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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 xx 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 45 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– x5 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. x2 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 b2 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 x2 x 4 Quadratic equations in one variable Solve by taking the square root of each side – Examplex 2 49 0 49 x 2 49 x7 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 37 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 b0 yx 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 mn 3 4 ex.2 2 2 3 4 2 7 Simplifying expressions with exponents B.) quotient of powers a a a m n mn 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 8b 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 w8 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 w8 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.