals or inverses prop.
Of multiplication
For every nonzero real number a, there is
a unique 1/a
1
a  1
a
and
1
a 1
a
Properties



Identity property of addition
There is a unique real number 0 such that for
every real number a
a+0=a 0+a=0
Properties


Identity property of multiplication
There is a unique real number 1 such that for
every real number a, a 1  a and 1 a  a
Properties

Multiplicative property of zero
a 0  0 and 0  a  0
Properties
Closure property of addition
 For all real numbers a and b:
a+b is a unique real number

Properties
Closure property of Multiplication
 For all real numbers a and b:
ab is a unique real number

Properties

Product of powers property
k k  k
5
4
9
Properties

Power of a product property
(ab)  a b
7
7 7
Properties

Power of a power property
(a )  a
2 4
8
Properties


Quotient of powers property
Subtract the exponents
5
x
3

x
2
x
Properties

Power of a quotient property
3
3
a
a
 3
b
b
()
Properties

Zero Power
Property
(4ab)  1
0
Properties

Negative power property
a
2
1
 2
a
Properties


Zero product property
If (x+3)(x-2)=0, then (x+3)=0 or (x-2)=0
Properties

Product of roots property
20  4  5
Properties

Quotient of roots property
45
3
5
Properties

Root of a power property
3
x
3
x
Properties

Power of a root property
( 7)  49  7
2
Solving first power inequalities in one
Variable

Examples of a first
power inequalities–
x5
When something is
equal to another number,
then you use a dark
circle, but when it isn’t
equal to, you use a a
non dark circle.
x2
5
2
Solving first power inequalities in one
Variable

Disjunction
–
A Disjunction uses the
word or

Example-
x  3 orx  1
1
3
Solving first power inequalities in one
Variable

Conjunctions
–
conjunctions include and


Example- x<3 and x>1
Or 3>x>1
1
3
Linear equations in two variables

Slope of lines
–
–
–
Horizontal: 0
Vertical: Undefined
Linear: rise over run
Linear equations in two variables

Equations of lines
–
–
–
–
Slope intercept form- Y=mx+b
Standard form: ax+by=c
vertical X= a constant
Horizontal y=a constant
Linear equations in two variables

In order to graph a line you need
–
–
–
A point and slope
Or two point
Or an equation
y  2x 1
slope
Y intercept
Y
intercept
Linear equations in two variables

How to find intercepts
–
–
X intercept- look for a point on the graph where y
equals zero
Y intercept- look for a point on the graph where x
equals zero
Linear equations in two variables
y y

How and when to use the point slope
formula–
You use the point slope formula when you don’t
know the y-intercept
Linear systems

Substitution Method–
Example-
Plug 15-x in for y
x  45  38
x  7
4x  45  3x  38
x  y  15
4 x  3 y  38
x  y  15
4 x  3(15  x)  38
y  15  x
Linear systems

Addition and
Subtraction
Method
(Elimination)
–
Example-
5 x  y  12
3x  y  4
Since the y’s
already
cross each
other out
there is no
need to use
the least
common
denominator
8 x  16  x  2
Linear systems

You can use
graphing but it only
gives an estimate
Linear systems

Check for understanding of terms–
–
–
Dependent system- Infinite set or all points (if
same line is used twice)
Inconsistent system-Null set (if they are parallel)
Consistent system-One point (if they cross)
Factoring

Methods
–
–
–
–
–
–
GCF- always look for the GCF first
Difference of Squares- used for binomials
Sum or Difference of cubes- used for binomials
PST- For trinomials
Reverse of FOIL- Trinomials
Grouping- Grouping
Factoring

GCF
–
Example
-
2x  8x  8
2
2( x 2  4 x  4)
2( x  2)( x  2)
Factoring

Difference of Squares
75 x  108 y
4
2
3(25 x  36 y )
4
2
3(5x – 6y) (5x  6y)
2
2
Factoring

Sum or difference of cubes
x3  y 3
( x  y)( x  xy  y )
2
2
Factoring

Perfect Square Trinomial
x  4x  4
2
( x  2)
2
Factoring

Reverse Foil–
Trial and error
ax 2  bx  c
( _  _ )( _  _ )
ax 2  bx  c
(_  _)(_  _)
ax 2  bx  c
(_  _)(_  _)
Factoring

Grouping–
Example-
b3  2b 2  ab  2a
b2 (b  2)  a(b  2)
(b2  a)(b  2)
Rational expressions

Simplify by factor and cancel-
x  x x( x  1)

x
x 1
x 1
2
Rational Expressions

Addition and Subtraction of rational
expressions
–
Addition-use LCM to cancel out the variable
a  2b  1

a  4b  5
6b  6
b 1
Rational Expressions

Subtraction of rational
expressions
–
–
Use LCM to cancel out the variablesExample-
6a  4b  5

6a  2b  1
2b  4
b2
6a  8  5
8 8
6 a  3
a
1
2
Rational Expressions

Multiplication and division of rational
expressions
–
Example-
2xy
2
3
4
4x y z
3
  2 xy
2 xyz
Quadratic equations in one variable

