FINAL EXAM REVIEW JAMES mccroy Algerbra-hartdke 14 may 2010

Addition Property (of Equality) Multiplication Property (of Equality) Example: If a=b then a+c =b+c Example: If a = b then a·c = b·c.

Reflexive Property (of Equality) Symmetric Property (of Equality) Transitive Property (of Equality) Example: a=a Example: If a=b then b=a Example: If a=b and b=c then a=c

Associative Property of Addition Associative Property of Multiplication Example: c+(a+b)=a(c+b) Example: c(ab)=a(cb)

Commutative Property of Addition Commutative Property of Multiplication Example: a+b=b+a Example:

Distributive Property (of Multiplication over Addition Example: a(b+c) ab+ac

Prop of Opposites or Inverse Property of Addition Prop of Reciprocals or Inverse Prop. of Multiplication Example: =0 Example:

Identity Property of Addition Identity Property of Multiplication Example: a+0=a Example:

Multiplicative Property of Zero Closure Property of Addition Example: Closure property of real number addition states that the sum of any two real numbers equals another real number. Example Closure Property of Multiplication : Closure property of real number multiplication states that the product of any two real numbers equals another real number. Example:

Product of Powers Property Power of a Product Property Power of a Power Property This property states that to multiply powers having the same base, add the exponents. Example: This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them. Example: (3t)4 = 34 · t4 = 81t4 This property states that the power of a power can be found by multiplying the exponents. Example:

Quotient of Powers Property Power of a Quotient Property This property states that to divide powers having the same base, subtract the exponents. Example: This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. Example:

Zero Power Property Negative Power Property If any number is raised to the 0 power the answer is automatically 0 Example: If any number is raised to the 0 power the answer is automatically 0 Example: If any number is raised to a negative power the answer is the number’s reciprocal Example:

Zero Product Property If the product of two or more factors is zero, then at least one of the factors must be zero Example: If XY = 0, then X = 0 or Y = 0 or both X and Y are 0.

Product of Roots Property The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. Example: Quotient of Roots Property

Root of a Power Property Power of a Root Property Example: Example:

Now you will take brief a quiz! Look at the sample problem and give the name of the property illustrated. 1. a + b = b + a Click when you’re ready to see the answer. Answer: Commutative Property (of Addition)

Brief Quiz Cont… 2. If a = b then a·c = b·c Click when you’re ready to see the answer. Answer: Addition Property (of Equality)

Brief Quiz Cont… 3. a(bc)=(ab)c Click when you’re ready to see the answer. Answer: Associative Property of Multiplication

Brief Quiz Cont… 4. 6(c+d)=6c+6d Click when you’re ready to see the answer. Answer: Distributive Property of Multiplication Over Addition

Brief Quiz Cont… Click when you’re ready to see the answer. Answer: Negative Power Property

Brief Quiz Cont… Click when you’re ready to see the answer. Answer: Quotient of a Powers Property

Brief Quiz Cont… 7. ab=ba 7. ab=ba Click when you’re ready to see the answer. Answer: Commutative Property of Multiplication

Brief Quiz Cont… Click when you’re ready to see the answer. Answer: Quotient of Roots Property

Brief Quiz Cont… Click when you’re ready to see the answer. Answer: Identity Property of Multiplication

Brief Quiz Cont… Click when you’re ready to see the answer. Answer: Closure Property of Multiplication

1 st Power with Only One Inequality Sign

1 st Power- Conjunction The word conjunction means there are two conditions in a statement that must be met. The word conjunction means there are two conditions in a statement that must be met. The greater than or less than signs will always be pointing in the same direction The greater than or less than signs will always be pointing in the same direction Look out for statements that cannot be true, such as the following: 10 < x < 5 Look out for statements that cannot be true, such as the following: 10 < x < 5 Always uses the word “and” Always uses the word “and”

1 st Power- Disjunction Work as two separate inequalities Work as two separate inequalities Always uses the word “or” Always uses the word “or”

Slopes of All Types of Lines Positive slope (when lines go uphill from left to right) Positive slope (when lines go uphill from left to right) Negative slope (when lines go downhill from left to right) Negative slope (when lines go downhill from left to right) Undefined slope (when lines are vertical) Undefined slope (when lines are vertical) Zero slope (when lines are horizontal) Zero slope (when lines are horizontal)

