~ Q ~ -a m ~~~~" .PA R TS .1.& C.O.M BI.N.E.D C O V.E R P.R C IN I.L.S B~ P.E FOR IA SIC,,~TE ' IN ~.RpM.D E.I.A.E T & C .O.L.L.E.G.E.C O U.RS E S (~ II1II1 1=1 Z ~rj;'.":J:1I REAL NUMBER LINE SLOPE OF A LINE Chart of the graphs, on the real number line, of solutions to one-variable equations: SYMBOL & GRAPHIC NOTATION - SYMBOL - CLOSED CIRCLE Ex X = -2 • I I • -4 -2 I I I I I II > SYMBOL - OPEN CIRCLE AND A RAY Ex x>4 ·1 -2 I I I I I ED I I~ < SYMBOL - OPEN CIRCLE AND A RAY Ex x 0, a #- 1, and the constant real number, a, is called the base Example of an • PROPERTIES I The graph always 1I1tersects the y-aXiS at (0,1) Exponent Function because aO = ~ The domain is the set of all real numbers _r The range is the set of all positive real numbers Y x because a is always positive When a > 1, the function is increasing; when a < I, the function is decreasing Inverses of exponential functions are logarithmic functions • HORIZONTAL LINE TEST I Indicates a one-to-one function ifno horizontal line intersects the graph of the function in more than one point NOTATION • f(x) IS READ AS "f of x" I Does not indicate the operation of multiplication Rather, it indi cates a function of x LOGARITHMIC FUNCTIONS a f(x) is another way of writing y in that the equation y = x + may also be written as f(x) = x + and the ordered pair (x,y) may also be written (x,f(x)) • DEFINITIONS: A logarithm is an exponent, such that for all posi Exampleofa tive numbers a, where a #- 1, Y = log a x if and on Logarithmic Function ly ifx = a Y; notice that this is the logarithmic func b To evaluate f(x), use whatever expression is found in the set of tion of base a parentheses following the f to substitute into the rest of the equa The common logarithm, log x, has no base indi tion for the variable x, then simplify completely cated and the understood base is always 10 The natural logarithm, In x, has no base indicat • COMPOSITE FUNCTIONS: f [g(x)l ed, is written In instead of log, and the lmderstood Composition of the function f with the function g, and it may also be base is always the number e • PROPERTIES WITH THE VARIABLE a REPRESENTING A written as f g(x) POSITIVE REAL NUMBER NOT EQUAL TO ONE: The composition, f [g(x»), is simplified by evaluating the g function alog"x= x logaa X = x log_a = I first and then using this result to evaluate the f function If logau = log_v, then u = v log_I = If log"u = 10gbu and u #- 1, then a = b log" xy = log" x + log" y • (f + g)(x) EQUALS f(x) + g (x) Y~I'o~:~ lf That is, it represents the sum of the functions • (f - g)(x) EQUALS f(x) - g (x) ° 10ga(~ )= log x -log Y IOg.(+)=-IOg x 10 log x" = n(log x), where n is a real number 11 Change of Base Rule: If a> 0, a #- 1, b > 0, b #- I , and x> 0, then log x = (10g bx) ~ (lOgb a ) (log x) 12 Fmdmg Natural Loganthms: In x = (10 e)' • COMMON MISTAKES! g I log_ (x+y) = log x+logaY FALSE! log x" = (log_x)" FALSE! That is, it represents the difference of the functions • (fg)(x) EQUALS f(x) • g(x) That is, it represents multiplication of the functions • (f/g)(x) EQUALS f(x)/g(x) That is, it represents the division of f(x) by g(x) (log,x) =log (.x-y) FALSE! • (log y) NOTICE TO STUDENT: This QUICKSTUDY'" guide is the second of guides outlining the major topics taught in Algebra courses Keep it handy as a quick reference source in the classroom, while doing homework and use it as a memory refresher when reviewing pri or to exams It is a durable and inexpensive study tool that can be re peatedly referred to during and well beyond your college years Due to its condensed format, however, use it as an Algebra guide and not as a replacement for assigned course work • SOLVING LOGARlTHM1C EQUATIONS Put all logarithm expressions on one side of the equals sign Use the properties to simplity the equation to one logarithm statement on one side of the equals sign Convert the equation to the equivalent exponential form Solve and check the