Mrs. Greenwood's Math Page - Home



Chapter 1 – Connections to Algebra Variables in AlgebraWarm-up: Ken-Ken problemVariables – letters that represent 1 or more #sExpression – letters & #s for a meaningful purpose; tells us what we are doing i.e. adding, multiplyingTerm pieces – -4x2“Evaluate” the expression – find the #s that expressionPg. 4-5 ex.Practice: Pg. 6 #11,17,23,27,31,33 Ken-Ken #2Exponents and PowersWarm-up: ACT #34,1746 is a power base raised to an exponentExponents tell us “how many are you multiplying” (remember how multiplication tells us how many groups we are adding)46 = 4?4?4?4?4?4Order of operations is importantPg. 10 ex. 3,4,5,6 Pg. 12 4-7Practice: Pg. 12 #11,21,31,35,43,57,61,67,69 Order of OperationWarm-up: Number Enigmas #31 & ACT #460Grouping symbols first, then“Please expect my dear aunt sally”“left to right rule” – if all the same operation**Division sign can act like a grouping symbol – Pg. 17Groups: do Pg. 18 ex 5 if calculators Practice: Pg. 19 #7,21,29,31,39,41,51,53,56Equations and InequalitiesWarm-up: Number Enigmas #58Equation – an expression with an “=”Solution – value that makes both sides of “=” equal. This is “checking a possible solutionSolve – find all the solutionsMental math – figuring it out in your head, estimatingInequality – read from the variable outGroup: Pg. 27 #10-13Practice: Pg. 27 #9,21,25,35,49,55,59,67,69,75 Start Review Pg. 54 1.1 – 1.4A Problem Solving Plan Using ModelsWarm-up: ACT example #1Make “Secret Math Code” – on index cardPaper clip pg. 29 for formulasModeling – real life that translates into algebraic expression or equationPg. 33 ex 2,3 pg. 35 1-12Practice: Pg. 35 #23,27,33,37,43,47,51,61-103Tables and GraphsWarm-up: Math logic –“20?”Pg. ex 1-3 show different graphs. ACT #12 Practice: Pg. 43 9,11,13,17,19 Logic “You’re Busted”-103An Introduction to FunctionsWarm-up: C=5/9 x (F0 – 32) what are the following temperatures in C0 ? 0, 212, 70, 90 F0 Wouldn’t it be nice to have a machine that would calculate the temps?Function – rule that establishes a relationship between 2 quantities (input, output)For each input, there is exactly one outputGroup: Pg. 49 #4-6, 10-12 X Y-136Input/output table shows the relationshipsDomain – input valuesRange – output valuesPg. 47 ex. 1-3Practice: Pg. 49 #17,21,25,37 Puzzle math “Identify Function”Review Pg. 54 1.5 – 1.7Chapter 2 – Properties of Real Numbers2.1 The Real Number LineWarm-up: What are all the types of numbers included in Real numbers and what is not?Number systems chartNumber line – neg, 0, pos integers, fractions, decimalsPlotting the point – putting a value where it should go on the number line<> plotting - 〇 ≤≥ plotting Pg. 64 ex. 4Opposite #s – equidistance from the center 0.Absolute value – always returns a positive valuePg. 66 ex. 8 Although velocity might be in a downward motion, speed is always + (doesn’t care about direction)Counter example – only need to find one example to prove something false. (a true statement must withstand all examples)Practice: Pg. 67 #11,23,31,39,49,53,63,75,77,79 2.2 Addition of Real NumbersWarm-up: 5 ? 19 ? 6 ? 1 = 4 (+ ÷ ÷or×)Properties of addition – Commutative (order), Associative (groups), identity (0), Zero property (opposite) Pg. 74 ex. 4 Practice: Pg. 75 #11,29,31 don’t use calculator #39,47,51,53,57Index card2.3 Subtraction of Real NumberWarm-up: Why is 5-(-3) = 8?Subtraction of a negative #: 7-(-2) = 7+2Subtraction of a term: -9 – 2x when x=2: -9 – (2)(2) = -9-4 = -13Use parenthesis!!Difference between highest and lowest: 2.26 – (-1.35) = 3.61 why? Show on a number line.Practice: Pg. 82 #9,19,35,39,49,61,71 & 73 but use the chart below:#73 was written in 2000. (1) What is the change in