Equations of All Types of Lines Standard/General Form Standard/General Form Ax + By = C, where A > 0 and, if possible, A, B, and C are relatively prime integersAx + By = C, where A > 0 and, if possible, A, B, and C are relatively prime integers Point-Slope Form Point-Slope Form y=mx+b or f(x)=mx+b (synonyms)y=mx+b or f(x)=mx+b (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 m The slope of the line is m The line crosses the y-axis at b The line crosses the y-axis at b The point where the line crosses the y-axis is called the y-intercept The point where the line crosses the y-axis is called the y-intercept

Inconsistent System Parallel lines don’t intersect Parallel lines don’t intersect Null setNull set

Consistent System Line intersect at one point Line intersect at one point

Dependent System Infinite Set or All Pts on the Line Infinite Set or All Pts on the Line same line is used twicesame line is used twice Is consistent but supersedes itIs consistent but supersedes it

Linear Systems -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 variableIsolate That variable Substitute this expression into that variable in Equation BSubstitute this expression into that variable in Equation B Solve for the remaining variableSolve for the remaining variable Back-substitute this coordinate into Step 2 to find the other coordinateBack-substitute this coordinate into Step 2 to find the other coordinate

Linear Systems Addition/Subtraction Method (Elimination) Combine equations to cancel out one variable Combine equations to cancel out one variable Steps: Steps: Look for the LCM of the coefficients on either x or yLook for the LCM of the coefficients on either x or y Multiply each equation by the necessary factorMultiply each equation by the necessary factor Add the two equations if using opposite signs (if not, subtract)Add the two equations if using opposite signs (if not, subtract) Solve for the remaining variableSolve for the remaining variable Back-substitute this coordinate into any equation to find the other coordinate (Look for easiest coefficients to work with)Back-substitute this coordinate into any equation to find the other coordinate (Look for easiest coefficients to work with)

Factoring Methods-GCF Used for any # of terms Used for any # of terms Factor GCF of equation Factor GCF of equation DONE DONE

Factoring Methods-Sum/Diff of Cubes Used for binomials Used for binomials The opposite of the product of the cube roots The opposite of the product of the cube roots DONE DONE

Factoring Methods-PST Used for trinomials Used for trinomials If 1st & 3rd terms are squares and the middle term is twice the product of their square roots If 1st & 3rd terms are squares and the middle term is twice the product of their square roots DONE DONE

Factoring Methods-Reverse FOIL Trinomials Trinomials Trial and Error Trial and Error DONE DONE

Factoring Methods-Grouping 4 or more terms 4 or more terms 2X2 Grouping2X2 Grouping Look for two small factorable groups Look for two small factorable groups Pull the final GCF out in front of the leftover factors Pull the final GCF out in front of the leftover factors DONE DONE 3X1 Grouping3X1 Grouping PST + perfect square PST + perfect square Write as Write as DONE DONE

Radical Expressions 9x 2 – 30x x 3 – 10x 2 – 5x 75x y 2 a 3 - b 3 9x 2 – 30x x 2 – 17x + 12

Functions 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 x values which will make the function "work" and will output real y- values The domain of a function is the set of all possible x values which will make the function "work" and will output real y- values The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness" but they all have the same basic "U" shape. All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness" but they all have the same basic "U" shape. All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.

Simplifying Expressions with Exponents To simplify with exponents, you DON’T have to work only from the rules for exponents. It is often easier to work directly from the definition and meaning of exponents To simplify with exponents, you DON’T have to work only from the rules for exponents. It is often easier to work directly from the definition and meaning of exponents Keep in mind The Negative Power Property and The Zero Power Property Keep in mind The Negative Power Property and The Zero Power Property

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

WORD PROBLEMS!!!! Eleven cards and three boxes of candy cost $ Twelve cards and four boxes of candy cost $ How much do two cards cost? Eleven cards and three boxes of candy cost $ Twelve cards and four boxes of candy cost $ How much do two cards cost? Lois rides her bike to visit a friend, She travels at 10mph 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 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 friends house? Lois rides her bike to visit a friend, She travels at 10mph 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 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 friends house? Al's father is 45. He is 15 years older than twice Al's age. How old is Al? Al's father is 45. He is 15 years older than twice Al's age. How old is Al? Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen? Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen?

Line of Best Fit or Regression Line Is the BEST process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to limitations Is the BEST process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to limitations Easy to do on a graphing calculator because it does it automatically Easy to do on a graphing calculator because it does it automatically EXAMPLE: EXAMPLE: