1. GeometricGeometric ADVANCED ALG/TRIG Chapter 11 – Sequences and Series Sequences (an ordered list of numbers) Geometric R = common ratio Geometric mean = square root of product of 2 numbers Recursive formula a n = a n-1  r; a 1 given Explicit formula a n = a 1  r (n-1) Arithmetic D = common difference Arithmetic mean = sum of 2 numbers divided by 2 (the average) Recursive formula a n = a n-1 + d; a 1 given Explicit formula a n = a 1 + (n-1)d Series (sum of terms in a sequence) Geometric FINITE – ends; has a sum INFINITE Converges when |r|<1; approaches a limit Diverges when |r|> 1; does not approach a limit Sum of a Finite Geometric Series S n = a 1 (1-r n ) 1 - r Sum of an Infinite Geometric Series S n = a 1 1 - r Arithmetic Finite – ends Infinite – does not end… Summation Notation uses Sigma  ; has lower and upper limits Sum of a Finite Arithmetic Series S = n/2(a 1 + a n )

2. Conclusion about these features Cubes are Three Dimensional Features Main ideasFeatures CubeSquare Conclusion about these features Squares are One Dimensional Count the sides A cube has ______ sides. Count the sides A square has ______ sides. Sides Conclusion about this main idea The sides are shaped like squares Hold the block and count the corners A cube has _______ corners. Touch each corner with your pencil A square has ______ corners. Corners Conclusion about this main idea A cube has more corners Build something Make a design on paper What can you do with it? Conclusion about this main idea A square is easier to draw and use

3. Divide Divide both numbers and start again 3 3-3 =1 6 6-3 =2 Is there a number that will divide into both numbers? No, the fraction is reduced 4 7 Yes, keep going Is the numerator 1 less than the denominator? Yes, the fraction is reduced 5/6 No, keep going 5/3 = 1 2/3 Is the numerator a “1”? Yes, the fraction is reduced 1/6 No, keep going 2/4 = 1/2 How to tell if a fraction is reduced to lowest terms Reducing Fractions Is about …

4. 1. Identify the slope (m) and the y- intercept (b). Example: m = 2, b = -1 3. Use the slope the locate a second point. 4. Draw a line through the two points. 2. Graph the y- intercept on the y- axis. Example: 2. Find the y- intercept. Let x=0 and solve the equation for y. Example: 3(0) – 2y = 6 -2y = 6 y=-3 1. Find the x- intercept. Let y=0 and solve the equation for x. Example: 3x – 2(0) = 6 3x = 6 x=2 Slope-intercept form: y = mx + b Example: y= 2x - 1 Standard Form: Ax + By = C Example: 3x – 2y = 6 4. Draw a line through the two points. 3. Graph the x- intercept on the x- axis and the y- intercept on the y- axis. Graphing Linear Equations

6. SYSTEMS OF LINEAR INEQUALITIES First Inequality 1 Graph: a.Solve for y and identify the slope and y-intercept. b.Graph the y-intercept on the y- axis and use the slope to locate another point. -or- find the x and y intercepts and graph them on the x and y axis.) 2 Determine if the line is solid or dashed. 3 Pick a test point and test it in the original inequality. If true, shade where the point is. If false, shade on the opposite side of the line.) Second Inequality 1 Graph: a.Solve for y and identify the slope and y-intercept. b.Graph the y-intercept on the y- axis and use the slope to locate another point. -or- find the x and y intercepts and graph them on the x and y axis.) 2 Determine if the line is solid or dashed. 3 Pick a test point and test it in the original inequality. If true, shade where the point is. If false, shade on the opposite side of the line.) Solution 1 Darken the area where the shaded regions overlap. 2 If the regions do not overlap, there is no solution. Check 1 Choose a point in the darkened area. 2. Test in both original inequalities. 3. Correct if both test true. Copyright 2005 Edwin Ellis