gold prices since then? (2) Find the year with the greatest change overall in the gold price. Estimate the change.2.4 Adding and Subtracting MatricesWarm-up: Group – decide on a #code for 4 colors (assign them each a unique #) Get the cubes in order – 1 color per sideMatrix – rectangular arrangement of numbers into rows and columns. A table that organizes data.Work with the color cubesEntry or element – each # in the matrixSize of matrix is "rows by columns” or – “3 by 4”Equal Matrices – entries in same position are equalAdd/subtract Matrices - +/- corresponding entries (can only combine if they are the same size matrix)Pg. 87 ex. 2,3,4Practice: Pg. 89 #9,15,19,21,23,25,29a Review Pg. 122 2.5 – 2.82.5 Multiplication of Real NumbersWarm-up: Solve 12 ?612 as many different ways as you canMultiplication – repeated addition + ? + = + - ? - = +Odd # of negative signs – answer will be negativeEven # of negative signs – answer will be positiveYou can determine the sign before or after you multiply the #sDisplacement - change in position of an object (+ or -)Pg. 95 ex. 4,5Practice: Pg. 96 #21,29,33,41,43,47,57,61,65,692.6 The Distributive PropertyWarm-up: Distributive property – distribute the work so 3?68 = 3(60+8)**Make sure you watch the signs! 4(x-3) = 4?x + 4?(-3) = 4x-12Pg. 101 ex. 2-4“Like terms” – same base and exponent“Constant terms” – 4x0 or just 4Simplifying equations – combining all like terms and distributing everything possiblePractice: Pg. 103 #9,17,23,35,43,65,73,75,79,922.7 Division of Real NumbersWarm-up: Reciprocal – x and 1xInverse prop of mult. - for every #x (≠ 0), there is a number 1x that when multiplied together = 1Division rule – a ÷ b = a ? 1bPg. 109 ex. 2Group: Pg. 109 ex. 3 a-c work: 10 ÷ (-15) and 16-29 and -2b7 ÷ 79Pg. 110 ex. 4 d=rate displacementtimeDomain changes with division – we have to worry about what is in the denominator of a fraction ( denominator ≠0)Practice: Pg. 111 #25,27,37,43,51,57,61,63 (d = rt ), 692.8 Probability and OddsWarm-up: Probability of an event – measure of the likelihood the event will occurOutcomes – different possible resultsFor N equally likely outcomes, probability = 1N that one will occur - a 5 coming up on a dice, P= 16Event – One action – roll a dice – and it’s collection of outcomesP = number of favorable outcomestotal number of outcomes even # on a dice = 36 = 12Experimental Probability – starts with a survey or experiment. Based on the results, predictions are made as to another event.Odds – equally likely outcomes, odds = # of favorable outcomes# of unfavorable outcomesIf you know the P of that event, odds = P event occurs1-P event occursAll examplesPractice: Pg. 117 #7,11,15,17,19,21,23,31 Review Pg. 122 2.5 – 2.8Chapter 3 – Solving Linear Equations3.1 Solving Equations Using Addition & Subtraction REVERSE THE + or – to get x aloneSame steps we used to evaluate -5 + (-4) – x 1. Simplify the expression using PEMDAS -5 – 4 – x2. Combine like terms-9 – xEquations have the added “=”-8 = n – (-4)-8 = n + 4-4 -8 = n +4 -4-12 = n actually find what n equals Try it: t – (-4) = 4|2| - (-b) = 6What is the equation for: The temperature rose 15 degrees to 70 F. What was the original temperature?Practice: Pg. 135 #25,29,31,33,35,41,43,49,51 Make sure you are writing the steps you are using for the first problem and any problems that use new or different steps!3.2 Solving Equations Using Multiplication & Division REVERSE the x or ÷ (after you handle the + or - )Sometimes x is not by itself: 2x = 4 or x2 = -42x = 4 (reverse 2?x by dividingx2 = -4 (reverse x ÷ 2 by multiplying)2x = -4 (first rewrite 2x = -41 ) Flip both fractions x2 = -14 Multiply by 2 to reverse the divisionSimilar Triangles & Ratios – ways we solve real life problems.Practice: Pg. 141 #7,11,13,21,33,35,41,51,59 Make sure you are writing the steps you are using!3.3 Solving Multi-Step Equations ALWAYS take care of the + or – before the x or ÷ to get x aloneThese are COMBINATION equations:13x+6=-8 7x – 3x – 8 = 24Practice: Pg. 148 #7,11,17,23,27,33,35,43,51,533.4 Solving Equations with Variables on Both Sides GET VARIABLES on one side and CONSTANTS on the other3(x + 2) = 3x + 6Don’t forget: Simplify by using PEMDAS, get x’s on one side (+ - ) then get x’s alone.Practice: Pg. 157 #13,15,17,23,31,37,41,43,453.5 Linear Equations and Problem Solving 1. Change the problem into algebra 2. Draw it! 3. Check it!Package size: Pg. 163 #6-8Total Girth = 108 = L + 4s (P of the square end)L = 36 inchesPractice: Pg. 163 #9,11,13,23-263.6 Solving Decimal Equations Rounding – sometimes common sense: $34.23 uses 2 decimal places(always check the next decimal place in line and see if you need to round - $34.236 would round to $34.24) EASY way to solve 7.2x-3.5=2.6-3.4x since each term has one decimal place, multiply whole equation by 10. 72x – 35 = 26 – 34x Every time you round, you put in a little bit of error. ≈ is the symbol for “approximately”Practice: Pg. 169 3,4,11,15,27,35,39,41,533.7 Formulas and Functions Common formulas: A =LW, A = ? bh, P=Irt, d=rt, S=L-rL, C=2πr, A = πr2 Function form: replace y with f(x) where x is the variable in the equation. If y = 2x+1, rewrite __________________If b = 2a +1 rewrite _________________If a=-1 what would the equation look like? Practice: Pg. 177 #3,5,11,15,23,25,29,353.8 Rates, Ratios and Percents Rates – comparison of 2 quantities with different units.d= rt exampleIf we solve for r (the rate) we get r = dt Rates are also used for conversions3 ft per yard = 3ft1yd30 miles per gallon = 30 miles1 galAverage rates – we are generalizing or estimating “per person” or “per day”***When you say the rate, you will know how to write it! Pg. 180 Example 1: you want to find “money spent per person” so $ spent in US totalpeople in US total=avg.$/personRatio – comparison of 2 numbers (rates are ratios)Looks like 412 or 41:2 (said “41 to 2)How many boys vs. girls in class? 14 boys3 girls or 14:3How many boys in class? 14 boys17 students or 14 boys:17 students Proportions: comparison of 2 ratiosAlways has “=” sign in the middlePg. 181 example 3: exchange $180 US for pesos. $1 = 9,990 pesos (that’s a rate of exchange). Set up 2 ratios with matching units: $1 US9,990 pesos=$180 USx pesos or “if $1 = 9,990 pesos how many pesos for $180?***Any time you have a proportion (2 fractions with an “=”), you can cross multiply to solve.Write the cross multiplication here Percents – always think “parts of 100” or “parts of the total” Pg. 182 example 5: what percentage of 15-17 year olds said they are dating? So, 15-17 year olds datingtotal # 15-17 surveyed***Remember! When you see the word “percent” the decimal needs to be moved sometime during the problem. If I were to model this problem: “what percentage” = x%“of” = “times”“15-17 year olds” = 482“are” or “is” = “=”“total” or “out of” is implied by percentx% ? 