College Algebra MATH 1130
pellissippi state community college
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Notes for 12 Applications and Modeling of Linear Name Equations pp 92 100 Date 39 Instructor Toplcs General Problem Solvmg Geometry Appllcatlons and Modeling of Linear Information I Solving Applied Problems pp92 7 93 General Steps for Solving Applications 1 and reread the problem carefully and slowly until you understand and 2 to represent the unknown value Use diagrams or tables to organize the information Express other unknowns in terms of the variable using the variable expressions 4 the equation 5 to the problem Does it seem reasonable in the words of the original problem 7 Write the answer to the question in a complete sentence being careful to use the correct units required II Geometry Problems p 93 Example 1 page 93 Use two figures to represent the old square and the new square Label each completely Recall that the perimeter of a square is the sum of all 4 sides or 43 Read the problem and look for key math words that can be used later to make an equation Find the verb of the key math sentence FIRST That will anchor the entire equation Put the left phrase of the sentence on the left side of the equals sign and the right phrase of the sentence on the right side of the equals sign Don t scramble the order of the words of the problem Example Problem 13 page 101 Remember that operation phrases like less than and fewer than and even more than must be translated in reverse order of the English statement Ex 5 less than a number x 7 5 Recall that perimeter is the sum of all the sides III Motion Problems pp 94 7 95 Example Problem 21 page 103 Use a chart to organize the information Units of time must be consistent from one entry to the other and also be consistent with the context of the problem Miles per hour dictates what the units of time in problem must be written as Use the calculator to help with the fractions7MATH l and MATH 2 will go back and forth from fraction to decimal forms IV Rate of work pp 95 7 96 ZEROS 0F POLYNOMIAL FUNCTIONS Summary of Properties 1 The function given by fx anx n an1xquot 1 xn2xquot 2 a2x2 a1x an is called a polynomial function ofx with degree n where n is a nonnegative integer and an an1 an2 a2 a1 an are real numbers with an 0 2 The graphs of polynomial functions are continuous and have no sharp corners The sign of the leading coef cient an determines the end behavior of the function The degree n determines the number of complex zeros ofthe function The number of real zeros ofthe function will be less than or equal tothe number of complex zeros 3 The real zeros of a polynomial function may be found by factoring where possible or by nding where the graph touches the xaxis The number of times a zero occurs is called its multiplicity If a function has a zero of odd multiplicity the graph ofthe function crosses the xaxis at that xvalue However if a function has a zero of even multiplicity the graph of the function only touches the xaxis at that xvalue 4 The graphing calculator has a builtin function for nding a zero or root of a function As an alternative method you can graph y 0 the xaxis as a second function and use the intersection function to nd the zero While this latter method is somewhat easier to use on some calculators it may not work for finding zeros of even multiplicity Finding the Zeros of Polynomial Functions Find the real zeros and state the multiplicity of each for the following polynomial functions Algebraic solution Graphical solution 1 fx4x2 3x 7 4x2 3x 70 4x 7x10 4 x 70 or x10 X1 or x 1 E0 4 x17s 0 Each zero has multiplicity one Repeat to nd other zero algebraic solution graphical solution 2 fxx41 x410 x4 1 has no real solutions This function has no real zeros algebraic solution 3 fx x72x5 x3 x7 2x5 x30 x3x4 2x210 x3x212 X3X 1x1X 1x10 x 0 or x 120 or x120 x0 or x1 or x1 graphical solution l The zeros of the function are 0 multiplicity 3 1 multiplicity 2 and 1 multiplicity 1 Writing Polynomial Functions with Specified Zeros Write an equation of a polynomial function of degree 3 which has zeros of 0 2 and 5 General solution Any function oftheform fx axx 2x 5 where a 0 will have the required zeros Speci c solutions fx xx 2x 5 x3 3x2 10x gx 3xx 2x 5 3x3 9x2 30x Write an equation of a polynomial function of degree 7 which has zeros of 0 multiplicity 2 2 multiplicity 3 and 5 multiplicity 2 General solution Any function oftheform fx ax2x 23x 52 where a 0 will have