8. These are the steps to … Ratio Method Factor Ax 2 + Bx + C Five Steps Copyright 2003 Edwin Ellis www.GraphicOrganizers.com Write the first ratio in the first binomial and the second ratio in the second binomial. 8(3x – 1) ( x – 1) Check using FOIL. Step 5 Write the ratio A/ Factor. Write the ratio Factor / C. Reduce. 3/1 1/1 are reduced. Step 4 Example: 24m 2 – 32m + 8 Factor out the GCF 8(3m 2 – 4m + 1) Write two binomials 8 ( ) ( ) Signs: + C signs will be the same sign as the sign of b. - C negative and positive 8( - ) ( - ) Find AC AC = 3 Find the factors of AC that will add or subtract (depends on the sign of c) to give u B. 3 and 1 are the factors of AC that will add (note c is +) to give B. Step 1 Step 2 Step 3 Why are these steps important? Following these steps will allow one to factor any polynomial that is not prime. Factoring a quadratic trinomial enable one to determine the x- intercepts of the parabola. Factoring also enables one to use the x- intercepts to graph. menu

9. Key Topic is about... Essential Details Divisibility two (2) - if a number ends with 0,2,4,6,8 dividing numbers that don’t have remainders five (5) - if a number ends with 0 or 5 ten (10) - if a number ends with 0 three (3) - if the sum of the digits can be nine (9) - if the sum of the digits can be It’s an easier way to divide BIG numbers! Lesson by Tuwanna McGee So what? What is important to understand about this? divided by 3 9

11. Word Walls New Word DefinitionPicture Knowledge Connection triangle 3 sides 3 angles “tri” means 3 Looks like a pyramid quadrilateral 4 sides 4 angles “quad” means 4 Looks like a box hexagon 6 sided box; 6 angles “hex” means 6 Looks like a stop sign pentagon 5 sided box; 5 angle “pent” means 5 Looks like a house

12. Graphing a Quadratic Function by Hand Is about … These options allow one to graph any quadratic function. A quadratic function models many of the physical, business, and area problems one see in real world situation. For example: maximize height of a projectile; maximize profit or revenue; minimize cost; Maximize/ minimize area or volume Main Idea Details 1Complete the square in x to write the quadratic function in the form f(x) = a(x – h) 2 + k. 2Graph the function in stages using transformations. Option 1 Main Idea Details 1Determine the vertex ( -b/2a, f(-b/2a)). 2Determine the axis of symmetry, x = _b/2a 3Determine the y-intercept, f(0). 4a) If the discriminant > 0, then the graph of the function has two x-intercepts, which are found by solving the equation. b) If the discriminant = 0, the vertex is the x- intercept. c) If the discriminant < 0, there are no x- intercepts. 5Determine an additional point by using the y- intercept the axis of symmetry. 6Plot the points and draw the graph. menu Option 2

14. menu Copyright 2003 Edwin Ellis Graphicorganizers.com Product property log­ b xy = log b x + log b y Power property log­ b N x = x log b N Quotient property log­ b x / log b y = log b x - log b y PROPERTIESPROPERTIES Exponential y = ab x Asymptote = x-axis, y = 0 y-intercept (0,1) For both graphs, relate to parent function and label intercepts. Logarithms Log b N = P Asymptote = y-axis, x = 0 x-intercept = (1,0) GRAPHSGRAPHS Exponential y = ab x b>1 = growth, 0<b<1 = decay Common Log = base 10; log Natural Log = base e; ln Logarithms Log b N = P B = base, N = #, P = power EXPRESSIONSEXPRESSIONS Exponential y = ab x Take log of both sides. Use properties while solving and simplify. Logarithms Log b N = P Write in exponential form. EQUATIONSEQUATIONS Advanced Alg/Trig Chapter 8 EXPONENTIALS AND LOGARITHMS

