final exam review james mccroy algerbra-hartdke 14 may 2010

FINAL EXAM REVIEWFINAL EXAM REVIEW

JAMES mccroy JAMES mccroy

Algerbra-hartdkeAlgerbra-hartdke

14 may 201014 may 2010

Addition Property (of Equality)Addition Property (of Equality)

Multiplication Property (of Multiplication Property (of Equality)Equality)

Example: Example: If If a=ba=b then then a+c =b+ca+c =b+c

Example: Example:

If If a = ba = b then then a·c = b·ca·c = b·c..

Reflexive Property (of Reflexive Property (of Equality)Equality)

Symmetric Property (of Equality)Symmetric Property (of Equality)

Transitive Property (of Equality)Transitive Property (of Equality)

Example: Example: a=aa=a

Example: Example: If a=bIf a=b then b=athen b=a

Example: Example: If a=b and b=c then a=cIf a=b and b=c then a=c

Associative Property of Associative Property of AdditionAddition

Associative Property of Associative Property of MultiplicationMultiplication

Example: Example: c+(a+b)=a(c+b)c+(a+b)=a(c+b)

Example: Example: c(ab)=a(cb)c(ab)=a(cb)

Commutative Property of Commutative Property of AdditionAddition

Commutative Property of Commutative Property of MultiplicationMultiplication

Example: Example:

a+b=b+aa+b=b+a

Example: Example: a•b = b•a

Distributive Property (of Distributive Property (of Multiplication over AdditionMultiplication over Addition

Example: Example: a(b+c) a(b+c) ab+acab+ac

Prop of Opposites or Inverse Prop of Opposites or Inverse Property of AdditionProperty of Addition

Prop of Reciprocals or Prop of Reciprocals or Inverse Prop. of Inverse Prop. of MultiplicationMultiplication

Example: Example: 11+ -11=011+ -11=0

Example: Example: 16• =1

6

Identity Property of Identity Property of AdditionAddition

Identity Property of Identity Property of MultiplicationMultiplication

Example: Example: a+0=aa+0=a

Example: Example: a•1= a

Multiplicative Property of ZeroMultiplicative Property of Zero

Closure Property of AdditionClosure Property of Addition Example: Example:

Closure property of real number addition states that Closure property of real number addition states that the sum of any two real numbers equals another real the sum of any two real numbers equals another real

number.number. Example Example

Closure Property of MultiplicationClosure Property of Multiplication:: Closure property of real number multiplication states that the product of Closure property of real number multiplication states that the product of

any two real numbers equals another real number.any two real numbers equals another real number.

Example:Example:

a•0 = 0

2+5 = 7

2•7 =14

Product of Powers PropertyProduct of Powers Property

Power of a Product Property Power of a Product Property

Power of a Power PropertyPower of a Power Property

This property states that to multiply powers having the same base, add This property states that to multiply powers having the same base, add the exponents.the exponents.

Example: Example:

This property states that the power of a product can This property states that the power of a product can be obtained by finding the powers of each factor and be obtained by finding the powers of each factor and

multiplying them.multiplying them. Example:Example:

(3(3tt)4 = 34 · )4 = 34 · tt4 = 814 = 81tt44

This property states that the power of a power can be found by multiplying This property states that the power of a power can be found by multiplying the exponents.the exponents.

Example:Example:

n m n+ma •b = ab

m n m•n(a ) = a

Quotient of Powers PropertyQuotient of Powers Property

Power of a Quotient Property Power of a Quotient Property

This property states that to divide powers having the This property states that to divide powers having the same base, subtract the exponents.same base, subtract the exponents.

Example:Example:

This property states that the power of a quotient can This property states that the power of a quotient can be obtained by finding the powers of numerator and be obtained by finding the powers of numerator and

denominator and dividing them.denominator and dividing them. Example: Example: 2 2 2(a÷ b) = a ÷ b

44-3

3

a= a

a

Zero Power Property Zero Power Property

Negative Power Property Negative Power Property

If any number is raised to the 0 power the answer is If any number is raised to the 0 power the answer is automatically 0automatically 0

Example: Example:

If any number is raised to a negative power the If any number is raised to a negative power the

answer is the number’s reciprocalanswer is the number’s reciprocal

Example: Example:

0a =1

-bb

1a =

a

Zero Product Property Zero Product Property

If the product of two or more factors is zero, then at If the product of two or more factors is zero, then at least one of the factors must be zero least one of the factors must be zero

Example: Example:

If XY = 0, then X = 0 If XY = 0, then X = 0 oror Y = 0 Y = 0 oror both X and Y are 0. both X and Y are 0.

Product of Roots Property Product of Roots Property

The product of the roots of a quadratic equation is equal to The product of the roots of a quadratic equation is equal to thethe

constant term divided by the leading coefficient.constant term divided by the leading coefficient.Example:Example:

1 3(-3) = -

2 2

Quotient of Roots Property Quotient of Roots Property The square root of the quotient is the same as the quotient of the square roots

a a=

bb

Example :

Root of a Power Property Root of a Power Property

Power of a Root Property Power of a Root Property

Example: Example:

Example:Example:

2a

2a

Now you will take brief a quiz!Now you will take brief a quiz!Look at the sample problem and give Look at the sample problem and give

the name of the property illustrated.the name of the property illustrated.

