Advanced Algebra SS Semester 2 Final Exam Study Guide Mrs. Dunphy

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1 Advaced Algebra SS Semester 2 Fial Exam Study Guide Mrs. Duphy My fial is o at Iformatio about the Fial Exam The fial exam is cumulative, coverig previous mathematic coursework, especially Algebra I. All problems will be multiple choice with oe correct aswer out of 4. You will be allowed to use a otecard (3 x 5 or 4 x 6 ) ad you may put aythig o it. You will tur i the idex card o the day of the fial exam. Calculators are permitted o the Guide ad o the Fial Exam. The study guide below has all of the topics we have covered ad you will be workig selected review problems. The problems may ot cover every cocept, so a good strategy is to study those topics you do ot immediately kow. You will be workig o the study guide review problems every day at school ad for homework. Each day we will review ay questios that you have geerated from the day before, ad the you will work o the ext chapter test. All Exam Prep work is to be doe o separate paper with oly the aswer writte i the space provided. O the Fial Exam, write the aswer correspodig to the letter of the correct aswer with BLOCK CAPITAL LETTERS. You will tur i the packet with all correspodig work o the day of the fial exam. The packet will be worth 10% of your fial exam grade. The fial exam accouts for 20% of your total semester average. While your grade is ot just a umeric computatio, your average does play a huge part i determiig your grade. Study Aids/Resources Notes Chapter quizzes/tests Homework assigmets Textbook (Chapters 6, 7, 8, 10, 11, ad 15) Peers Iteret Resources such as Kha Academy ( Fial Exam Dates & Times Period 2: Tuesday, May 22 d at 10:30 a.m. (You may start at 10:05 but you MUST fiish by 12:15) Period 3: Wedesday, May 23 rd at 8:15 a.m. (You may start at 8:00 but you MUST fiish by 10:00) Period 4: Wedesday, May 23 rd at 10:30 a.m. (You may start at 10:05 but you MUST fiish by 12:15) You MUST be preset o the specified day ad time. There will NOT be ay make-ups! You must retur your textbook o the day of your exam. BRING A PENCIL, YOUR TEXTBOOK, COMPLETED EXAM REVIEW PACKET WITH ANSWER SHEET, AND SOMETHING TO DO WHEN YOU FINISH!!!

2 Chapter 6: Irratioal ad Complex Numbers Sectios 6-1 ad 6-2: Simplifyig Radicals (square, cube, ad higher order roots) o A solutio of a equatio x = b is called a th root of b. o The radical b deotes the pricipal th root of b. ì ( b) ïb if is odd = b b = í îï b if is eve o A expressio cotaiig th roots is i simplest radical form if: No radicad cotais a factor (other tha 1) that is a perfect th power, ad Every deomiator has bee ratioalized, so that o radicad is a fractio ad o radical is i a deomiator. Sectios 6-3 ad 6-4: Arithmetic with Radical Expressios (+, -, x, ) o Radical expressios with the same idex ad radicad ca be added i the same way as like terms. o Biomials cotaiig radicals are multiplied usig the BOX ad/or FOIL method. a b +c d ad a b - c d are called cojugates. Whe multiplied, they act like Differeces of Squares (a 2 b c 2 d), so their product is always a ratioal umber. Sectio 6-5: Solvig Radical Equatios o To solve a radical equatio ivolvig square roots, first isolate the radical term ad the square both sides of the equatio. Repeat this process if ecessary. Be sure to check all possible solutios i the origial equatio to determie if ay extraeous roots were itroduced. Sectio 6-6: Ratioal ad Irratioal Numbers o A real umber that ca be expressed as a quotiet of two itegers is ratioal. Otherwise it is irratioal. o Covert a fractio ito a decimal ad idetify as termiatig or o-termiatig o Covert a termiatig or repeatig decimal ito a ratioal fractio. o Fid a ratioal ad a irratioal umber betwee two give umbers. Sectio 6-7: The Imagiary Number i o A complex umber is a umber of the form a + bi, where a ad b are real umbers ad i = -1. The umber a is called the real part of the complex umber, ad b is called the imagiary part. ( ) o Use the patter of i i = -1, i 2 = -1, i 3 = - -1, i 4 =1 to simplify expressio cotaiig imagiary umbers. o The complex cojugate of a + bi is a bi. The product of these complex cojugates is the real umber a 2 + b 2. o To simplify a expressio cotaiig square roots of egative umbers, first rewrite each radical usig i. o Kow how to simplify radical expressio that result i imagiary umbers or that ivolve imagiary umbers o Kow how to solve simple quadratic equatios that result i imagiary solutios Sectio 6-8: The Complex Number o Uderstad the etire hierarchy of umbers, startig with the complex set. o Add, subtract, multiply, ad divide complex umbers.