15. Find the vertex and make a table. General form: y = │mx + b│ + c So what? What is important to understand about this? The graph is always a “V”. A minus sign outside the absolute value bars cause the “V” to be flipped upside down. The m value in the equation affects the slope of the sides of the “V”. 1Graph the parent function. Its vertex is usually the origin. 2Translate h units left (if h is positive) or right (if h is negative). 3Translate k units up (if k is positive) or down (if k is negative). Example: y = -│2x - 4│+ 1 Parent function: -│2x│ Translate the parent function. Parent function: y = │mx │ Translated form: y = │mx ± h│ ± k Graphing Absolute Value Equations 1Find the vertex using (-b/m, c). 2Make a table of values. 3Choose values for x to the left and to the right of the vertex. Find the corresponding values of y. 4Graph the function. Example: y =│2x - 4│+ 1 Vertex (-(-4)/2, 1 ) = (2, 1) x y 3 │2(3) - 4│+ 1=│2│+1=3 1 │2(1) - 4│+1=│-2│+1=3

16. Hot Dog Gist & Details © 2003 Edwin Ellis www.GraphicOrganizers.com Exponents Zero Exponents n 0 = 1 -n 0 = -1 (-n) 0 = 1 Details Gist Multiplying Like Bases a m  a n = a m + n Details Gist Quotient to a Power (a/b) n = a n / b n Details Gist Dividing Like Bases a m = a m-n a n Details Gist Negative Exponents n -1 = 1/n 1/n -1 = n Details Gist Power to a Power (am) n = a mn Details Gist Product to a Power (ab) n = a n b n Details Gist Details Gist

18. Polynomials Algebraic expressions of problems when the task is to determine the value of one unknown number… X So what? What is important to understand about this? The concept is important for the AHSGE Objective I-2. You may add or subtract polynomials when determining or representing a customer’s order at a store. Main idea Monomials Example: 4x 3 Cubic 5 Constant Number of terms : One = “mon” 4x 3 Degree: Sum the exponents of its variables Main idea Binomials Example: 7x + 4 Linear 9x 4 +11 4 th degree Number of terms: Two = “bi” 7x + 4 Degree: The degree of monomial with greatest degree Main idea Trinomials Example: 3x 2 + 2x + 1 Quadratic Number of terms: Three =”tri” 3x 2 + 2x + 1 Degree: The degree of the monomial with greatest degree Main idea Polynomials Example: 3x 5 + 2x 3 + 5x 2 + x - 4 5 th degree Number of terms: Many (> three) =”poly” Degree: The degree of the monomial with greatest degree

19. Why are these steps important? Factor A 2 - C 2 “DOTS” = Difference of Two Squares Check for DOTS in DOTS3(x 2 + 4 ) ( x 2 - 4) 3(x 2 + 4 ) ( x + 2)(x - 2) Check using FOIL. Step 5 Write the numbers and variables before they were squared in the binomials. (Note: Any even power on a variable is a perfect square... just half the exponent when factoring) x 2 + 4 ) ( x 2 - 4)3( Step 4 Example: 3x 4 - 48 Factor out the GCF. 3( x 4 – 16) Check for : 1.) 2 terms 2.) Minus Sign 3.) A and C are perfect squares Write two binomials Signs: One + ; One - 3( + ) ( - ) Step 1 Step 2 Step 3 Following these steps will allow one to factor any polynomial that is not prime. Factoring a quadratic trinomial enables one to determine the x-intercepts of a parabola. Factoring also enables one to use the x-intercepts to graph.

20. Function notation /evaluating functions Domain Range Function – vertical line test Relations and Functions Point-slope form y – y 1 = m(x – x 1 ) Standard form Ax + By = C Slope-intercept form y = mx + b Linear Equations y = y x Constant of variation, k y = kx Direct Variation Equation of line of best fit Scattergram Line of best fit or trend line Linear Models h is horizontal movement and k is vertical movement Always graphs as a “V” y = |x – h| + k Vertex (h,k) Absolute Value Functions If test is true, shade to include point Graph line first – called the boundary equation Test a point not on the line – best choice is (0,0) Graphing Two- Variable Inequalities Functions, Equations, and Graphs