1. a + b = b + a1. a + b = b + a

Click when you’re ready to see the answer.Click when you’re ready to see the answer.

Answer: Answer: Commutative Property (of Commutative Property (of

Addition)Addition)

Brief Quiz Cont… Brief Quiz Cont…

2. 2. If If a = ba = b then then a·c = b·ca·c = b·c


Answer: Answer: Addition Property (of Equality)Addition Property (of Equality)


3. a(bc)=(ab)c3. a(bc)=(ab)c


Answer: Answer: Associative Property of Associative Property of

MultiplicationMultiplication


4. 6(c+d)=6c+6d4. 6(c+d)=6c+6d


Answer: Answer: Distributive Property of Distributive Property of

Multiplication Over AdditionMultiplication Over Addition


5. 5.


Answer: Answer: Negative Power PropertyNegative Power Property

22

1a

a


6. 6.


Answer: Answer: Quotient of a Powers Property Quotient of a Powers Property

52

3

aa

a


7. ab=ba 7. ab=ba


Answer: Answer: Commutative Property of Commutative Property of

Multiplication Multiplication


8. 8.


Answer: Answer: Quotient of Roots PropertyQuotient of Roots Property

a a=

bb


9. 9.


Answer: Answer: Identity Property of MultiplicationIdentity Property of Multiplication

a•1= a


10. 10.


Answer: Answer: Closure Property of MultiplicationClosure Property of Multiplication

2•7 =14

11stst Power with Only One Power with Only One Inequality SignInequality Sign

11x

0 11

11stst Power- Conjunction Power- Conjunction

The word The word conjunctionconjunction means there are two conditions in a means there are two conditions in a statement that must be met.statement that must be met.

The greater than or less than signs will always be pointing The greater than or less than signs will always be pointing in the same direction in the same direction

Look out for statements that cannot be true, such as the Look out for statements that cannot be true, such as the following: 10 < x < 5 following: 10 < x < 5

Always uses the word “and”Always uses the word “and”

and

11stst Power- Disjunction Power- Disjunction

Work as two separate inequalitiesWork as two separate inequalities Always uses the word “or”Always uses the word “or”

or

Slopes of All Types of LinesSlopes of All Types of Lines

Positive slopePositive slope (when lines go uphill from left to right) (when lines go uphill from left to right)

Negative slopeNegative slope (when lines go downhill from left to right)(when lines go downhill from left to right)

Undefined slopeUndefined slope (when lines are vertical)(when lines are vertical)

Zero slopeZero slope (when lines are horizontal)(when lines are horizontal)

Equations of All Types of LinesEquations of All Types of Lines

Standard/General Form Standard/General Form • Ax + By = CAx + By = C, where , where AA > 0 and, if possible, > 0 and, if possible, AA, , BB, ,

and and C C are relatively prime integersare relatively prime integers Point-Slope FormPoint-Slope Form

• y=my=mxx+b or f(+b or f(xx)=m)=mxx+b +b (synonyms) (synonyms) The graph of this equation is a straight line The graph of this equation is a straight line The slope of the line is The slope of the line is mm The line crosses the y-axis at The line crosses the y-axis at bb The point where the line crosses the y-axis is called The point where the line crosses the y-axis is called

the y-interceptthe y-intercept

Inconsistent SystemInconsistent System4

2

-2

-4

Parallel lines don’t Parallel lines don’t intersect intersect • Null setNull set

Consistent SystemConsistent System

4

2

-2

-4

Line intersect at Line intersect at one pointone point

Dependent SystemDependent System

4

2

-2

-4

Infinite Set or All Infinite Set or All

Pts on the Line Pts on the Line • same line is used same line is used

twicetwice

• Is consistent but Is consistent but

supersedes itsupersedes it

Linear Linear SystemsSystems-Substitution Method-Substitution Method

Replace one variable with an equal expression Replace one variable with an equal expression

Steps:Steps:

• Look for a variable with a coefficient of one.Look for a variable with a coefficient of one.• Isolate That variable Isolate That variable • SubstituteSubstitute this expression into that variable in Equation this expression into that variable in Equation

B B • Solve for the remaining variableSolve for the remaining variable• Back-substitute this coordinate into Step 2 to find the Back-substitute this coordinate into Step 2 to find the

other coordinateother coordinate

Linear Systems Linear Systems Addition/Subtraction Addition/Subtraction Method (Elimination)Method (Elimination)

Combine equations to cancel out one Combine equations to cancel out one variablevariable

Steps:Steps:

• Look for the LCM of the coefficients on either x Look for the LCM of the coefficients on either x or y or y

• Multiply each equation by the necessary factor Multiply each equation by the necessary factor • AddAdd the two equations if using opposite signs the two equations if using opposite signs