solution ~'1 ::::{PIIJ =t ~ Lei ~ =t:1 I~ RATIONAL FUNCTIONS Definition: f(x) = ~~:~, where P(x) and Q(x) are polynomials that are relatively prime (lowest terms), Q(x) has degree greater than zero, and Q(x) :F- O TO GRAPH ·DOMAIN I The domain is all real numbers, except for those numbers that make Q(x) = O • INTERCEPTS y-intercept: Set x = and solve for y; there is one y-intercept; if Q(x) = o when x = 0, then y is undefined and the function does not intersect the y-~xis P(x) x-mtercepts: Set y = 0; SInce f(x) = Q(x) can equal zero only when P(x) = 0, the x-intercepts are the roots of the equation P(x) = O ASYMPTOTES A line that the graph of the function approaches, getting closer with each point but never intersecting • HORIZONTAL ASYMPTOTES Horizontal asymptotes exist when the degree of P(x) is less than or equal to the degree of Q(x) The x-axis is a horizontal asymptote whenever P(x) is a constant and has degree equal to zero Steps to find horizontal asymptotes: a Factor out the highest power of x found in P(x) b Factor out the highest power of x found in Q(x) c Reduce the function; that is, cancel common factors found in P(x) and Q(x) d Let Ixl increase, and disregard all fractions in P(x) and in Q(x) that have any power of x greater than zero in the denominators, because these fractions approach zero and may be disregarded completely e When the result of the previous step is: i a constant, c, the equation of the horizontal asymptote is y = c ii a fraction such as c/xn, where c is a constant and n :F- 0, the asymptote, is the x-axis iii neither a constant nor a fraction, there is no horizontal asymptote • VERTICAL ASYMPTOTES I Vertical' asymptotes exist for values of x that make Q(x) = 0; that is, for values of x that make the denominator equal to zero and, therefore, make the rational expression undefined There can be several vertical asymptotes Steps to find vertical asymptotes: a Set the denominator, Q(x), equal to zero b Factor if possible c Solve for x d The vertical asymptotes are vertical lines whose equations are ofthe form x = r, where r is a solution of Q(x) = because each r value will make the denominator, Q(x), equal to zero when it is substituted for x into Q(x) SYMMETRY • DESCRIPTION I Graphs are symmetric with respect to a line if, when folded along the drawn line, the two parts of the graph then land upon each other Graphs are symmetric with respect to the origin if, when the paper is folded twice, the first fold being along the x-axis (do not open this fold before completing the second fold) and the second fold being along the y-axis, the two parts of the graph land upon each other • GRAPHS ARE SYMMETRIC WITH RESPECT TO: The x-axis if replacing y with -y results in an equivalent equation The y-axis ifreplacing x with -x results in an equivalent equation The origin if replacing both x with -x and y with -y results in an equivalent equation • DETERMINE POINTS I Create a few points, by substituting values for x and solving for f (x), that make the rational function equation true 2.lnclude points from each region created by the vertical asymptotes (choose values for x from these regions) Include the y-intercept (if there is one) and any x-intercepts Apply symmetry (if the graph is found to be symmetric after testing for symmetry) to find additional points; that is, if the graph is symmetric with respect to the x-axis and point (3,-7) makes the equation f (x) true, then the point (-3,-7) will be on the graph and should also make the equation true • PLOT THE GRAPH I Sketch any horizontal or vertical asymptotes by drawing them as broken or dashed lines Plot the points, some from each region created by the vertical asymptotes, that make the equation f(x) true Draw the graph of the rational function equation, f(x) = P(x)/Q(x), applying any symmetry that applies DEFINITIONS • INFINITE SEQUENCE is a function with a domain that is the set of positive integers; written as aI, a2, a3, , with each aj representing a term o FINITE SEQUENCE is a function with a domain of only the first n positive integers; written as a., a2, aJ, , an-., an o SUMMATION: f ak = al+a2+ + a m_ + am, where k is the 1:=1 index of the summation and is always an integer that begins with the value found at the bottom of the summation sign and increases by until it ends with the value written at the top of the summation sign o nTH PARTIAL SUM: Sn = ak = a + a2 + + an_.