482 = 992 or x% = 992482 or x% = .48588… ***if in doubt what to do replace % with “over 100”x% = x100 so x100 = .48588… or x=.48588 ?100x= ________%***ALWAYS include UNITS with these problems!!Practice: Pg. 183 #5,9 (write the algebra!), 13,17,21,23,27,29,33,39Chapter 4 – Graphing Linear Equations & Functionsy4.1 Coordinates & Scatter PlotsIIIGraphing – done on a coordinate plane Ordered Pair – tells us where the point should gox(x,y) (2,3)x axis – shows horizontal movementy axis – shows vertical movementIVIIIOrigin – point (0,0) – where the x,y axis meetQuadrants – x,y axis cuts the graph into 4 pieces, we label those with Roman numeralsScatter Plots – taking data & plotting the individual points on a graph- sometimes we can see trends or patterns in the graphPractice: Pg. 206 #7,11,15,19,25,27,29,31,354.2 Graphing Linear EquationsGraph – visual solution to an equation in two variables (x,y). It show the set of all answers that are solutions of the equation.Linear equation – looks like y=mx+b (mslope or slant, bwhere the line hits the y-axis). It’s graph is always a line.How many points do you need before you can draw a line??xyy + 2 = 3xGet in the habit of putting your equation into y=mx+b formWhat are the easiest #s to plug in for x?y=3 Line that hits the y-axis at 3x=4 Line that hits the x-axis at 4Function form – just write the y=mx+b like this: f(x) = mx+b**Match the graphs on Pg. 215 #52-55Practice: Pg. 214 #7,13,17,18,21,31,41,43,634.3 Quick Graphs Using InterceptsEasiest way to graph – plug in x=0 and y=0 gives you the x,y intercepts**Pg. 220 Example 4 – Zoo tickets – how to graph something with many solutions.**Pg. 222 #41-43 match the graphs**Groups: Pg.222 School play, Marathon and Movie tickets problems Practice: Pg. 221 #4,9,15,21,31,35,47,574.4 The Slope of a LineLines are not always flat or straight up/down.Slanted lines have a “slope” – how far the line rises off “flat”NEVER MESS UP THE SLOPE – if you move up/down first (with the sign) then right.-23 slope would be down 2, right 3Slope - riserun or up/downright or change in ychange in x or ?y?x or y2-y1x2-x1Slope is the “rate of change” for your graphWhat is the slope of a horizontal line?What is the slope of a vertical line? Practice: Pg. 230 #5,11,13,25,35,39,47,51,534.5 Direct VariationDirect Variation – means that x & y are multiples of each othery=2x or y= ? x or y=7x or y=kxWhat point will this line always go through?**Pg. 236 Example 4 – Animal studies Practice: Pg. 237 #3,7,13,27,34-35,36-374.6 Quick Graphs Using Slope-Intercept FormSlope-intercept form – y=mx+b (m=slope, b=intercept)Parallel lines – have the same slope, but different y-interceptPerpendicular lines – have the negative reciprocal slopey=2x+4 y= - ? x -4**Pg. 245 #52-55 Practice: Pg. 244 #15,23,37,45,47,51,59,65,674.7 Solving Linear Equations Using GraphsGraphing a line shows us the “answer” or solution. The x-intercept is the solution to the equation, i.e. x=?Graph y=mx+b and find the x-interceptGraph both sides of the equation, i.e. 2.5 = .055t + 1.26 would produce 2 graphs: y=2.5 and y=.055t + 1.26. The intersection would be the answer.**Pg. 253 #11-13 Practice: Pg.253 15,23,25,33,37,474.8 Functions and RelationsA relation is an equation where an input can have more than one output value associated with it, i.e. x=y2 If x=4 then y=2 or -2A function is a relation between inputs and outputs where each x has only one y output value#1 RULE – a function is predictable! If you put a value in for x, you better only have one path that it will follow (output possibility)Which one is a function: (2,3) (3,3) (2,4) (4,5)(2,1) (3,2) (4,1) (5,2)(1,1) (2,1) (3,4) (4,2) Which is a function? Do the vertical line test.Function notation substitutes F(x) for yf(x) = 2x + 3 or g(a) = 4a + 1 When we plug a value in for the variable it becomes: g(2) = 4(2) + 1 f(2) = 9 or (2,9) is a point on the graphf(2) = 2(2) + 3 f(2) = 7 or (2,7) is a point on the graphGraph with slope and y-intercept (y=mx+b)** Pg. 260 #29-31 in class Practice: Pg. 259 9,13,15,16,18,21,35,37,45Chapter 5 – Writing Linear Equations5.1 Writing Linear Equations In Slope-Intercept FormSlope-intercept form – y=mx + b gives us slope (m) and intercept (b)Slope - ?y?x - the slant of the line“Model” is