3 Chapter 7: Quadratic Equatios ad Fuctios Sectio 7-1: Solve a quadratic equatio ax 2 + bx + c = 0 by completig the square Sectio 7-2: The Quadratic Formula 2 o Be able to derive the quadratic formula from the equatio ax bx c 0 by completig the square. o Solve a quadratic equatio usig the quadratic formula. o Kow how to fid the roots of a quadratic equatio usig the quadratic formula. The coefficiets may be itegers, imagiary umbers, or irratioal (e.g. square roots). o Be able to approximate roots with a calculator ad correctly roud to a give umber of decimal places. o Kow how to setup ad solve word problems ivolvig quadratic equatios Sectio 7-4: Equatios i Quadratic Form o Kow how to solve a equatio i quadratic form usig a variable substitutio "y = ". For example, i the equatio x 4 3x = 0, let y = x 2 ad the equatio becomes y 2 3y + 5 = 0. Fid the y roots, ad the use the variable substitutio to fid the x roots. o If the variable substitutio cotais a radical, you MUST check for extraeous roots. They are easy to spot. Remember, x b has NO SOLUTION for b > 0 because the square root represets a positive umber! o Also remember that the umber of roots is the same as the highest power of the polyomial. Sectio 7-5: Graphig y k = a(x h) 2 o Graph parabolic equatios i vertex form: y k ax h 2. This is really just the graph of 2 y ax startig at the "ew origi" ( h, k ), which is the vertex of the parabola. The vertex is the parabola shifted h uits horizotally ad k uits vertically. o Kow how to create a equatio i vertex form whe give iformatio about the parabola such as its vertex ad aother piece of iformatio (e.g. y-itercept, or a x-itercept). o Fid the vertex, axis of symmetry, x-itercepts, ad y-itercept whe give a equatio i vertex form. Sectio 7-6: Quadratic Fuctios 2 o Kow how to covert a geeral form fuctio f ( x) ax bx c ito a stadard form fuctio f ( x) a x h 2 k ad back ito vertex form y k ax h 2. o Kow how to fid the vertex, maximum or miimum, axis of symmetry, domai, rage, ad zeros of a quadratic fuctio i geeral form by covertig it ito stadard form (complete the square). Chapter 8: Polyomial Equatios Sectio 8-3: Dividig Polyomials o Divide polyomials usig log divisio o To divide oe polyomial by aother, fid the quotiet ad remaider usig the divisio algorithm: Divided Remaider = Quotiet +. Divisor Divisor o Use the divisio algorithm to fid missig parts of a divisio problem. Sectio 8-4: Sythetic Divisio o Divide a polyomial by a first-degree biomial x c usig sythetic divisio. Sectio 8-5: The Remaider ad Factor Theorems o Remaider theorem: Whe P(x) is divided by (x c), the remaider is P(c). o Evaluate polyomial fuctios at x = c usig sythetic substitutio o Factor theorem: If P(x) has (x r) as a factor, the r is a root of P(x) = 0. o Determie if a give biomial (x c) is a factor of P(x) by sythetic divisio o Create a polyomial equatio with iteger coefficiets usig give roots. o Solve a third-degree polyomial give oe of its roots usig sythetic divisio.