the required zeros Write an equation ofa polynomial function of degree 2 which has zero 4 multiplicity 2 d and opens downwar Atypical solution is fx 3x 42 The leading coef cient must be negative Write an equation of a polynomial function of degree 3 which has zeros of 2 2 and 6 and which passes through the point 3 4 Solution fx ax 2x 2x 6 hasthe required zeros f3a323 23 64 15a4 a fX X 2X 2X 6 has the required zeros and passes through the specified point Exercises A Algebraically find the exact real zeros and state the multiplicity of each 1 fX 2x2 9x 5 2 fX 9x2 24x 16 3 fX 9x2 4 4 fX 9x2 4 5 fX 2x2 4x 1 6 fX 8x3 27 7 fX 3x5 5x4 x3 8 fX 32x3 4 9 fX 1 x4 10 fX 2x4 26x2 72 11 fX 4x4 36 12 fX x32x 13x4 16 B Graphically find the real zeros and state the multiplicity of each Round answers to 4 decimal places 13 fXX2X 1 14 fXX3 3X22X 4 15 fX x3 4x22x1 Notes for 41 Inverse Functions pp 390 398 Name Date Toplcs OnetoOne Functlons Inverse Functlons Instructor I Inverse Functions and One 7 to 7 One Functions pp390 7 Inverses of arithmetic are addition and subtraction multiplication and division etc A function takes a value of the domain x and operates on it using the function formula then gives you an answer that is in the range y To find an inverse of the function we want to undo the function and have the range value y give back its answer x in the domain EX fx5x and gx For f6 5 6 30 and g30 6 These two functions are inverses of each other How to determine whether a function is onetoone p 391 1 Work several examples and see if one y value is determined by more than one x value EX Determine whether the function de ned by f x 25 x2 is onetoone Substituting 3 and 73 f3 and f3 Since these answers are the same positive 4 then this function is not 1 7 l 2 Use the Horizontal Line Test on the graph This is very similar to the Vertical Line Test that determines whether the graph is a function or not It states that any horizontal line passes through the graph in only one place at any part of the graph Omit and stop the video from 734 7 9 10 II Finding the Inverse Function pp 394 7 397 If a function is a one 7 to 7 one function then it has an inverse To find the inverse of a function you p 394 1 Write the function using the variable y instead of the x label 2 Interchange the x and y in the equation This will do it generically for all the values of x and y instead of you having to do them one at a time from the TABLE 3 Solve for the new y 4 Write the answer using the f 391x notation Notes for 22 Functions pp 197 7209 am Toplcs RelatlonsFuncuonsDomamRangeEva1uauon Increasmg Decreasmg and Constant Funcuons ate Instructor 1 Relauons and Funcuons pp197 e199 A cornponent of the ls a of ordered patrs ls arelatron tn whrch for each value of the al of the second cornponent A orderedpatrs there ls v ue The e are no repeatedx values Ex Gwen a set of ordered patrs declde whether each relauon de nes afunctron a 2 4 0t 2 2 5gt b 73t14t1t72t7gt Ex Gwen agraph declde whether the relatron de nes a funcuon Sketch the graph tn th vldeo Name e two pornts that are hrghhghted as the reason that the graph does not represent a funcuon and E vvhrch states that lfa lme 0 wt Gwen an equatron declde whether the relatron ls afunctron ornot Choose values that 11 Domam and Range of aFuncuon pp 199 e 204 The values that x can have ls calledthe Sonneurnes ths ls called the lnput valuequot The values thaty can have ls calledthe Sometrrnes ths ls called the output value quot EX State the domain and the range for the functions below Write the answer using interval notation a y 2x 7 6 Domain Range b x y6 Domain Range 111 Notation for Functions pp 204 7 207 Functions are written in a more formal style using fx and is read fofx This is not multiplication but is a more concise way of writing an equation with the particular characteristic of no repeated X coordinates When you evaluate a function you re being asked to substitute the value from the domain the X coordinate and nd the value of the range that results the y coordinate Pay attention to the instructions that s where the functions are defined to be used throughout the parts of the problem EX Let fx 7 3x 4 andg x 7 7x2 4x 1 Find a g 2 b f 2m 7 3 EX An equation that defines y as a lnction of x is given Solve for y in terms of x and replace y with the function notation Find f 3 The replacement of y with the function notation will dress the equation up a bit makes it more formal 4x 7 3y 8 When the x value is 3 then the y