a function that is used to model real life situationsCompany T – initial charge of ______After that, they charge a constant rate of _________Company S – initial charge of ______After that, they charge a constant rate of _________What does the slope represent? _______________________What does the y-intercept represent? ___________________Who would benefit from each phone company’s service? Why? Practice: Pg. 276 9-11,15,20,23,28-29,31Extra Credit (on separate paper): #40 a,b,c5.2 Writing Linear Equations Given the Slope & a PointWhat are the 2 things we need to write the equation of a line?? ____________________________ ____________________________y = mx + b (when we find m & b) will work for every point on the lineIf we don’t know m or b, but we know a point on the line, we can find the missing piece: (1,4) is a point on the line, m=3y = mx + b (b is the only unknown)Find b:Check by graphing it!Find the equation of the line: x-intercept = 2, m = - 2/3 (don’t be tricked!)“rate”, “miles per hour”, “change in”, “increase”, “decrease” – clues that we are talking about the “slope” or rate of change between 2 #s. Practice: Pg. 282 2,4,9,13,23,25,29,31,33Extra Credit: #50 a,b,c5.3 Writing Linear Equations Given Two PointsSlope - ?y?x All we need is 2 points to find a lineWe can find more than just the equations of lines, we can draw shapes and tell whether or not they are parallelograms, squares, rectangles, etc.Find the equations of the 4 lines hereWhat can you tell about this shape??Practice: Pg. 288 1 (list the steps for each process) 2,3,4,7,11,19,27,2,39,47,495.4 Fitting a Line to Data“Line of best fit” – a line that is close to representing graphed dataTRICK: plot the points and then move your ruler around until you have as many points above the ruler as you do below it. Correlation – how well a line represents the data graphed.Positive – points follow closely a line with positive slopeNegative – points follow a negative slopeNo correlation – points don’t follow a straight line patternGroups: try Pg. 296 10-12 and 17-22 then worksheet Practice: Pg. 296 13,17-22, worksheet(s)5.5 Point-Slope Form of a Linear EquationPoint slope formula comes from m= ?y?x Rework:y1 – y2 = _______________Group activity: Pg. 301 Practice: Pg. 303 3,5,6,15,19,25,33,43 Extra Credit #57-595.6 The Standard Form of a Linear EquationStandard form – ax + by = c useful when equations have fractionsy = 15x- 75 becomes ________________________in standard form Practice: Pg. 311 3,9,13,16-17,27,29,35,63 5.7 Predicting With Linear ModelsSome data can be represented by a linear graph. Plot data on a graphCan the data be approximated with a line of fit?Draw the line (original points don’t have to be on the line)Pick two points on the line and determine the slopeFind the y-intercept and build the y=mx + b equationYou now have a linear model for predicting points that are not given originally Linear interpolation – estimating a point between 2 given pointsLinear extrapolation – estimating a point that lies to the right or left of all the points given originally Practice: Pg. 319 6-10,17-22Chapter 6 – Solving & Graphing Linear Inequalities6.1 Solving One-Step Linear InequalitiesLinear inequalities – have more than one answer or more points than fit on a straight line. They have a set of numbers that do not work, also.Number line is used to graph simple problemsArrow points at the small # & opens its mouth to eat the large oneStart from the variable and read out: x > 2 is “x is greater than 2”x < 2 & 2 > x are both “x is less than 2”when you graph, <> get 〇; ≤≥ get ?Rules for inequalities like 2x + 4 > 20Math rules are the same except for one thing: if you ? or ÷ by a negative #, then the <> flips-2x + 4 > -20 works out to: -2x > -24 x < 12As a group: Pg. 338 match the graphs 55-60 Practice: Pg. 337 7,15,19,20-21,29,41,45,53,616.2 Solving Multi-Step Linear Inequalities- still want the linear equations in y=mx+b format- arrange what’s given and then solve for y Practice: Pg. 343 3,5,11,15,19,27,32,37,43,446.3 Solving Compound Inequalities“and” inequalities look like -2<3x-8<10solve as 2 separate equations or solve all pieces at once“or” inequalities look like x<-1 or x>4 Practice: Pg. 349 2,3,9,15,21,25,29,33,376.4 Solving Absolute-Value Equations & Inequalities|x| = 8 x=±8 (inside expression can be positive or negative) |x-2| =5 break into 2 equations x-2 =5 and x-2 = -5 solve bothAbsolute value <> - are split into 2 equations also: < “or” / > “and” Practice: Pg. 356 3,10,13,17,27,31,38,44,53,636.5 Graphing Linear Inequalities in Two VariableLinear equations like 2x – 3y ≤ -2 has many (x,y) solutions - graph just like a line and then test to find the side to shade- <> will have a dotted line- ≤≥ will have a solid line (includes the points of the line) Groups: Pg. 364 43-48,61-63 Practice: Pg. 363 3,9,13-14,19,23,27,636.6 Stem-and-Leaf Plots & Mean, Median, ModeLike a plant – branches of data that help organize Central tendency – “most typical” value for a given set of dataMean – average (sum of #s / #of items)Median – in a set of ordered #s, it’s the centerMode – most frequent # (or n1-n2/2 if no center)Bell Curve – distribution of data in the shape of a bell Practice: Pg. 371 1,4,7,9,11,21,336.7 Box-and-Whisker Plots- all about medians Practice: Pg. 378 7,9, draw both the leaf/stem and the box/whisker for these: 11,17,19,25-29Chapter 7 – Systems of Linear Equations & Inequalities7.1 Solving Linear Systems by GraphingLinear systems – is there a point that works in more than one linear equation or model? Two linear equations can intersect at one (x,y) pointSystem can have no answers (line are parallel and never intersect)Systems can have all points that work in both equationsPractice: Pg. 401 2,3-6,11,15,17,19,23,31,35,437.2 Solving Linear Systems by SubstitutionSolve the easiest equation for one variable. Substitute what the variable equals into the second equation.- if y = 2x+1 then 2x + 3y = 7 would be 2x +3(2x+1) = 7Practice: Pg. 408 1-5,17,21,27,31,33 (day 1) 9,19,25,33,35,43,44 (day 2)-x + 3y = 18 x – y = 4 add down 2y = 14 y = 77.3 Solving Linear Systems by Linear CombinationPerform addition horizontally on equations Mixed Review & Quiz 1 Pg. 416Practice: Pg. 414 3,5,7,9,11,13,15,17,19,21 (day 1) 23,25,27,29,31,33,35,44,48 (day 2)7.4 Applications of Linear SystemsGraphingSubstitutionCombinationDecide which is easiest for each problem.Practice: Pg. 421 3-9,11,17,19,25,31,33 (day 1) 35-45 odds, 47,49-50, 52-54 (day 2)7.5 Special Types of Linear SystemsPractice: Pg. 429 1-5,7,11,12-17 all (day 1) 19,21,23,31,32,33,39,41 (day 2)7.6 Solving Systems of Linear InequalitiesPractice: Pg. 435 1-5, 7, 9-14 (day 1) 15-21 odds, 31, Quiz 2 oddsChapter 8 – Exponents & Exponential Functions8.1 Multiplication Properties of ExponentsADDing bases and exponents – only if they are the same animal!!2x5 + x5 = 3x5 - x5 is a group of 5 x’s all multiplied (like groups can be added). x5 + x4 are NOT like groups – both the exponent and base must match to add or subtract groupsOther math operations are done for like basesx5 ? a5 x and a are the bases, but are not alike so they don’t multiply together!3c4 only c is the base(3c)4 everything inside ( ) is the base2x4 ? 