4 Chapter 10: Expoetial ad Logarithmic Fuctios Sectio 10-1: Ratioal Expoets m m o Express b b o Covert a radical expressio ito expoetial form o Covert a expoetial expressio ito radical form o Solve equatios i fractioal expoetial form usig ratioal roots b i radical form: m Sectio 10-2: Real Number Expoets o Simplify expressios havig irratioal expoets o Defie the expoetial fuctio: b x ad kow its properties: b 1, b > 0, oe-to-oe o Solve expoetial equatios usig the oe-to-oe property Sectio 10-4: Defiitio of Logarithms o Defie the logarithmic fuctio log b N = k as the iverse of the expoetial fuctio b k = N log Because these are iverses, we get b x x b x ad logb b x. o Recogize the logarithmic fuctio as givig the expoet for the base to geerate a umber N: if b k = N, the log b N = k. The log ad expoet fuctios "udo" each other! o Write a logarithmic expressio i expoetial form ad a expoetial expressio i logarithmic form o Simplify logarithms by writig i expoetial form o Solve simple logarithmic equatios Sectio 10-5: Laws of Logarithms o Use the three laws of logarithms (product rule, quotiet rule, ad power rule) to expad ad codese logarithm expressios o Solve logarithm equatios usig the oe-to-oe property of logarithms Evaluate a logarithm usig the three rules ad kow values of logarithms. Solve logarithmic equatios by applyig the three rules ad covertig ito expoetial form. Sectio 10-6: Applicatios of Logarithms o Evaluate commo logarithms by covertig to scietific otatio ad usig a table of logarithms. o Fid atilogarithms usig a table of logarithms. o Approximate radical ad expoetial expressios usig a calculator. o Solve logarithmic ad expoetial equatios by puttig ito calculator-ready form, the approximatig with a calculator. o Evaluate logarithms usig the chage of base formula. Sectio 10-7: Expoetial Growth ad Decay o Solve word problems related to compoud iterest: future value, time to double (or triple, etc.), ad amout to ivest. o Solve word problems related to doublig time or half-life: future amout ad time to double/half. Sectio 10-8: The Natural Logarithm Fuctio o Defie the umber e as the value that f x 1 ( ) 1 reaches whe x gets very large. x o Do everythig above for the logarithm base e, writte as l x. x

5 Chapter 11: Sequeces Sectio 11-1: Type of Sequeces o A sequece is a set of umbers arraged i a defied order accordig to their positio i the list. o Each value i the sequece is a term. o A fiite sequece has a limited umber of terms ad a ifiite sequece has a ulimited umber of terms. o Sequeces ca be arithmetic, geometric, or either. Sectio 11-2: Arithmetic Sequeces o If there is a commo differece d betwee ay two successive terms, the the sequece is arithmetic. o I ay arithmetic sequece with first term t 1 ad commo differece d, the th term is give by t = t 1 + d( -1). o The arithmetic mea of two umbers is simply the average of the two umbers. If there is more tha oe arithmetic mea betwee two umbers, fid them by creatig a system of liear equatios. Sectio 11-3: Geometric Sequeces o If ratio of successive terms is costat, the the sequece is geometric. The ratio is the commo ratio ad is deoted as r. o I ay geometric sequece with first term t 1 ad commo ratio r, the th term is give by t = t 1 r (-1). o The geometric mea of two umbers a ad b is ab (if a, b > 0) or - ab (if a, b < 0). If there is more tha oe geometric mea betwee two umbers, fid them by creatig a system of liear equatios.

6 Chapter 15: Statistics ad Probability Sectio 15-1: Presetig Statistical Data o Kow how to create ad read all forms of statistical data display tools, icludig frequecy tables, bar graphs, circle graphs, lie graphs, lie plots, histograms, stem-ad-leaf plots, twoway frequecy tables, ad box-ad-whiskers plots. Sectio 15-2: Aalyzig Statistical Data o Measures of Cetral Tedecy are values that describe the typical value i the distributio of data The three most commo are: Mea, Media, ad Mode. The Mea of a set of data is the average value of all the umbers. It is derived by takig the sum of the data ad dividig it by the umber of pieces of data. The Media is umber i the middle whe the data are arraged i order. Whe there are two middle umbers, the media is their mea. This occurs if there are a eve umber of data i the set. The Mode of a set of data is the umber or item that appears most ofte. There ca be oe mode, more tha oe mode, or o mode i the data set. o Measures of Variatio are quatities that describe the spread of the values i a set of data. The Rage of a set of data is the differece betwee the largest ad smallest values. It is the size of the smallest iterval which cotais all the data ad provides a idicatio of statistical dispersio. The Quartiles of a data set are the values that separate the data ito four equal subsets, each cotaiig oe-fourth of the data. The media separates the data ito two equal parts, so Q2 is the media. The Lower Quartile (Q1 or LQ) divides the lower half of the data ito two equal parts. The Upper Quartile (Q3 or UQ) divides the upper half of the data ito two equal parts. The Iterquartile Rage (IQR) is the differece betwee the upper ad lower quartiles. A Outlier is a value that is much smaller or larger tha most of the other values i a set of data. Outliers are defied as ay elemet of a set of data that is at least 1.5 iterquartile rages less tha the lower quartile or greater tha the upper quartile. Sectio 15-3: Variace ad Stadard Deviatio o Stadard Deviatio The most commo measure of spread that relates to the mea is called the stadard deviatio. The basic idea is to fid out how far, o average, a data poit is from the mea. The process of fidig the stadard deviatio is: 1. Calculate the mea. 2. Fid each deviatio from the mea. 3. Square the deviatios. 4. Add up the squares of the deviatios. 5. Divide the sum by 1, where is the umber of data poits. 6. Take the square root. Variace is the Stadard Deviatio Squared.