value is Notes for 42 Exponential Functions pp 402 414 Name Date I Exponents and Thelr Propertles pp 402 7 404 Instructor Def An exponential function has its variable as an exponent It is written as fx a where a gt 0 a at l and x is any real number Ex Recall 3 8 vs 2 8 is an exponential equation Calculator reminders To put an exponent into the calculator use the A key If the exponent is a fraction you must put parentheses around the entire fraction so that the calculator will see the entire exponent 3 Ex 23 2 A 314 ENTER and 22 2 A 3 2 ENTER Ex If fx 2 nd fl f3 f 52 f492 Use calculator Recall Properties of Exponents 1 am 0aquot amH1 2 am 1 amquot 3 a0 1 4 m L 5 WW 1 11 Exponential Functions pp 404 7 407 A For fx ax where a gt 1 p 405 Ex fx 2 Graph is an upwards swoop Characteristics of the exponential when a the base gt 1 1 Domain oooo and Range 000 graph lies on top ofthe xaxis 2 Graph has no x intercept because it doesn t cross the x axis only approaches it 3 Graph has a y intercept at 0 1 since 20 l in fact a0 1 so that point is on all exponential functions 4 Graph is increasing 5 Graph is continuous 6 Any graph contains the points 0 l la and l l a 7 The larger the value of the base the steeper the graph 1 2 B For fx ax where a lt 1 fraction base Ex fx E gx g Graph 1s a downwards swoop Characteristics of an exponential where the base a is lt 1 1 Domain oooo and Range 000 graph lies on top ofthe xaxis 2 Graph has no x intercept because it doesn t cross the x axis only approaches it 3 Graph has a y intercept at 0 1 4 Graph is decreasing 5 Graph is continuous 6 Any graph contains 0 1 1 a and 1l a 7 The smaller the value of a the steeper the graph Translations still work III Solving Exponential Equations pp 407 408 1 This type of exponential uses the property that if am aquot the bases are the same then m n the exponents will be the same Ex 3 1 81 Since 81 34 we can write the equation as 3 1 34 Now that the bases 3 are the same then x 1 4 An x 3 IV Compound Interest pp 408 7 410 Compound interest has a time frame associated with it annually 1 semiannually2 quarterly4 monthly12 daily365 It has a onetime deposit with the interest gured every time period tm The formula is A P lLj where A Future Value P Present Value or m Principal r rate written as a decimal m number of compounding periods in the time frame above and t time in years so 6 months 12 for time Calculator Hint Put the info to the power that you get when you do the easy multiplication in your head for the exponent then ENTER Then multiply that by the Principal Don t try putting it in as it s written too many to trip you up getting it from that direction Calculator Hint Enter the first then ENTER to make the calculator catch its breath and get that answer then multiply or divide by the Principal or Future Value Calculator Hint When dividing to get the Principal alone use 2quotd to get the ANS pasted where you want it Should look like 1000 ANS EX p 409 EX 7 P r m t Write the formula every time so you ll do the correct substitution EX p 415 63a P 7 r m n Notes for 36 Variatlon pp 368 372 Name D t Topics Direct Indirect and Joint Variation Intuct0r When a circumstance pairs numbers with a ratio that is constant then there is a direct variation That ratio is called the constant of proportionality These problems always fall into a pattern there is one sentence that has a complete fact relating the values of the story and there is one sentence with incomplete information that you must complete This sets up the order of solution to a problem link this 1 Find the constant of proportionality k 2 Write a general formula using k value and the variables of the problem 3 Substitute the values from the incomplete statement and answer the question I Direct Variation pp 369 7 370 Direct Variation can be stated as y varies as x or y is directly proportional to x This means that there is a nonzero real number k such that y k x The constant of proportionality is EX The total cost of making copies varies directly as the number of copies made If the cost of making 520 copies is 2080 what is the cost of 178 copies The first sentence sets up the type of variation of the problem C k w where C is the cost and w is the number of copies Find the k for the problem by substituting the complete information 2080 k520 Now write the formula using the value of k and substitute the partial information Variations can happen in other situation besides linear EX y varies directly as the cube of x EX