21. 1 Graph the linear equations on the same coordinate plane. 2 If the lines intersect, the solution is the point of intersection. 3 If the lines are parallel, there is no solution. 4 If the lines coincide, there is infinitely many solutions. 1Solve one of the linear equations for one of the variables (look for a coefficient of one). 2Substitute this variable’s value into the other equation. 3Solve the new equation for the one remaining variable. 4Substitute this value into one of the original equations and find the remaining variable value. 1Look for variables with opposite or same coefficients. 2If the coefficients are opposites, add the equations together. If the coefficients are the same, subtract the equations, by changing the sign of each term in the 2 nd equation and adding. 3Substitute the value of the remaining variable back into one of the orig. equations to find the other variable. 1 Choose a variable to eliminate. 2 Look the coefficients and find their LCD. This is the value you are trying to get. 3 Multiply each equation by the needed factor to get the LCD. 4 Continue as for regular elimination GRAPHING SUBSTITUTION ELIMINATIONELINIMATION VIA MULTIPLICATION The solution (if it has just one) is an ordered pair. This point is a solution to both equations and will test true if substituted into each equation. Method 1Method 4Method 3Method 2 SUMMARY Topic Solving Systems of Linear Equations Finding solutions for more than one linear equation by using one of four methods.

22. Main Idea Details Five (5) If a number ends with O or 5 Topic Details Divisibility Dividing numbers that don’t have remainders Main Idea Details Two (2) If a number ends with O, 2, 4, 6, 8 Main Idea Details Ten (10) If a number ends with O Main Idea Details Nine (9) If the sum of the digits can be divided by 9 Main Idea Details Three (3) If the sum of the digits can be divided by 3

23. Exponents are important for the graduation exam. Exponents are also used in exponential functions which model population growth, compound interest, depreciation, radio active decay, and the list goes on an on..... So what? What is important to understand about this? Exponent Rules Zero Exponents n 0 = 1 -n 0 = -1 (-n) 0 = 1 Negative Exponents n -1 = 1/n 1/n -1 = n Multiplying Like Bases a m  a n = a m + n Power to a Power (a m ) n = a mn Product to a Power (ab) n = a n b n Dividing Like bases a m = a m-n a n Quotient to a Power (a/b) n = a n / b n

24. Synonyms Intersection, Overlap, Common, Same Union, Every, All Main ideas Comparing…. So what? What is important to understand about this? Topic Graph  ----------│ ----------------│---------------  -4 0 1  ---------- │-----│----------│---------------  --2/3 0 5 And Or To solve real world problems involving chemistry of swimming pool water, temperature, and science. Notation 2x + 3 -5 -5 < 2x + 3 < 5 3x – 2 > 13 or 3x – 2 < - 4

25. TOPIC Order of Operations - What we should do in an equation situation… AHSGE: I-1 Apply order of operations Main Idea Details P LEASE P arenthesis Do all parentheses first. Main Idea Details E XCUSE E xponents Do all exponents second. Main Idea Details MYMY M ultiplication Do all multiplication from left to right next. Main Idea Details D EAR D ivision Do all division from left to right next. Main Idea Details A UNT A ddition Do all addition from left to right. Main Idea Details S ALLY S ubtraction Do all subtraction from left to right. Note: Complete multiplication and division from left to right even if division comes first. Complete addition and subtraction from left to right even if subtraction comes first.

26. Polynomial Products Multiplying Polynomials and Special Products So what? What is important to understand about this? AHSGE Objective: I -3. Applications include finding area and volume. Special products are used in graphing functions by hand. Punnett Squares in Biology. Main idea Polynomial times Polynomial Monomial times a Polynomial: Use the distributive property Example: - 4y 2 ( 5y 4 – 3y 2 + 2 ) = -20y 6 + 12y 4 – 8y 2 Polynomial times a Polynomial: Use the distributive Property Example: (2x – 3) ( 4x 2 + x – 6) = 8x 3 – 10x 2 – 15x + 18 Main idea Binomial times Binomial FOIL: F = First O = Outer I = Inner L = Last Example: (3x – 5 )( 2x + 7)= 6x 2 + 11 x - 35 Main idea Square of a Binomial (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 1 st term: Square the first term 2 nd term: Multiply two terms and Double 3 rd term: Square the last term Example: ( x + 6) 2 = x 2 + 12 x + 36 Main idea Difference of Two Squares DOTS: (a + b) ( a – b) = a 2 – b 2 Example: (t 3 – 6) (t 3 + 6) = t 6 – 36