Solve by factoring
–
Example
x2  2x  8
( x  2)( x  4)  0
x2
x  4
Quadratic equations in one variable

Solve by taking the square root of each side
–
Examplex 2  49  0
49
x 2  49
x7
 49
Quadratic equations in one variable

Solve by completing the square
–
Examplex2  6 x  2  0
Take half
of x and
square it
x 2  6 ____  2 _____
x2  6 x  9  2  9
( x  3) 2  11
x  3  11
Quadratic equations in one variable

Quadratic
formula
–
Example
Quadratic
Equation
b  b2  4ac
2a
x 2  3 x  10  0
3
3
9  (40)
2
49
2
3 7
5
2
or
37
 2
2
Quadratic equations in one variable
b 2  4ac

What does the discriminant tell you?
–
Discriminant is the value of
b  4ac
2
Functions

What does f(x) mean?
–
–
–
–
F(x)= name of independent variable or argument
Usually equal to “Y”
Not all relations are functions (those that are
undefined)
Ex.
f ( x)  3 x  1  y
2
Functions




range and domain of a function
Domain- set of all x values
Range- set of all y values
Ex. f (0)  let x 0 Ex.(2) f ( x)  0 when y  0
f ( x)  5 x  10 x
f (0)  (0, 0)
2
0  5 x 2  10 x
5 x( x  2)  0
x 0x  2
Functions

Ordered pairs
–
–
Ex. (1,1) (5,5)
Slope equals
5 1
1
5 1
y  1x  b
1  1 b
b0
yx
Functions


Quadratic functions
How to graph a parabola
–
–
–
–
–
If A>0 then it opens up
If A<0 then it opens down
Vertex- is equal to a –b/2a to find x
Plug into f(x) to find y
Axis of symmetry- vertical through the vertex so
x= -b/2a
Functions

How to graph a parabola cont.
–
–
–
Y int. let x=0 or f (0)
X int. let y=0 or f (x) (0)
Factor and find solutions
Simplifying expressions with
exponents
A.) Product of powers
a a a
m
n
mn
3 4
ex.2  2  2
3
4
2
7
Simplifying expressions with
exponents
B.) quotient of powers
a a  a
m
n
mn
Ex. 2  2  2
4
2
4 2
2 4
2
Simplifying expressions with
exponents
C.) Power of a Power
(a )  a
m n
mn
Ex. (2 )  2  512
3 3
9
Simplifying expressions with
exponents
D.) Power of a Product
(ab)  a b
m
m m
Ex. (2 x)  2 x  16 x
4
4
4
4
Simplifying expressions with
exponents
E.) Power of a Quotient
m
a m a
( )  m
b
b
2
4 2 4
16 1
Ex. ( )  2 

8
8
64 4
Simplifying expressions with radicals
A.) Root of a Power
x

x
B.) Power of a Root
3
3
x x
2
Ex. 7  7
2
Simplifying expressions with
radicals
C.) Rationalizing the Denominator
–
Use the multiplication identity property
7
2
7 2
Ex.
( )
2
2 2
Word Problems

Example 1–
A baseball game has 1200 people attending. Adult tickets are 5
dollars an student tickets are two dollars. The total amount of
money made a tickets was 3660 dollars. The visiting team is
entitled to half of the adult tickets sales. How much money does
the visiting team get?
x  y  1200
y  x  1200
5 x  2 y  3660
5 x  2 x  2400  3660
3 x  1260
x  420 adults
other school gets $1050
Word Problems

Example 2–
Al left MUHS at 10:30 AM walking 4 mi/hr. Bob left MUHS
at noon running to catch up with Al. If Bob overtakes Al at
1:30 PM how fast was he running.
Step 1- label variables
rate time distance
mi
4
Al
3 hrs 12 mi
hr
Bob
b
mi
hr
3
hrs
2
3
b mi
2
Step 3- solve for the variable
2
2 3
 12   b
3
3 2
8b
Step 2- write an equation
Equal
distance
3
12  b
2
Step 4
Bob’s rate- 8
mi
hr
Word Problems

Example 3–
A serving of beef has 320 more calories than a serving of
chicken. The calories in 3 servings of beef is equal to the
calories in seven servings of chicken. Find the number of
calories in a serving of each meat.
chicken : c
beef : c  320
3c  960  7c
3c
 3c
960 4c

4
4
240  c
3(c  320)  7c
chicken : 240 calories
beef : 560 calories
Word Problems
6 w  6  34  w
w

Example 4–
w
5w  6  34
6
6
5w 40is 3 cm less then twice the width.
The length of a rectangle

5
5
The perimeter is 34 cm
more
then the width. Find the
w8
length and width of the rectangle?
2w-3
6w  6  34  w
w
w
2w-3
6 w  6  34  w
w
w
5 w  6  34
6
5w
40

5
5
w8
8cm
6
13 cm
Line of Best fit or Regression line


You use to the line of best fit to estimate
what the average is for the data
Your TI-84 calculator can determine the line
of best fit for you
Line of Best fit or Regression line

What is the best fit line here?
Draw a line on the
graph if you want.