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Do Now!! In your composition notebook THOROUGHLY explain how you would solve the equation below: -3(x – 3) ≥ 5 – 4x.

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Presentation on theme: "Do Now!! In your composition notebook THOROUGHLY explain how you would solve the equation below: -3(x – 3) ≥ 5 – 4x."— Presentation transcript:

1 Do Now!! In your composition notebook THOROUGHLY explain how you would solve the equation below: -3(x – 3) ≥ 5 – 4x

2 EOC Review

3 Objective SWBAT make connections with content from Chapters 1 – 4.

4 1-1 Variables and Expressions A variable is a symbol, usually a letter, that represents values of a variable quantity. For example, d often represents distance. An algebraic expression is a mathematical phrase that includes one or more variables. A numerical expression is a mathematical phrase involving numbers and operations symbols, but no variables.

5 1-1 Variables and Expressions What is an algebraic expression for the word phrase 3 less than half a number x? You can represent “half a number x” as x/2. Then subtract 3 to get: x/2 – 3.

6 1-4 Properties of Real Numbers You can use properties such as the ones below to simplify and evaluate expressions. Commutative Properties: = 7 + (-2) 3 × 4 = 4×3 Associative Properties: 2× (14×3) = (2×14) × (12 + 2)= (3 + 12) + 2 Identity Properties: = × 1 = 21 Zero Property of Multiplication: -7 × 0 = 0 Multiplication Property of -1: 6 ∙ (-1) = -6

7 2-1 and 2-2 Solving One- and Two-Step Equations To solve an equation, get the variable by itself on one side of the equation. YOU can use properties of equality and inverse operations to isolate the variable. For example, use multiplication to undo its inverse, division.

8 2-1 and 2-2 Solving One- and Two-Step Equations What is the solution of _y_ + 5 = 8 2 _y_ + 5 – 5 = 8 – 5Subtract to undo + 2 _y_ = 3Simplify 2 2 * _y_ = 3*2Multiply 2 y = 6Simplify

9 2-3 Solving Multi-Step Equations To solve some equations, you may need to combine like terms or use the Distributive Property to clear fractions or decimals.

10 2-3 Solving Multi-Step Equations You do! What is the solution of 12 = 2x + _4_ – _2x_ ? = x

11 2-4 Solving Equations With Variables on Both Sides When an equation has variables on both sides, you can use properties of equality to isolate the variable on one side. An equation has no solution if no value of the variable makes it true. An equation is an identity if every value of the variable makes it true.

12 2-4 Solving Equations With Variables on Both Sides What is the solution of 3x – 7 = 5x + 19 ? 3x – 7 – 3x = 5x + 19 – 3xSubtract 3x -7 = 2x + 19Simplify -7 – 19 = 2x + 19 – 19Subtract = 2xSimplify -26 = 2xDivide by = x Simplify

13 2-5 Literal Equations and Formulas A literal equation is an equation that involves two or more variables. A formula is an equation that states a relationship among quantities. You can use properties of equality to solve a literal equation for one variable in terms of others.

14 2-5 Literal Equations and Formulas You Do! What is the width of a rectangle with area 91 ft 2 and length 7 ft? 13 = w

15 3-1 Inequalities and Their Graphs A solution of an inequality is any number that makes the inequality true. You can indicate all the solutions of an inequality on the graph a closed or dot indicates that the midpoint is a solution. An open dot indicates that the midpoint is not a solution.

16 3-1 Inequalities and Their Graphs What is the graph of x ≤ - 4? -4

17 3-2 Solving Inequalities Using Addition or Subtraction You can use the addition and subtraction properties of inequality to transform an inequality into a simpler, equivalent inequality.

18 3-2 Solving Inequalities Using Addition or Subtraction What are the solutions of x + 4 ≤ 5 ? x + 4 ≤ 5 x + 4 – 4 ≤ 5 – 4 Subtract 4 x ≤ 1Simplify

19 3-3 Solving Inequalities Using Multiplication or Division You can use the multiplication and division properties of inequality to transform an inequality. When you multiply or divide each side of an inequality by a negative number you have to reverse the inequality symbol.

20 3-3 Solving Inequalities Using Multiplication or Division What are the solutions of -3x > 12 ? -3x > 12 -3x < 12Divide each by Reverse Inequality Symbol x < -4Simplify

21 3-4 Solving Multi-Step Inequalities When you solve inequalities, sometimes you need to use more than one step. You need to gather the variable terms on one side of the inequality and the constant terms on the other side.

22 3-4 Solving Multi-Step Inequalites You do! What are the solutions of 3x + 5 > -1 ? x > -2

23 3-5 Working With Sets The complement of a set A (A’) is the set of all elements in the universal set that are not in A.

24 3-5 Working With Sets Suppose U = {1, 2,3,4,5,6} and Y = {2,4,6}. What is Y’? The elements in U that are not in Y are 1, 3, and 5. So Y’ = {1, 3, 5}

25 3-8 Unions and Intersections of Sets The union of 2 or more sets is the set that contains all elements of the sets. The intersection of 2 or more sets is the set of elements that are common to all the sets. Disjoint sets have no elements in common.

26 4-4 Graphing a Function Rule A continuous graph is a graph that is unbroken. A discrete graph is composed of distinct, isolated points. In real-world graph, show only points that make sense.

27 4-4 Graphing a Function Rule The total height h of a stack of cans is a function of the number n of layers of 4.5-in. cans used. This situation is represented by h = 4.5n. Graph the function.

28 4-5 Writing a Function Rule To write a function rule describing a real-world situation, it is often helpful to start with a verbal model of the situation.

29 4-5 Writing a Function Rule At a bicycle motocross (BMX) track, you pay $40 for a racing license plus $15 per race. What is the function rule that represents your total cost? Total cost = license fee + fee per race ∙ # of races C = ∙ r A function rule is C = r

30 4-6 Formalizing Relations and Functions A relation pairs numbers in the domain with numbers in the range. A relation may or may not be a function.

31 4-6 Formalizing Relations and Functions Is the relation [(0,1), (3,3), (4,4), (0,0)] a function? The x-values of the ordered pairs form the domain, and the y-values form the range. The domain value 0 is paired with two range values, 1 and 0. So the relation is not a function.

32 4-7 Sequences and Functions A sequence is an ordered list of numbers, called terms, that often forms a pattern. In an arithmetic sequence, there is a common difference between consecutive terms.

33 4-7 Sequences and Functions Tell whether the sequence is arithmetic …… The sequence has a common difference of -3, so it is arithmetic

34 Summary Our Objectives were that: SWBAT make connections with content from Chapters 1 – 4.

35 Homework In TEXTbook NC EOC Test Practice Chapter 1 pg. 74 – 76 Problems 1 – 20 even Chapter 2 pg Problems 1 – 18 even Chapter 3 pg. 228 – 230 Problems 1 – 20 even Chapter 4 pg Problems 1 – 14 even

36 Class Assignment In the paper back NC Algebra 1 EOC Test Workbook Complete Problems: 1-9; 11-12; 14-22; 24-26; 28-29; 31 – 34 On pages 1-7 SHOW ALL WORK If there is no computation to answer the question, EXPLAIN your reasoning for getting your answer choice. YOU MAY write in the text book.

37 EARLY BIRDS Review Released EOC test booklet and choose questions from the booklet you need to go over. Have these questions ready for Thursday’s review.

38 Objective SWBAT make connections with content from Chapters 5 – 8.

39 Do Now!! Factor each expression: 1)h 2 + 8h )d 2 – 20d )m m + 81

40 5-1 Rate of Change and Slope Rate of change shows the relationship between two changing quantities. The slope of a line is the ratio of the vertical change (the rise) to the horizontal change (the run). slope = rise = y 2 – y 1 runx 2 – x 1 The slope of a horizontal line is 0, and the slope of a vertical line is undefined.

41 5-1 Rate of Change and Slope What is the slope of the line that passes through the points (1, 12) and (6, 22)? Slope = y 2 -y 1 = 22 – 12 = 10 = 2 x 2 -x 1 6 – 1 5

42 5-2 Direct Variation A function represent a direct variation a direct variation if it has the form y = kx, where k ≠0. The coefficient k is the constant of variation.

43 5-2 Direct Variation Suppose y varies directly with x, and y = 15 when x = 5. Write a direct variation equation that relates x and y. What is the value of y when x = 9? y = kx 15 = k(5) 3 = k y = 3x The equation y = 3x relates x and y. When x = 9, y = 27

44 5-3, 5-4, and 5-5 Forms of Linear Equations The graph of a linear equation is a line. You can write a linear equation in different forms. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y- intercept. The point-slope form of a linear equation is y – y1 = m(x – x1), where m is the slope and (x1,y1) is a point on the line. The standard form of a linear equation is Ax+By = C, where A, B, and C are real numbers, and A and B are not both zeros.

45 5-6 Parallel and Perpendicular Lines Parallel lines are lines in the same plane that never intersect. Two lines are perpendicular if they intersect to form right angles.

46 5-6 Parallel and Perpendicular Lines Are the graphs of y = 4/3 x + 5 and y = -3/4x + 2 parallel, perpendicular, or neither? Explain. The slope of the graph of y = 4/3x + 5 is 4/3 The slope of the graph of y = -3/4x + 2 is -3/4 (4/3)∙(-3/4) = -1 The slopes are opposite reciprocals, so the graphs are perpendicular. What type of slopes do parallel lines have? The same slope

47 5-7 Scatter Plots and Trend Lines A scatter plot displays two sets of data as ordered pairs. A trend line for a scatter plot shows the correlation between the two sets of data. The most accurate trend line is the line of best fit. To estimate or predict values on a scatter plot, you can use interpolation or extrapolation.

48 6-1 Solving Systems by Graphing One way to solve a system of linear equations is by graphing each equation and finding the intersection point of the graph, if one exists.

49 6-1 Solving Systems by Graphing What is the solution of the system? y = -2x + 2 y = -0.5x – 3 The solution is (2, -2)

50 6-2 Solving Systems Using Substitution 6-3 Solving Systems Using Elimination You can solve a system of equations by solving one equation for one variable and then substituting the expression for that variable into the other equation. You can add or subtract equations in a system to eliminate a variable. Before you add or subtract, you may have to multiply one or both equations by a constant to make eliminating a variable possible.

51 6-5 and 6-6 Linear Inequalities and Systems of Inequalities A linear Inequality describes a region of the coordinate plane with a boundary line. Two or more inequalities form a system of inequality. The system’s solutions lie where the graphs of the inequalities overlap.

52 6-5 and 6-6 Linear Inequalities and Systems of Inequalities What is the graph of the system? y > 2x – 4 y ≤ -x + 2

53 7-1 Zero and Negative Exponents You can use zero and negative integers as exponents. For every nonzero number a, a 0 = 1. For every nonzero number a and any integer n, a -n = 1/a n. When you evaluate an exponential expression, you, you can simplify the expression before substitution values for the variables.

54 7-3 and 7-4 Multiplication Properties of Exponents To multiply powers, with the same bases, add the exponents a m ∙a n = a m+n, where a≠0 and m and n are integers. To raise a power to a power, multiply the exponents. (a m ) n = a mn, where a≠0 and m and n are integers. To raise a product to a power, raise each factor in the product to the power. (ab) n = a n b n, where a≠0, b≠0, and n is an integer

55 7-5 Division Property of Exponents To divide powers with the same base, subtract the exponents. a m = a m-n, where a ≠ 0 and m and n are integers a n To raise a quotient to a power, raise the numerator and the denominator to the power. _a_ n = _a n where a≠0 & b≠0; n is an integer b b n

56 7-3 and 7-4 Multiplication Properties of Exponents 7-5 Division Property of Exponents You Do a)3 10 ∙3 4 b)(x 5 ) 7 c)(pq) 8 d) 5x 4 3 z 2

57 7-6 Exponential Functions An exponential function involves repeated multiplication of an initial amount a by the same positive number b. The general form of an exponential function is y = a ∙ b x, where a≠0, b>0, and b≠1.

58 7-6 Exponential Functions What is the graph of y = ½ ∙5 x ? Make a table of values. Graph the ordered the pairs.

59 7-7 Exponential Growth and Decay When a > 0 and b >1, the function y = a ∙ b x models exponential growth. The base b is called the growth factor. When a >0 and 0

60 7-7 Exponential Growth and Decay You Do! The population of a city is 25, 000 and decreases 1% each year. Predict the population after 6 years. The population will be about 23,537 after 6 years.

61 Rule for an Arithmetic Sequence The nth term of an arithmetic sequence with first term A(1) and common difference d is given by: A(n) = A(1) + (n – 1)d Where n = nth term A(1) = the first term n = term number d= common difference

62 Arithmetic Sequence and Recursive Formula Arithmetic Sequence: – Sequence with a constant difference between terms. Recursive Formula: – Formula where each term is based on the term before it Recursive Formula for an Arithmetic Sequence:

63 Arithmetic Sequence and Recursive Formula If you buy a new car, you might be advised to have an oil change after driving 1000 miles and every 3000 miles thereafter. Then the following sequence gives the mileage when oil changes are required:

64 Arithmetic Sequence and Recursive Formula You DO Briana borrowed $870 from her parents for airfare to Europe. She will pay them back at the rate of $60.00 per month. Let a n be the amount she still owes after n months. Find a recursive formula for this sequence.

65 8-1 Adding and Subtracting Polynomials A monomial is a number, a variable, or a product of a number and one or more variables. A polynomial is a monomial or the sum of two or more monomials. The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. To add 2 polynomials, add the like terms of the polynomials. To subtract a polynomial, add the opposite of the polynomial.