7 Sectio 15-4: Scatter Plots ad Correlatio o A Scatter Plot is a graph that shows the geeral relatioship betwee two sets of data. It is the graph of a series of ordered pairs i a coordiate plae. There are three possible relatioships betwee the two sets of data. A Positive Correlatio A Negative Correlatio No Correlatio o Lies of Best Fit Whe two variables are liearly related, you ca use a lie to describe their relatioship. This lie is kow as the Liear Model. You ca also use the equatio of the lie to predict the value of the y-variable based o the value of the x-variable. Residuals Oe way to thik about how useful a lie is for describig a relatioship is to use the lie to predict the y-values for the poits i the scatter plot. These predicted values could the be compared to the actual y-values. This predictio error is called a residual. Residual = actual-value predicted value. Whe you use a lie to describe the relatioship betwee two umerical variables, the best lie is the lie that makes the residuals as small as possible overall. The most commo choice for the best lie is the lie that makes the sum of the squared residuals as small as possible. The lie that has a smaller sum of squared residuals for this data set tha ay other lie is called the least squares lie. This lie ca also be called the best-fit lie or the lie of best fit (or regressio lie) o Correlatio Coefficiet The correlatio coefficiet is a umber betwee 1 ad +1 (icludig 1 ad +1) that measures the stregth ad directio of a liear relatioship. The correlatio coefficiet is deoted by the letter r. Whe a lie of best fit is produced, it usually comes with a correlatio coefficiet that tells us how good is the best. The closer the poits are to beig o the lie, the closer the correlatio coefficiet is to ±1. Sectio 15-5: Fudametal Coutig Priciple o A outcome is the result of a sigle trial of a experimet. o The sample space is the list of all possible outcomes. o A evet is ay collectio of oe or more outcomes. o There are two ways to determie the sample space: Tree Diagrams ad the Fudametal Coutig Priciple o Kow how to use the fudametal coutig priciple to solve "umber of ways" problems o Kow how to fid "umber of ways" for mutually exclusive situatios usig the additio priciple.

8 Sectios 15-6 ad 15-7: Permutatios ad Combiatios o Fid the umber of permutatios of objects take at a time: P!! o Fid the umber of permutatios of objects take r at a time P r r!! o Fid the umber of permutatios whe there are multiple idetical items: P 1! 2!! o Fid the umber of combiatios of objects take r at a time: C r r! r! Sectios 15-8 ad 15-9: Sample Space ad Evets ad Probability o Experimet: A actio for which the outcome is ucertai. o Outcome: The result of a experimet. o Sample Space: The set of all possible outcomes. o Evet: A sub-set of the sample space. It is usually somethig i which we are iterested o Fid the sample space for a experimet ad the umber of elemets i a evet items i evet o Fid the probability of a evet occurrig: P = items i sample space = (E) (S). Sectios 15-10: Mutually Exclusive ad Idepedet Evets o UNION: A or B is the same as A B (uio) A elemet of the sample space is i A oly, B oly, or both A & B. Both circles are shaded for the uio. o INTERSECTION: A ad B is the same as A B (itersectio) A elemet of the sample space is i both A & B at the same time. Oly the commo sectio is shaded for a itersectio. o MUTALLY INCLUSIVE: P(A B) = P(A) + P(B) P(A B) "Probability of beig i A + Probability of beig i B mius the probability of beig i BOTH A & B simultaeously." The last part prevets double coutig! o MUTUALLY EXCLUSIVE: If two evets A ad B are mutually exclusive, P(A B) = 0 because there is o itersectio. P(A B) = P(A) + P(B) for mutually exclusive evets o Idepedet evets meas the outcome of the first evet does ot affect the outcome of the secod evet. If A ad B are idepedet evets, the P(A B) = P(A) P(B). If P(A B) = P(A) P(B), the A ad B are idepedet evets. (i.e., the coverse is also true)

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.

( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q. A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The