3x2 = 6 (multiply constants first) x4+2 or 6x6 Practice: Pg. 453 1-21 all (day 1) 23-59 odds (day 2) 8.2 Zero & Negative Exponentsa0 is always 1 anything to the zero power is 1a-n says “flip me!” and becomes 1an Practice: Pg. 459 3-10 all, 15-21 odds, 31-37 odds, 538.3 Division Properties of ExponentsRemember – like bases only!! 6563=65-3=62=36(3y)3=33y3=27y3Practice: Pg. 466 19-47 odds8.4 Scientific NotationUsed for very large or very small numbersAlways 1 decimal place when you are done276.3 = 2.763 x 102 (moved the decimal 2 left - # will get bigger).000458 = 4.58 x 10-4 (moved 4 right - # will get smaller)***easy to use exponent rules when using scientific notation (handle constant part normally and then the 10n part like all like basesPractice: Pg. 473 3-11 odds, 14, 15, 37-47 odds8.5 Exponential Growth FunctionRabbit populations, bacteria, our savings account – all things that can grow exponentially fast.y = C(1+r)t y=future amount C=starting amount r=rate t=timerate usually a %, so change to a decimalrate will be > 1if rate “triples” then (1+r) = 3 or r=2 or 200%Practice: Pg. 480 4-5, 7-21 odds8.6 Exponential Decay FunctionDepreciation of a car, devaluation of the dollar, sports tournament bracket, declining enrollment are examples of decay.y = C(1-r)t y=future amount C=starting amount r=rate t=timerate will be < 1Practice: pg. 488 5,8,9,13,17-21 odds, 27-30 allChapter 9 – Quadratic Equations & Functions9.1 Solving Quadratic Equations by Finding Square RootsQuadratics always have an x2 term – which means that we have to x2 to get it back to x = ___.Only x2 positive #sEach x2 yields a positive and a negative answer 9=±3Perfect squares have integer answers with no remainderIrrational Numbers – numbers written with the a (not the quotient of 2 integers)50=252=552=52 2 of the same a one “comes out of jail”Standard form for quadratic: ax2+bx+c = 0 (a,b,c are constants)a is the “leading coefficient” – in front of the highest exponent termPractice: Pg. 507 4-22 all (day 1) (day 2) 23-41 odds, 51-61 odds9.2 Simplifying RadicalsSimplest form – all perfect squares have been factored outRULES – not many for adding/subtracting Only add/subtract “like” terms, apples/oranges: 3+3 RULES – multiplication is in 2 stepsOutside x - constants outside can be multiplied togetherInside x - numbers can be multiplied or factored apartEx: 23?45=2?4?3?5=815 RULES – division can be all together or in parts455=9=33=3 or455=455=595=9=3 Practice: Pg. 514 4-7 all, 9, 11-17 odds, 23-29 odds(day 2) 31-49 odds, 50-54 all9.3 Graphing Quadratic FunctionsParabola – an x2 graph y = x2 Vertex – lowest or highest point of the curve - -b2a Axis of symmetry – “fold” or line that divides the parabola in ?Practice: Pg. 521 1-4 all, 5-19 odds, 20(day 2) 21-35 odds, 45-49 odds (day 3) 65-71 all, 769.4 Solving Quadratic Equations by Graphing“Solutions” or “roots” or “zeros” are what x equals when you solve for x (can be 2 answers)Where the graph crosses the x-axis1. If there is an x2 and an x term – get everything in standard form, find the vertex and some points on the graph2. If there is only an x or x2 term – solve for x and find the vertex Practice: Pg. 529 3-5 all, 7-17 odds, 18-20 all(day 2) 21-43 every other odd9.5 Solving Quadratic Equations by the Quadratic FormulaIf a quadratic isn’t easily factored, use the big nasty x=-b±b2-4ac2a Practice: Pg. 536 5,9,11,13,23,27,33,37,53Factoring – another method to try to find the zerosExtra worksheets on factoring9.6 Application of the Discriminantb2 – 4ac is the discriminant – it tells how many real zeros there will be+ means 2 solutions, - means no solutions, 0 means 1 solutionPerfect square means zeros are integersNegative means there are 2 imaginary roots (not seen on graph)Practice: Pg. 544 1-11 all, 15-17 all(day 2) 18-20 all, 21,23,25-26,319.7 Graphing Quadratic InequalitiesSame rules