66 8-1 Adding and Subtracting Polynomials You Do! What is the difference of 3x 3 – 7x and 2x 2 – 9x – 1? 3x 3 – 9x 2 + 9x + 6

67 8-2 Multiplying and Factoring You can multiply a monomial and a polynomial using the Distributive Property. You can factor a polynomial by finding the greatest common factor (GCF) of the terms of the polynomial.

68 8-3 and 8-4 Multiplying Binomials You can use algebraic tiles, tables, or Distributive Property to multiply polynomials. The FOIL method (First, Outer, Inner, Last) can be used to multiply two binomials. You can also use rules to multiply special case binomials.

69 8-3 and 8-4 Multiplying Binomials What is the simplified form of (4x + 3)(3x + 2)? = 12x x + 6

70 8-5 and 8-6 Factoring Quadratic Trinomials You can write some quadratic trinomials as the product of two binomials factors. When you factor a polynomial, be sure to factor out the GCF first.

71 8-5 and 8-6 Factoring Quadratic Trinomials What is the factored form x 2 + 7x + 12? List the pairs of factors of 12. Identify the pair with a sum of 7. x 2 + 7x + 12 = (x+3)(x+4) Factors of 12Sum of Factors 1, , 68 3,47

72 8-7 Factoring Special Cases When you factor a perfect-square trinomial, the two binomial factors are the same. a 2 + 2ab + b 2 = (a+b)(a+b) = (a+b) 2 a 2 – 2ab + b 2 = (a-b)(a-b) = (a-b) 2 When you factor a difference of squares of 2 terms, the 2 binomial factors are the sum and the difference of the two terms. a 2 – b 2 = (a+b)(a-b)

73 8-7 Factoring Special Cases What is the factored form of 81t 2 – 90t + 25 ? (9t – 5) 2

74 Summary Our objective was: SWBAT make connections with content from Chapters 5 – 8.

75 Homework Chapter 5 EOC pg. 354 – 356 Problems Chapter 6 EOC pg. 408 – 410 Problems 1 – 17 Chapter 7 EOC pg. 468 – 470 Problems odd Chapter 8 EOC pg. 528 – 530 Problems 1–21 odd

76 Assignment Resume yesterday’s assignment: In the paper back NC Algebra 1 EOC Test Workbook Complete Problems: 1-9; 11-12; 14-22; 24-26; 28-29; 31 – 34 On pages 1-7 SHOW ALL WORK If there is no computation to answer the question, EXPLAIN your reasoning for getting your answer choice. YOU MAY write in the text book.

77 Do Now Find the length of the hypotenuse. Write your answer in simplified radical form. 3√2

78 Objective SWBAT make connections with the content from Chapters

79 9-1 and 9-2 Graphing Quadratic Functions A function of the form y = ax2 + bx + c, where a ≠0, is a quadratic function. Its graph is a parabola. The axis of symmetry of a parabola divides it into two matching halves. The vertex of a parabola is the point at which the parabola line intersects the axis of symmetry.

80 9-1 and 9-2 Graphing Quadratic Functions What is the vertex of the graph of y = x 2 + 6x – 2? The vertex is (-3, -11)

81 9-3 and 9-4 Solving Quadratic Equations The standard form of a quadratic equation is ax 2 + bx + c = 0, where a ≠ 0. Quadratic equations can have two, one, or no real-number solutions. You can solve a quadratic equations by graphing the related function and finding the x-intercepts. Some quadratic equations can also be solved using square roots. If the left side of ax 2 +bx+c = 0 can be factored, you can use the Zero-Product Property to solve the equation.

82 9-3 and 9-4 Solving Quadratic Equations What are the solutions of 2x 2 – 72 = 0 ? x = + 6

83 9-5 Completing the Square You can solve any quadratic equation by writing it in the form x 2 + bx = c, completing the square, and finding the square roots of each side of the equations.

84 9-5 Completing the Square What are the solutions of x 2 + 8x = 513 x = 19 or x = -27

85 9-6 The Quadratic Formula and the Discriminant You can solve the quadratic equation ax 2 +bx+c = 0 where a ≠ 0, by using the quadratic formula x = -b + √(b 2 – 4ac) 2a Discriminant: is the expression under the radical sign in the quadratic formula, it tells how many solutions the equation has. x = -b + √(b 2 – 4ac) 2a The discriminant

86

87 9-6 The Quadratic Formula and the Discriminant How many real-number solutions does the equation x = 2x have? Because the discriminant is negative, the equation has no real-number solutions.