y varies directly as the n3911 power of x Note the units of each value in the formula ft inches hrs etc as a hint of where that value needs to be substituted II Inverse Variation pp 370 7 371 Inverse Variation is a situation where the product is constant between two numbers This can be stated as y varies inversely as the n1 power of x or y is inversely proportional to the n3911 n power of x This means that there is a nonzero real number k such that y x EX The illumination produced by a light source varies inversely as the square of the distance from the source The illumination of a source at 5 m is 70 candela What would it be at 12 m Again the first sentence sets up the relationship the second is the complete information and the third is the incomplete with the question Notes for 27 Operations on Functions Composition of Name Functions pp 268 275 Date 39 Instructor Topics Add Subtract Multiply Divide and Compose Two Functions 1 Arithmetic Operations of Functionspp 268 7 270 Given two functions f and g then for all values of x for which both x and gx are de ned as A f 90 Combine similar terms B f7 gx Distribute the negative 1 through the second function then combine similar terms 0 fg x Multiply the two functions together using FOIL distributive property or just writing them sidebyside as on the video L g Exclude all values that would make the denominator become 0 when substituted in for x D x Write as a single fraction and reduce when appropriate Omit problems concerning the domain of the result This is not tested in this course Sorry that it s scattered throughout the video Always write out the operation s definition above as the rst step of the solution then evaluate at a number or clean up as outlined above EX fxx2 4 and gx 3 0rg9 b f gx 0 0590 d fig x 6 01 f fg x g 1 g h LJOQ g Notes for 31 Quadratic Functions and Models pp 294 303 Name Date Instructor Topics Quadratic Functions and Their characteristics Graphing and Mode ing A function is a quadratic lunction if fx or bx c where a b and r are real numbers with a t 0 This is the algebrazc fonn ofthe function I Graphing Techniques pp 295 e 296 299 e 300 We begin with the basic quadratic function Enter Y1 x2 lou Plotz not X W1 tIX2 s 2 2 39I 3 01 3 ME 1 1 I E E 39s 7 X393 The axis of symmetry is x 0 since that is the line ofreflection from the left side ofthe graph to the right side ofthe graph Notice the symmetry ofthe points in the TABLE also The domain is moo andthe range is 0cc Mistake on the video graph The quadratic function can also be written in the form fx oas h2 we where h is the horizontal translation left or right and k is the vertical translation up or down Ex Graph fx x7 2 State its domain and its range lou Plotz not X W1 iICXZ2 1 5 2 o 39I x a l q 3 1 39I g h s 7 X5 The domain is moo andthe range is 0cc Ex Graph fxx3274 State its domain andits range lou Plotz not X W1 I III 4 l 5 2 395 3 I 391 2 s 391 s 7 X396 11 Completing the Square omit pp 297 e 298 l 39 L r39 39 the q 39 391 39 Againthereaiebeuei rquotLthe sameresult III Graphing a Quadratic Using the Algebraic Form of the Function The Vertex Formula pp 299 7 300 Recall The quadratic function can be written as f x axz bx c algebraic form and also f x ax h2 k graphing form The connection between the two forms is made by the vertex h k where h g b and k f h which is found by substituting the h value into the a quadratic Characteristics of a Quadratic Function 1 The graph is a with the vertex h k and the vertical line x h as its axis of symmetry It opens up if a gt 0 has a positive leading coefficient or it opens down if a lt 0 has a negative leading coefficient N E It is broader than the basic graph of x2 if lal lt l is a proper fraction or it is narrower than the basic graph of x2 if lal gt 1 is an improper fraction or whole number 4 The y intercept is f 0 c and is written as 0 c V39 The x intercepts are found from the quadratic formula x bib2 4ac 2a If b2 4610 lt 0 is negative then there are no x intercepts The graph oats above the x axis or remains below the x axis so that it doesn t cross there if b2 4610 Z 0 is positive or 0 Ex Find the axis of symmetry and the vertex of the parabola having the equation fx2x2 4x5 First find the vertex x 2a Now the y coordinate is found by substituting and finding f 1 yf1 2 12 41 5 Note that finding y is easily done by using the STO gt feature of the calculator II Using Quadratics to Model Data Ex page 300 Example 5 a 7 f a Video b The calculator WINDOW at the