+ an ~ i.e rate; : in hop arol :t,-, o ARITHMETIC SEQUENCE OR ARITHMETIC PRO GRESSION is a sequence in which each term differs from the preceding term by a constant amount, called the common dif ference; that is, an = an- + d where d is the common difference o GEOMETRIC SEQUENCE OR GEOMETRIC PROGRESSION is a sequence in which each term is a constant multiple of the preceding term; that is, an = n_., where o r is the constant multiple and is called the common ratio n! = n(n - 1)(n - 2)(n - 3) (3)(2)(1); this is read "n factorial." NOTE: O! = PROPERTIES OF SUMS, SEQUENCES & SERIES ,-, 1:=1 :t ca, = ct 10;=1 nth ,-, a, , where c is a constant l=1 = nc, where c is a constant t c k=1 The nth term of an arithmetic sequence is an = a + (n -l)d, where d is common difference ralto The sum of the first n terms of an arithmetic sequence, with a as the first term and d as the common difference, is So n n = 2(a, + an)orsn = 2[2a, + (n - I)d] The nth term of a geometric sequence, with a as the first term and r as the common ratio, is an = alr n-I The sum of the first n terms of a geometric sequence, with a as the first term and r as the common ratio and r :F- is , S 0 [a,(1 - rn)] (I-r) 8.The sum of the terms of an infinite geometric sequence, with a as the first term and r as the common ratio where Irl< Lis 1~ r ; if Irl > or Irl = , the sum does not exist 9.The rth term of the binomial expansion of (x + y)n is n! xn-(r-Ij (r- ) [n-(r-l)]!(r-l)! y fi ts fl O nio n nd a CONIC SECTIONS The charts below contain all general equation forms ofconic sections; these general forms can be used both to graph and to determine equations ofconic sections; the values for hand k can be any real number, including zero DESCRIPTION Conic sections represent the intersections of a plane and a right circular cone; that is, parabolas, circles, ellipses and hyperbolas; in addition, when the plane passes through the vertex ofthe cone, it may determine a degenemte conic section; that is, a point, line or two intersecting lines GENERAL EQUATION The general form of the equation of a conic section, with axes parallel to the coordinate axes, is: Ax2 +Bxy + Cy2 + Dx + Ey + F = 0, where A and C are not both zero TYPE: CIRCLE TYPE: LINE GENERAL EQUATION: y = fiX +b GENERAL EQUATION: (X - h)Z + (y - k)2 = r2 Notation: I m is slope b is y-intercept Values: I m > 0, then the line is higher on the right end m < 0, then the line is higher on the left end x x (h,k) Notation: X2 tenn and y' term , both with the same positive coefficient r' is a positive number (h,k) is center r is radius Values: None y (O ,b) TYPE: HORIZONTAL LINE / TYPE: ELLIPSE GENERAL EQUATION: y = b Notation: GENERAL EQUATION: b is y-intercept y Values: ) m = 0, then the line is horizontal through (O,b) i '( Z ~ ~ A x (h,k) y TYPE: VERTICAL LINE (c,O) GENERAL EQUATION: X = \ c Notation: c is X-intercept Values: = I No slope Vertical line through (c,O) (x-h)' +(y-k)' =1 a' b' Notation: x' term and y' term with different coefficients (h,k) is center a is horizontal distance to left and right of (h,k) b is vertical di stance above and below (h,k) Values: a> b, then major axis is horizontal and foci are (h ± c, k), where c'= a'- \)2 b> a, then major axis is vertical and foci are h, k ± c), where c' = b' - a' TYPE: HYPERBOLA GENERAL EQUATION: (y-k)' _ (x-h)' = a' b' TYPE: PARABOLA GENERAL EQUATION: y = a(x - h)2 Notation: l.x' term and y' term, with a negati ve coeffici ent for x' term 2.