as graphing linear inequalitiesTest points to find out where the graph is shadedPractice: Pg. 4-6, 7-11 odds, 17-22 all9.8 Comparing Linear, Exponential & Quadratic ModelsPractice: Pg. 557 3-8 all, 9, 15, 17, 23-26 allChapter 10 – Polynomials & Factoring10.1 Adding & Subtracting PolynomialsPolynomials – “many terms” – we group terms by what variable/exponent mix they have. Group all x3 and all x2 separately, for instance.Classify terms by:Degree – if x3 is the highest term, it is a degree 3 or quarticTerms – how many terms there are (binomial = 2 terms)Always put in standard form (highest degree variable first and then step down as the equation goes to the right)Leading coefficient – variable in front of the highest term (includes sign)Practice: Pg. 579 1-6 all, 7-17 odds (day 1)31-35 odds vertical, 39-45 odds horizontal, 47-53 odds (day 2)10.2 Multiplying PolynomialsDistributive way: (x+2)(x-3) = x(x-3) + 2(x-3) = x2 -3x +2x – 6 = x2 –x -6 FOIL – multiply first terms, add rainbow multiplication, multiply last termsVertical way: used less, but just like multiplying 121 x 12Horizontal way: a lot like distributing, then combinePractice: Pg. 587 1-8 all, 9-17 odds (day 1)19-43 odds, 52,53 (day 2)10.3 Special Products of Polynomials“Special” cases that can make solving easier (a+b)(a-b) the middle term always drops out!(a+b)2 and (a-b)2 have a pattern“Box” multiplication – organizes the multiplicationPractice: Pg. 593 1-3 all, 7-35 odds (day 1)39-47 odds, 49-52 all (day 2)10.4 Solving Polynomial Equations in Factored FormFactoring is the hard part!! Once you factor, you can find the solutions easily by setting each one = 0 (this gives you where the polynomial crosses the x-axis – or where y=o)Called factors, roots, zeros, x-intercepts or solutionsZero-Product Property – in a product, if one of the terms = zero, then the entire answer is zero. Zero times anything = zero. So, find what values would make y=o.Practice: Pg. 600 1-17 all (day 1)19-29 odds, 35-47 odds (day 2)10.5 Factoring x2 + bx + cAlready learned easy factoring – now how do we do a little harder problems? Pull out GCF – greatest common factor2x3 +4x2 -6x = 2x(x2 +2x - 3) = 2x (x+3)(x-1) zeros o,-3,1Factor remaining quadraticFind all zerosUse x-box, discriminant or quadratic formula for harder ones (10.6)Practice will be on worksheets10.6 Factoring ax2 + bx + c = 0Quadratics where a > 1Find the factors of c and aTrial and error as you put together the factorsOr…use x-boxPractice: Pg. 614 5-17 all (day 1)19 – 47 every other odd (day 2) Good review: 48-63 all10.7 Factoring Special ProductsUse the special products (a+b)(a-b), (a+b)2 and (a-b)2 Practice: Pg. 622 4-17 all (day 1)19 – 59 every other odd (day 2) 10.8 Factoring Using the Distributive PropertyPolynomials are easier to factor if you pull out all the common pieces (GCF)Each factor that has a variable will give you a zero – or a place where the graph crosses the x-axisPractice: Pg. 629 15-49 every other odd, 55-57 (graph 2 problems) Graph 3x(x-7)(x+4) Chapter 11 – Rational Equations & Functions11.1 Ratio & Proportion11.2 Percents11.3 Direct & Inverse Variation11.4 Simplifying Rational Expressions11.5 Multiplying & Dividing Rational Expression11.6 Adding & Subtracting Rational Expressions11.7 Dividing Polynomials11.8 Rational Equations & FunctionsChapter 12 – Radicals & Connections to Geometry12.1 Functions Involving Square Roots12.2 Operations with Radical Expressions12.3 Solving Radical Expressions12.4 Completing the Square12.5 The Pythagorean Theorem & Its Converse12.6 The Distance & Midpoint Formulas12.7 Trigonometric Ratios12.8 Logical Reasoning: Proof ................
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