88 9-8 Systems of Linear and Quadratic Equations Systems of linear and quadratic equations can have 2 solutions, one solution, or no solution. These systems can be solved graphically or algebraically.

89 9-8 Systems of Linear and Quadratic Equations What are the solutions of the system? y = x 2 – 7x – 40 y = -3x + 37 (11, 4) and (-7, 58)

90 10-1 The Pythagorean Theorem Given the lengths of 2 sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side. Given the lengths of all 3 sides of a triangle, you can determine whether it is a right triangle.

91 10-2 Simplifying Radicals A radical expression is simplified if the following statements are true: The radicand has no perfect-square factors other than 1 The radicand contains no fractions No radicals appear in the denominator of a fraction

92 10-2 Simplifying Radicals What is the simplified for of √(3x) ? √(2) √(3x) = √(3x) ∙ √(2)Multiply by √2 √(2) √(2) √(2) √2 = √(6x) = √(6x)Simplify √(4)2

93 10-5 Graphing Square Root Functions Graph a square root function by plotting points or translating the parent square root function y = √x. The graph of y = √x + k and y = √x – k are vertical translations of y = √x. The graphs of y = √(x+h) and y = √(x – h) are horizontal translation of y = √x

94 10-5 Graphing Square Root Functions What is the graph of the square root function y = √(x-2) ? The graph of y = √(x – 2) is the graph of y = √x shifted 2 units to the right.

95 11-5 Solving Rational Expressions A rational expression is an expression that can be written in the form: polynomial polynomial A rationale expression is in simplified form when the numerator and denominator have no common factors other than 1.

96 11-5 Solving Rational Expressions What is the simplified form of x 2 – 9 x 2 – 2x -15 x 2 – 9_= (x-3)(x+3) Factor numerator x 2 – 2x -15 (x+3)(x-5) and denominator (x-3)(x+3) Divide out common (x-5)(x+3)factor (x- 3)Simplify (x-5)

97 11-6 Inverse Variation When the product of 2 variables is constant, the variables form an inverse variation. You can write an inverse variation in the form xy = k or y = k/x, where k is the constant of variation.

98 11-6 Inverse Variation Suppose y varies inversely with x, and y = 8 when x = 6. What is an equation for the inverse variation? xy = k General form of inverse variation 6(8) = kSubstitute 48 = kSolve for k xy = 48Write equation

99 11-7 Graphing Rational Functions A rational function can be written in the form f(x) = polynomial. The graph of a rational polynomial function in the form y = _a_ + c has a vertical x – b asymptote at x = b and a horizontal asymptote at y = c. A line is an asymptote of a graph if the graph gets closer to the line as x or y gets larger in absolute value.

100 12-2 Frequency and Histograms The frequency of an interval is the number of data values in that interval. A histogram is a graph that groups data into intervals and shows the frequency of values in each interval.

101 12-3 Measures of Central Tendency and Dispersion The mean of a data set equals: sum of data values Total number of data values. The median is the middle value in the data set when the values are arranged in order. The mode is the data item that occurs the most times. The range of set of data is the difference between the greatest and least data values.

102 12-4 Box-and-Whisker Plots A box-and-whisker plot organizes data values into four groups using the minimum value, the first quartile, the median, the third quartile, and the maximum value.

103 1-7 Midpoint and Distance in the Coordinate Plane You can find the coordinates of the midpoint M of AB with endpoints A(x 1,y 1 ) and B(x 2,y 2 ) using the Midpoint Formula. M( x 1 +x 2, y 1 +y 2 ) 2 2 You can find the distance d between two points A(x1,y1) and B(x2,y2) using the Distance Formula. d = √(x 2 -x 1 ) 2 + (y 2 -y 1 ) 2

104 1-8 Perimeter, Circumference, and Area Circles have a circumference C. The area A of a polygon or a circle is the number of square units it encloses. Circle: Circumference = ∏d or Circumference = 2∏r Area = ∏r 2

105 11-4, 11-5, 11-6 Volumes of Cylinders, Pyramids, Cones, and Spheres Cylinder: V = Bh or V = ∏r 2 h Pyramids: V = 1/3Bh Cones: V = 1/3Bh; or V = 1/3∏r 2 h Spheres: V= 4/3∏r 3

106 Summary Our objective was: SWBAT make connections with the content from Chapters

107 Homework Chapter 9 EOC Practice pg. 594 – – 20 all Chapter 10 EOC Practice pg. 646 – – 17 odd Chapter 11 EOC Practice pg. 708 – – 21 odd

108 Assignment In TEXTbook Pg. 780 – 785 End-Of-Course Assessment Complete Problems: 1 – 47 ALL Do not do problems: 3, 27, 45


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