top sets the dimensions of the GRAPH screen You can adjust the dimensions using WINDOW XMIN is the left side of the x axis XMAX is the right side of the x axis XSCL is the scale that the calculator uses to write the hashmarks of the number line YMIN sets the depth of the y axis YMAX sets the height of the y axis YSCL sets the distance between hashmarks on the y axis number line XMIN XMAX XSCL x by YMIN YMAX YSCL is how some texts denote this information ZOOM 6Standard resets the WINDOW to 10 10 l x 710 10 l c Video Notes for 17 Inequahties pp 146 155 Name Date Top1cs Properties of Inequalities Interval Notation Types of Instructor Inequalities 39 I Properties of Inequalities ppl46 7 147 Summary of Properties 1 If a lt b then a c lt b c Addition Property of Inequalities 2 If c gt 0 is positive then a lt b and ac lt bc Multiplication Property of Inequalities 3 If c lt 0 is negative then a lt b and ac gt be are equivalent Switching the direction of the inequality has to be done when multiplying or dividing by a negative value II Solving Linear Inequalities p 147 A linear inequality in one variable can be written as where a 7i 0 EX 3p 2 S l EX 5x 3 lt 7 Ways to write the solution to an inequality 2 3 Interval notation description of the number line graph Uses numbers or symbols of 00 in nity to the positive side of the number line or 00 negative in nity to the left side of the number line along with a parentheses or a bracket to concisely describe the entire answer Rules for Interval Notation chart on p 148 1 Find critical values of the inequality which is are the solutions to the companion equality 2 Graph or imagine them on the number line 3 Place the left value on the left then a comma then the rightmost value 4a Use a or beside the value ifthe symbol beside the value is lt or gt Similar to the open circle at the point when you draw the number line b Use a or beside the value ifthe symbol beside the value is S or 2 Similar to the shaded circle at the point when you draw the number line 5 This notation is a description of the number line solution without the number line 6 If the number line was shaded over the arrows then you ll use either of the infinities depending on the direction You always need a or beside those because you never close off infinity 7 The interval notation description of the entire number line all real numbers is oo 00 EX xl2 lt x lt 5 means that x is between 2 and 5 The interval notation for that is 2 5 EX xll S x S 4 means that x is between 1 and 4 and includes the values of l and 4 The interval notation for this is 1 4 EX xl Z S x S 0 means The interval notation for this is EX xl Z lt x S 5 is sometimes called a interval because it has one lt and one S The interval notation for this is Athreepart inequality p 149 is quot called a r 39 39 q 39 This type is solve by working all three parts at the same time EX 3 S 2x 1 lt 9 Read as 2x 7 l is between 73 and 9 x 4 EX 3 S 5 lt 5 Multiply all three parts by to clear the fraction Applications of Linear Inequalities p149 Revenue Cost and Break Even Problems Recall at least translates into a Revenue 2 Cost inequality EX The cost to produce x units ofradios is C 20x300 while the revenue is R 30x Find the interval where the product will at least break even X III Quadratic Inequalities pp 150 7 153 A quadratic inequality is one that can be written as where a b and c are real numbers and a 7i 0 These inequalities also can be solved when they are lt gt or 2 types Recall Z 0 means positive and S 0 means negative Steps to solving a quadratic inequality 1 Solve the companion quadratic equation to get critical values 2 Identify the intervals determined by the solutions of the equation by placing the critical values on the number line and having them act as 0 s 3 Use the signs to the left and to the right to represent the number line signs above the line 4 Determine the signs of the products of each region and write that result below the number line 5 The solution is the regions that have the desired nal sign pattern Notes for Section R4 Factoring Polynomials Name pp37 43 Date Topics Factoring polynomials using various techniques Instructor Vocabulary to use factors prime factored form irreducible polynomials I Factoring the Greatest Common Factor Factor and rewrite the problem using the GCF then use the Distributive property in reverse to factor out the GCF EX 3X 15 EX 4y3 lOy2 2y EX 3m3 6m2 9 EX 3x3ycxcy Don t leave out the 1 when factoring out the entire term