27. These are the steps to … Factor A 2 - C 2 “DOTS” = Difference of Two Squares Five Steps Copyright 2003 Edwin Ellis www.GraphicOrganizers.c om Check for DOTS in DOTS3(x 2 + 4 ) ( x 2 - 4) 3(x 2 + 4 ) ( x + 2)(x - 2) Check using FOIL. Step 5 Write the numbers and variables before they were squared in the binomials. (Note: Any even power of a variable is a perfect square. Just half the exponent when factoring) 3(x 2 + 4 ) ( x 2 - 4) Step 4 Five Steps Example: 3x 4 - 48 Factor out the GCF. 3( x 4 – 16) Check for : 1.) Two terms 2.) Minus Sign 3.) A and C are perfect squares Write two binomials Signs: One + ; One - 3( + ) ( - ) Step 1 Step 2 Step 3 Name AHSGE : I – 4 Factor Polynomials. Why are these steps important? Following these steps will allow one to factor any binomial that is not prime. Factoring a quadratic trinomial enables one to determine the x- intercepts of a parabola.

28. These are the steps to … Factor Ax 2 + Bx + C Trial and Error. Five Steps Copyright 2003 Edwin Ellis www.GraphicOrganizers.c om Check using FOIL. 8(3x 2 – 3x – x + 1) =8(3x 2 – 4x + 1) =24x 2 – 32 x + 8 Step 5 If C is positive, determine the factor combination of A and C that will add to give B. If C is negative, determine the factor combination of A and C that will subtract to give B. Since C is positive add to get B: 8 (3x – 1) (x – 1) Step 4 Five Steps Example: 24m 2 – 32m + 8 Factor out the GCF 8(3m 2 – 4 m + 1) Write two binomials Signs: +C -- signs will be the same sign as the sign of B -C -- signs will be different: one negative and one positive 8( - ) ( - ) List the factors of A and the factors of C. A = 3C = 1 1, 3 1, 1 Step 1 Step 2 Step 3 Name AHSGE: I – 4 Factor Polynomials. Why are these steps important? Following these steps will allow one to factor any trinomial that is not prime. Factoring a quadratic trinomial enables one to determine the x- intercepts of the parabola.

29. These are the steps to … Factoring Polynomials Completely Five Steps Copyright 2003 Edwin Ellis www.GraphicOrganizers.c om Make sure that each polynomial is factored completely. If you have tried steps 1 – 4 and the polynomial cannot be factored, the polynomial is prime. Step 5 Four Terms: Grouping Group two terms together that have a GCF. Factor out the GCF from each pair. Look for common binomial. Re-write with common binomial times other factors in a binomial. Example: 5t 4 + 20t 3 + 6t + 24 = (t + 4) (5t 3 + 6) Step 4 Five Steps Factor out the GCF Example: 2x 3 - 6x 2 = 2x 2 ( x – 3) Always make sure the remaining polynomial(s) are factored. Two Terms: Check for “DOTS” A 2 – C 2 (Difference of Two Squares) Example: x 2 – 4 = (x – 2 ) (x + 2) See if the binomials will factor again. Check Using FOIL Three Terms: Ax 2 + Bx + C Check for “PST” m 2 + 2mn + n 2 or m 2 – 2mn + n 2. Factor using short cut. No “PST”, factor using trial and error. Example PST: c 2 + 10 c + 25 = (c + 5) 2 For an example of trial and error, see Trial/Error Method. Step 1 Step 2 Step 3 Name AHSGE: I – 4 Factoring Polynomials. Why are these steps important? Following these steps will allow one to factor any polynomial that is not prime. Factoring allows one to find the x-intercepts and in turn graph the polynomial.