(if not, (if not, subtractsubtract) ) • Solve for the remaining variable Solve for the remaining variable • Back-substitute this coordinate into any Back-substitute this coordinate into any

equation to find the other coordinate (Look for equation to find the other coordinate (Look for easiest coefficients to work with)easiest coefficients to work with)

Factoring Methods-GCFFactoring Methods-GCF

Used for any # of termsUsed for any # of terms

Factor GCF of equation Factor GCF of equation

DONEDONE

Factoring Methods-Sum/Diff of Factoring Methods-Sum/Diff of CubesCubes

Used for binomialsUsed for binomials

The opposite of the The opposite of the product of the cube roots product of the cube roots

DONEDONE

Factoring Methods-PSTFactoring Methods-PST

Used for trinomialsUsed for trinomials

If 1st & 3rd terms are squares If 1st & 3rd terms are squares and the middle term is twice and the middle term is twice the product of their square the product of their square roots roots

DONEDONE

Factoring Methods-Reverse FOILFactoring Methods-Reverse FOIL

TrinomialsTrinomials

Trial and Error Trial and Error

DONEDONE

Factoring Methods-GroupingFactoring Methods-Grouping

4 or more terms 4 or more terms • 2X2 Grouping 2X2 Grouping

Look for two small factorable groups Look for two small factorable groups Pull the final GCF out in front of the leftover factorsPull the final GCF out in front of the leftover factors DONEDONE

• 3X1 Grouping3X1 Grouping PST + perfect square PST + perfect square Write as Write as DONEDONE

2(glob) +PST

2(glob) +PST

Radical ExpressionsRadical Expressions

9x9x22 – 30x + 25 – 30x + 25 5x5x33 – 10x – 10x22 – 5x 75x – 5x 75x4+4+108y108y2 2

aa33 - b - b3 3 9x9x22 – 30x + 25 6x – 30x + 25 6x22 – 17x + 12 – 17x + 12

FunctionsFunctions

F(x) is a synonym for the variable y F(x) is a synonym for the variable y The domain of a function is the set of all possible The domain of a function is the set of all possible xx values values

which will make the function "work" and will output realwhich will make the function "work" and will output real y y--values values

The range of a function is the complete set of all possible The range of a function is the complete set of all possible resulting values of the dependent variable of a function, resulting values of the dependent variable of a function, after we have substituted the values in the domain after we have substituted the values in the domain

The graph of a quadratic function is a curve called a The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and parabola. Parabolas may open upward or downward and vary in "width" or "steepness" but they all have the same vary in "width" or "steepness" but they all have the same basic "U" shape. All parabolas are symmetric with respect basic "U" shape. All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the its axis of symmetry at a point called the vertex of the parabola. parabola.

Simplifying Expressions with Simplifying Expressions with ExponentsExponents

To simplify with exponents, you To simplify with exponents, you DON’T have to work only from the DON’T have to work only from the rules for exponents. It is often easier rules for exponents. It is often easier to work directly from the definition to work directly from the definition and meaning of exponents and meaning of exponents

Keep in mind The Negative Power Keep in mind The Negative Power Property and The Zero Power Property and The Zero Power PropertyProperty

Simplifying Expressions With Simplifying Expressions With RadicalsRadicals

Break down a number into its Break down a number into its smaller pieces, you can do smaller pieces, you can do the same with variables until the same with variables until the radical is a square rootthe radical is a square root

WORD PROBLEMS!!!!WORD PROBLEMS!!!! Eleven cards and three boxes of candy cost $35.39. Twelve Eleven cards and three boxes of candy cost $35.39. Twelve

cards and four boxes of candy cost $42.68. How much do cards and four boxes of candy cost $42.68. How much do two cards cost?two cards cost?

Lois rides her bike to visit a friend, She travels at 10mph Lois rides her bike to visit a friend, She travels at 10mph while she is there it starts to rain. Her friend drives her while she is there it starts to rain. Her friend drives her home in a car traveling at 25 mph. It takes Lois 1.5 hours home in a car traveling at 25 mph. It takes Lois 1.5 hours longer to go to her friends house than it does for her to longer to go to her friends house than it does for her to return home. How many hours did it take to ride to her return home. How many hours did it take to ride to her friends house? friends house?

Al's father is 45. He is 15 years older than twice Al's age. Al's father is 45. He is 15 years older than twice Al's age. How old is Al? How old is Al?

Karen is twice as old as Lori. Three years from now, the Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen? sum of their ages will be 42. How old is Karen?

Line of Best Fit or Regression LineLine of Best Fit or Regression Line

Is the BEST process of constructing a Is the BEST process of constructing a curve, or mathematical function, that curve, or mathematical function, that has the best fit to a series of data has the best fit to a series of data points, possibly subject to limitations points, possibly subject to limitations

Easy to do on a graphing calculator Easy to do on a graphing calculator because it does it automatically because it does it automatically

EXAMPLE:EXAMPLE:

final exam review james mccroy algerbra-hartdke 14 may 2010

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