Check by multiplying the factors back to get the original problem 11 Factoring by Grouping p 38 The Factoring by Grouping method groups the first two terms and the second two terms then follows with a greatest common factor of a parentheses within the two groups EX 3612 bc3b azc Notice that the problem doesn t begin so that you can find a Greatest Common Factor from the first two and second two terms Rearrange the terms so that a GCF shows up 111 Factoring Trinomials by TrialandErr0rp 39 Recall factoring by FOIL from factored form ax bcx d EX 3x2 5x 2 Find the factors of the product ac and the factors of the product bd Focus on the coefficient of the middle X term Notes for 18 Absolute Value Equations and Name Inequalities pp 160 163 Date 39 Instructor Topics Absolute Value Equations Inequalities Special Cases Distance Tolerance Recall l The absolute value gives the distance from a value a to zero on a number line 2 la bl or represents the distance between and I Rules for Absolute Value Equations 1 For b gt0 a bifand only ifa b or a b 2 lal lbl if and only ifa b or a b EX 395 3x 3 Must be 7 units from 0 to the left 01 the right Split the problem into 2 separate cases for each possibility EX 2k 35k4 s02k735k4OR2k735k4 EX x 2 3x 8 II Absolute Value Inequalities pp 161 7 162 Less than indicates values between the two values that make the absolute value true Greater than indicates values beyond the two values that make the absolute value true Solving Absolute Value Inequalities For any positive number b l lal lt b if and only ifib lt a lt b Graph is a segment 2 lal gt b if and only if alt b left side or a gt b right side Graph is looks like wings EX x 1 S 3 less than example EX I32 1 2 7 greater than example Remember to use U to hook the wings together Isolate the absolute value before splitting into 2 equations EX 4 2 lt 6 Hold the absolute value intact while moving other values to the other side 1 5x 2 EX 5x 2gt 20 111 Special Cases EX1 2x3 5 Absolute values cannot equal a negative value since there are no negative distances no negative numbers on the ruler Z 0 EX2 lx6 2 Any absolute value is greater than 0 no matter what the expression inside so the solution is the entire number line 00 00 Ex 3 6x9 0 This is an equation that 0 so there is only one value that makes it true IV Absolute Value Models of Distance and Tolerance p 163 Use absolute value for 1 Distance EX r is 5 units from 3 3 lr 3l 5 EX p is at least 5 units from 79 3 p9 Z 5 Recall at least translates Z EX m is no more than 8 units from 9 3 lm 9 S 8 Recall no more than is really less than or Notes for 24 Equations of Lines Curve Fitting pp 227 e 235 Topics Point slope slope Intercept Vertical Horizontal Parallel and Perpendicular Lines Applications 1 Writing the Equation of a Line Using Pointslope Form pp 227 7 228 T L 39 quot L 39 39 slopefmmula y but with only one point known x1 ofm 9 x2 The line with the slope m passing through m in has the equation y 7y mx in which is called the pointrslope form ofthe equation ofthe line Ex Find the equation ofa line through 5 2 with a slope 01 41 Ex Find the equation ofa line through 1 7 and 2 1 First you need to find a point on the line equation Also label the scale used on the axes 11 Writing the Equation ofa Line Using slopeIntercept Forrn pp 228 7229 The line with slope m andyinterceptb has an equation y nix b which is called the slope intercept form ofthe equation of the line Ex Find the slope and the yintercept onx e y 1 Isolate a positive y 111 Special Lines that are Vertical and Horizontal p 229 en Writing the equation of a vertical line or horizontal line plot the point on the line then stretch out the point so that it is a vertical up and down or a horizontal line across The equation of the vertical line is x the va ue on the xaxis that the stretched line hit The equation of the horizontal line is y the value on the yaxis that the stretched line hit Ex Write the equation of the vertical line that passes through 1 4 Plot the point then stretch it out vertically The graph hits the xaxis at 1 so the equation of the vertical line is Ex Write the equation of the horizontal line that passes through 1 3 Plot the point then stretch it out horizontally The graph hits the yaxis at 73 so the equation of the horizontal line is In general for a vertical line that contains a b the equation is x a and for a horizontal line that contains a b the equation isy b IV Equations of Parallel and Perpendicular Lines pp 229 r 231 Parallel