30. Projectile Trajectory Gravity Vocabulary -- Define and give an example Acceleration due to gravity More Vocabulary Parabola Fill in the blank. Horizontal and vertical motions are ___________( independent/ dependent) of each other. When was the snowboard invented? What is “goofy foot”? What is regular foot? What is “hang time”? Snowboarding Where does hang time occur on a parabola? Which motion is affected by gravity? Name sports that involve parabolas. Snowboarding and its relationship to math Is about … Big Air Rules

31. Standard form: ax2 + bx + c = 0 Quadratic Formula Pythagorean Theorem c = longest side Distance Formula Two points: (x1,y1) (x2,y2) Midpoint Formula RADICAL FORMULAS Most of these formulas involve simplifying a radical. Topic Applications: Finding the center of a circle, finding the perimeter of a figure, finding the missing side of a right triangle, solving quadratic equations are all areas where these problems are used in real-world situations. Q: Why should I know how to use these formulas? A: They are on the exit exam as well as in geometry and higher level math courses.

32. Conditional An if-then statementIf an angle is a straight angle, then its measure is 180. p → q If p, then q. Term DefinitionExampleSymbolic form/read it Negation (of p) Has the opposite meaning as the original statement. An angle is NOT a straight angle. ~p NOT p. Term DefinitionExampleSymbolic form/read it Inverse Negates both the if and then of a conditional statement If an angle is NOT a straight angle, then its measure is NOT 180. ~p → ~q If NOT p, then NOT q. Term DefinitionExampleSymbolic form/read it Contra-positive Switches the if and then and negates both. If an angle’s measure is NOT 180, then it is NOT a straight angle. ~q → ~p If NOT q, then NOT p. Term DefinitionExampleSymbolic form/read it

33. Name _____ _____ _____ ____ Conditional An if-then statementIf an angle is a straight angle, then its measure is 180. The part following the if is the hypothesis and the part following the then is the conclusion. p → q If p, then q. Term DefinitionExampleSymbolic form/read it Converse Switches the if and then of a conditional statement If the measure of an angle is 180, then it is a straight angle. q → p If q, then p. Term DefinitionExampleSymbolic form/read it Bi-conditional The combination of a conditional statement and its converse; usually contains the words if and only if. An angle is a straight angle if and only if its measure is 180. p ↔ q p if and only if q. Term DefinitionExampleSymbolic form/read it Term DefinitionPictureKnowledge Connection

34. The ratio of the sides is 1 : 2: √3 Side opposite the 30º angle = ½ hypotenuse Side opposite the 60º angle = ½ hypotenuse times √3 The larger leg equals the shorter leg times √3 30º- 60º- 90º 45º- 45º- 90º The ratio of the sides is 1 : 1 : √2 Side opposite the 45º angle = ½ hypotenuse times √2 Hypotenuse = s √2 where s = a leg Is about … Special Right Triangles 30° 60° 45°

35. Math Curse by Jon Scieszka + Lane Smith How do you get to school in the morning? What time do you leave and what time do you arrive? If you were 15 minutes late leaving your house for school, what time would you arrive? What would you not have time to do if you were late and why? If there were approximately 1300 students in the school, how many fingers would there be? How would you write that in scientific notation? If an M&M is about a centimeter long, then how many M&Ms long is your foot? If he bought an 80cent candy bar that was on sale for 25% off, how much would he have to pay? Would it be more or less than the candy bar that was on for sale for 50% off? Explain. Title 1. 2. 3. 4.

36. Writing Linear Equations … Three Forms of a Linear Equation Main Idea Details 1. Given slope = m, and y-int= b 2. Substitute m and b into the equation. 3. Transform to Standard if necessary. Slope-Intercept Form Y = mx + b Main Idea Details 1. Given an equation in slope- intercept form. a. Eliminate fractions b. Add or Subtract 2. Given an equation in point- slope form. a. D istribute b. Eliminate Fractions c. Add or Subtract Standard Form Ax + By = C Main Idea Details 1. Given slope = m&point (x1, y1) Substitute m and the point into the equation. 2. Given 2 points a. Find m b. Substitute into point slope form c. Simplify/change to Slope- intercept or Standard form if necessary. Point-Slope Form y – y1 = m(x – x1) So what? What is important to understand about this? Many real life situations can be described by an equation, for instance, payroll deductions, temperature...