(h,k) is center of a rectangle b is horizontal distance to left and right of (h,k) 4.a is vertical di stance above and below (h,k) to the vertices Values: +k STANDARD FORM: (X - h)2 = 4p(y - k) x :f :f 'f - ::::::=~ Notation: I X2 term and yl term (h,k) is vertex (h, k ± p) is center offocus, where P = 7;;a ~==::::::3-·4 y = k ± P is directrix equation, where P = 7;;a (h,k) a y - k = ± 1>(x - h) Value: a > 0, then opens up a < 0, then open~ down X= h is equation of line of symmetry Larger lal = thinner parabola; smaller lal = fatter parabola are equations of asymptotes TYPE: HYPERBOLA GENERAL EQUATION: (x - h )' _ (y - k )' = TYPE: PARABOLA b' Notation: x' term and y' term, with a negative coefficient for y' term (h,k) is center of a rectangle a is hori zontal di stance to left and right of (h,k) to the vertices b is vertical distance above and below (h,k) Values: k = ± Q( x - h) Y a are equations of asymptotes a' GENERAL EQUATION: x = a (y - k)' + h STANDARD FORM: (y - k)' = 4p (x - h) Notation: Xl term and y2 term x (h,k) is vertex / ===~ ~=~1iIt~3 (h ± p, k) is focus, where P = } 4a X= h ± p is directrix equation, where P = 7;;a Values: a > 0, then opens right a < 0, then opens left y = k is equation of line of symmetry PROBLEM SOLVING DISTANCE DIRECTIONS NOTATION ~ I Read the problem carefully d is distance; r is rate, i.e speed; t is time, value indicated in the speed, i.e Note the given information, the question asked and the value requested miles per hour has time in hours Categorize the given information, removing unnecessary mformatlOn NOTE: Add or subtract speed of wind or water current with the rate; Read the problem again to check for accuracy, to determme what, If any, (r ± wind) or (r ± current) III formulas are needed and to establish the needed variables Write the needed equation(s) and determine the method of solution to use; • FORMULAS " this will depend on the degree of the equations, the number of vanabies and I d = rt III the number of equations Example: John traveled 200 miles in hours ~ Solve the problem Check the solution Read the problem again to make Equation: 200 = r II.I- ~u~~the~==~rg~~ s~re == answe~ive~n~s i~~==o the ~ne ~~~~ requ~ste~ e7~d~==~~~~~~~~ -I dto = dreturning IIIIlI ODD NUMBERS, EVEN NUMBERS, MULTIPLES Example: With a 30-mph head wind, a plane can fly a certain distance in 'liliiii NOTATION hours Returning, flying in opposite direction, it takes one hour less d is the common difference between any two consecutive numbers of a Equation: (r - 30)6 = (r + 30)5 set of numbers d + d2 = dtotal FORMULAS First number = x Second number = x+d Example: Lucy and Carol live 400 miles apart They agree to meet at a shop Third number = x+2d Fourth number = x+3d; etc ping mall located between their homes Lucy drove at 60 mph, and Carol Example: The first multiples of3 are x, x+3, x+6, x+9, and x+l2 because d = drove at 50 mph and left one hour later Equation: 60t + 50(t-l) = 400 RECTANGLES Z ~ NOTATION SIMPLE INTEREST P is perimeter; I is length; w is width; A is area NOTATION FORMULAS P = 21 + 2w A = Iw I is interest; P is principal, amount borrowed, saved, or loaned; S is total amount, or I + P; r is % interest rate; Example: The length ofa rectangle is more than the width and the perimeter is 38 Equation: 38 = 2(w + 5) + 2w t is time expressed in years; p is monthly payment FORMULAS TRIANGLES I 1= Prt Example: Anna borrowed $800 for years and paid $120 interest Equation: 120 = 800 r(2) S = P + Prt Example: Alex borrowed $4,600 at 9.3% for months Equation: S = 4,600 + 4,600 (.093)(.5) NOTE: 9.3% = 093 and months = 0.5 year NOTATION P is perimeter; S is side length; A is area; a is altitude; b is base NOTE: Altitude and base must be perpendicular i e form 90° angles FORMULAS I P = S + S2 + S3 A = 112 ab Example: The base of a triangle is times the altitude and the area is 24 Equation: 24 = ·12 • a • 3a CIRCLE Z