lines have the same slopes and different yintercepts Perpendicular lines have slopes that have a product of 71 or negative opposite reciprocals Ex Find the equation in standard form of the line through 2 4 Which ispamllel to 3x 8y 7 First nd the slope of the given line then borrow it as is Ex Find the equation in standard form of the line through 2 4 Which isperpendicular to 3x 8 7 Find the slope of the given line then use the opposite reciprocal to Write the equation of your line Notes for R7 Radical Expressions pp 65 72 Name Date I Radical Notation and Simplifying Radicals pp 65 7 57 1 Instructor Recall a notation from R6 1 Def a 3 a is called the radicand expression under the radical n is called the index root J when n is even and a is positive is called the J is the Ex 169 Ex 3 64 Ex 5 243 Wquot negative number does not have a real number solution because Rational Exponents gt Radicals Rewrite a7 as V2 or lam where m and n are integers 3 4 Ex 81 i Ex 1253 2 3 Ex 125 v Note the difference the make in the two examples above The exponent applies for what it just in front of it a value of 125 or the 3 Ex 5x4 2 Ex x 2 y 3 Note that you cannot distribute an exponent over a sum or difference Radicals gt Rational Exponents powerl d Def J 12 a1 am M quot aquot lal when n 1s even 1quot aquot a when n 1s odd Ex 5y7 Ex 23 ab3 Simplifying Radicals Use absolute value notation when simplifying and even radical and exponent Vquot aquot lal when n is and quot aquot a when n is Exx 6 Summary of Properties for Radicals Examples p 68 1 Product Rule 0 M Indexes are the same 2 Indexes are the same b 3 3 Power Rule m 5 W 2 Quotient Rule 1 Summary of Simpli ed Radicals l The radicand has no factor The radicand has no No A 39 has Exponents in the radicand and the index have no All operations have been performed EX 13 270 Ex ll6x7ysz3 Elk59 Ex V 24a2b3cs Hint for simplifying Factor the smart way when simplifying a numberiuse a perfect square or cube etc as a factor Ex 24 4 6 smart way rather than 2 12 or 3 8 none of those factors is a perfect square 11 Operations with Radicals pp69 7 71 Like radicals are ones with the same and the same Use the distributive property to factor out the common radical and simplify combine the coefficients Recall 2x 3x x2 3 x 5 or 5x Likewise2J 3J1 J 23 J11 SorSJ Ex 63 2xy43 2992 Ex 63 2xy 43 2x EX V75a3b J12a Multiply two radical expressions can often be done by using EX 7J 5 J EX JE 4JE4 III Rationalizing Denominators pp 71 7 72 Radicals that are in a division problem cannot or We can rationalize the denominator by multiplying by When a denominator is a radical it s similar to a fraction being overreduced We have to multiply by a form of l to bring the number out from under the radical sign Rationalizing the denominator when the denominator is a monomial 2x 7 5 EX czle 9abc 7 Notes for 14 Quadratic Equations pp 119 123 Name Date Toplcs Solv1ng Quadratlc Equatlons by ZeroFactor Property Instructor Square Root Property and Quadratic Formula Solving for a 39 Variable Discriminant Restrictions and Domain The de nition of a quadratic equation is an equation that can be written as where a b and c are and A quadratic equation that is written in the form ax2 bx c 0 is in The degree ofa quadratic equation is 1 Solving a Quadratic Equation using the ZeroProduct Property p 166 The ZeroProduct Property states that if ab 0 then or or both This means that each factor can be set to 0 This method should be used when the equation can be easily factored EX 2x2 x 3 All quadratics must 0 before factoring Now split into 2 independent parts and solve separately 11 Solving a Quadratic Equation using the Square Root Property p 117 This property allows us to use a to undo a single square term or a parentheses that has been squared We always get 2 answers the and the of the radical This method should be used only when the quadratic has a single square term EX a2 20 Note that the number is positive Val iVZO This number under the radical 20 has a perfect square factor that can be used to help simplify this radical a i 405 i2xg EX b2 49 This cannot be done in the real number system since there are no twin factors that can multiply and give us a negative anything even a 49 Our solution for this is Q null set which means that it can t be done The text uses imaginary numbers here which is not part of our class content EX y l218 Note The smart way to factor 18 is as 9 2 not 6 3 EX p l2
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