p_P+Prt - t-12 Example: Evan borrowed $3,000 for a used car and is paying it otT month ly over years at 10% interest NOTATION C is circumference; A is area; d is diameter; r is radius; 1t is pi = 3.14 FORMULAS , l.C=1td 2.A=1tr2 3.d=2r Example: The radius of a circle is and the circumference is 25.12 Equation' p - (3 000 + 000 ( 1)(2)( / (2)(12) ·1_E~u~a~ti~o~n:~2~5~.1~2~1t~'~8 ~ -I~ ~ -~I~~~rn~~~~~~no~ -r~ · - ' ' ' " PROPORTION & VARIATION III PYTHAGOREAN THEOREM NOTATION ~ NOTATION a, b, c, d are quantities specified in the problem; k O II a is a leg; b is a leg; c is a hypotenuse FORMULAS IIIIlI NOTE: Hypotenuse is the longest side Proportion: ~ =~ ; cross-multiply to get ad = bc 'liliiii FORMULA b d a + b = c2 Direct Variation: y = kx k NOTE: Applies to right triangles only Example: The hypotenuse of a right triangle is times the shortest leg The other I.nverse Variation: y =-; r;; Examples: leg is ,, times the shortest leg Proportion: If 360 acres are dividcd between John and Bobbie in the ratio Equation: a +( -J3 a)2 = (2a)2 4~5, how many acres does each receive? III '* Equation: B~obhb~e so MONEY, COINS, BILLS, PURCHASES ~ = -36-g -J NOTATION V is currency value; C is number of coins, bills, or purchased items Direct Variation: If the price of gold varies directly as the square of its mass, and 4.2 grams of gold is worth $88.20, what will be the value of 10 grams of gold? Equation: 88.20 = k(4.2)2; solve to find k = 5; then, use the equation v = 5(10)2, where y is the value of 10 grams of gold inverse Variation: If a varies inversely as b and a = when b = 10, find a when b = FORMULA V.C + V2C2 =Vtotal Example: Jack bought black pens at $1.25 each and bluepens at $0.90 each He bought more blue pens than black pens and spent $36.75 E uation: I.25x + 0.90(x+5) = 36.75 MIXTURE fir~.~e~~~~?o~ution; V is first volume; PI is Y2 is second volume; P2 is second percent solution; VF.S fmal volume; PF.S fmal percent solutIOn NOTE: Water could be 0% solution and pure solution could be 100% Equation: = l~ , so k = 40; then, a= FORMULA ISBN-13: 978-157222922-8 ISBN-1D: 157222922-5 VIP +V2P2 =VFPF Example: How much water should be added to 20 liters of 80% acid solution to yield 70% acid solution? Equation: x(O) + 20(0.80) = (x+20)(0.70) 911~ lllJllll~llll11~11111111111111111 lIillll WORK NOTATION W I is rate of one person or machine multiplied by the time it would take for the entire job to be completed by or more people or machmes; W2 is the rate of the second person or machine multiplIed by tune for entIre Job; represents the whole job NOTE: Rate is the part of the job completed by one person or machine free & at nundiwn re~ad.s ol.tltles ~O to find a All ri2hls me rH':d No part of this publiclluon JllOI> be repr0 duced llr Inmsmitlcd in any form ()r b) uoy lTlCans, electronic Of mechanical inc luding tocopy R:cording or any inronnauon SfOfaall: and ret rieval ~ystcm without \\ r ilu."I1 pcmllSSlon from Iht publisher 21~)2 8I rCh ut~ Int Boea Raton f L 01()H Author: S.B Kizlik U.S $5.95 CAN $8.95 qUlc uuy.com FORMULA W +W2 = Example: J,ohn can paint a house in days, while Sam takes days How long would they take if they worked together? Equation: lx+!x=l Customer Hotline # 1.800.230.9522 We welcome your feedback so we can maintain and exceed your expectations ,1,.1, 20922 ... 978-15 722 2 922 -8 ISBN-1D: 15 722 2 922 -5 VIP +V2P2 =VFPF Example: How much water should be added to 20 liters of 80% acid solution to yield 70% acid solution? Equation: x(O) + 20 (0.80) = (x +20 )(0.70)... FORMULAS , l.C=1td 2. A=1tr2 3.d=2r Example: The radius of a circle is and the circumference is 25 . 12 Equation' p - (3 000 + 000 ( 1) (2) ( / (2) ( 12) ·1_E~u~a~ti~o~n: ~2~ 5~.1 ~2~ 1t~'~8 ~ ... of its mass, and 4 .2 grams of gold is worth $88 .20 , what will be the value of 10 grams of gold? Equation: 88 .20 = k(4 .2) 2; solve to find k = 5; then, use the equation v = 5(10 )2, where y is the