Formulas and Concepts Study Guide MAT 101: College Algebra

Transcription

1 Formulas and Concepts Stud Guide MAT 101: College Algebra Preparing for Tests: The formulas and concepts here ma not be inclusive. You should first take our practice test with no notes or help to see what material ou are comfortable with, and what material ou ma need to review more. Once ou have an idea of what material ou need to review more, go back to those sections to stud. Preparing for the Final Eam: Stud the individual practice test (Practice Test 1, Practice Test, etc.), and spend time on the sections in which ou had the lowest scores. After ou have prepared thoroughl then ou can take the Practice Final Eam. Keep in mind that the Practice Final Eam selects 40 questions at random from our previous practice tests, so it will not necessaril cover all of the material that is on the final. Things to keep in mind: Studing for the tests in this course is a process that will require a fairl significant amount of time, and should be completed over the course of a few das. Set aside stud time with fellow classmates each da in the das leading up to our eams. If ou are unsatisfied with our test scores, ou ma need to tr a different technique for studing. You should contact our instructor to discuss our stud habits. Review (1) Pthagorean Theorem: a + b = c () Motion formula: d = rt Test 1 ( ) (1) Equation of a circle: ( h) + ( k) = r where (h, k) is the center, and r is the radius () Distance formula: d = ( 1 ) + ( 1 ) (3) Midpoint formula: ( 1+ ), 1+ (4) Slope: m = 1 1 (5) Point-slope formula: 1 = m( 1 ) (6) Slope-interecept form: = m + b Five steps for problem solving: (a) Familiarize ourself with the problem situation. If the problem is presented in words, this means to read carefull. Some or all of the following can also be helpful. (i) Make a drawing, if it makes sense to do so. (ii) Make a written list of the known facts and a list of what ou wish to find out. (iii) Assign variables to represent unknown quantities. (iv) Organize the information. Look up a formula, Consult a reference book or an epert in the field, or do research on the internet. (v) Guess or estimate the answer and check our guess or estimate. (7) Translate the problem situation to mathematical language or smbolism. For most of the problems ou will encounter in algebra, this means to write one or more equations, but sometimes an inequalit or some other mathematical smbolism ma be appropriate. (8) Carr Out some tpe of mathematical manipulation. Use our mathematical skills to find a possible solution. In algebra, this usuall means to solve an eequation, an inequalit, or a sstem of euqations or inequalities. (9) Check to see whether our possible solution actuall fits the problem situation and is thus reall a solution of the problem. Although ou ma have solved an equation, the solution(s) of the equation might not be solution(s) of the original problem. (10) State the answer clearl using a complete sentence. Test (1.6,.-.5) (1) Function composition: (f g)() = f(g()) () Difference quotient: f(+h) f() h (3) Smmetr: Algebraic Tests of Smmetr: (a) -ais: If replacing wtih produces an equivalent equation, then the graph is smmetric with respect to the -ais. (b) -ais: If replacing with produces an equivalent equation, then the graph is smmetric with respect to the -ais. (c) Origin: If replacing with and with produces an equivalent equation, then the graph is smmetric with respect to the origin. Summar of Transformations of = f(): 1

2 (a) Vertical Translation: = f() ± b For b > 0 : the graph of = f() + b is the graph of = f() shifted up b units; the graph of = f() b is the graph of = f() shifted down b units. (b) Horizontal Translations: = f( ± d) For d > 0: the graph of = f( d) is the graph of = f() shifted right d units; the graph of = f( + d) is the graph of = f() shifted left d units. (c) Reflections: Across the -ais: the graph of = f() is the reflection of the graph of = f() across the -ais. Across the -ais: the graph of = f( ) is the reflection of the graph of = f() aross the -ais. (d) Vertical Stretching and Shrinking: = af() the graph of = af() can be obtained from the graph of = f() b stretching verticall for a > 1, or shrinking verticall for 0 < a < 1. For a < 0, the graph is also reflected across the -ais. (e) Horizontal Stretching and Shrinking: = f(c) the graph of = f(c) can be obtained from the graph of = f() b shrinking horizontall for c > 1, or stretching horizontall for 0 < c < 1. For c < 0, the graph is also reflected across the -ais. Test 3 (3.-3.5) (1) Standard form of the quadratic equation: () Quadratic formula: a + b + c = 0 = = b ± b 4ac (3) Properties of quadratic functions and graphing quadratic functions: a > 0 = h a > 0 = h Maimum= k (h, k) (h, k) Minimum= k (4) The verte of the graph of f() = a + b + c is ( b, f ( )) b, where b is the coordinate, and f ( b ) is the coordinate. The graph of the function f() = a( h) + k is a parabola that opens up if a > 0 and down if a < 0; has (h, k) as theverte; has = h as the ais of smmetr; hask as a minimum value if a > 0; has k as a maimum value if a < 0. (5) Absolute value equations: (6) Absolute value inequalities: X = a X = a or X = a X < a a < X < a X > a X < a or X > a

3 3 Test 4 ( ) (1) Leading term test If a n n is the leading term of a polnomial function, then the behavior of the graph as or as can be described in one of the four following was: n a n > 0 a n < 0 Even Odd The portion of the graph is not determined b this test. () Intermediate Value Theorem: If f(a) and f(b) have opposiote signs, then there is at least one real zero between a and b. (3) Dividing f() b a factor ( c) ields a quotient Q() and a remainder R. This can be summarized as: (4) Remainder theorem: f(c) = R f() = ( c)q() + R (5) Factor theorem: if R = 0, then ( c) is a factor of f() (6) Factors and zeros: if ( c) is a factor, c is a zero, and vice versa (7) Rational zeros theorem: Let P () = a n n + a n 1 n a 1 + a 0, where all the coefficients are integers. Consider a rational number denoted b p/q, where p and q are relativel prime (having no common factor besides -1 and 1). If p/q is a zero of P (), then p is a factor of a 0 and q is a factor of a n. (8) Descartes rule of signs: Let P () written in descending or ascending order be a polnomial function with real coefficients and a nonzero constant term. The number of positive real zeros of P () is either: (a) The same as the number of variations of sign in P (), or (b) Less than the number of variations of sign in P () b a positive even integer. The number of negative real zeros of P () is either: (a) The same as the number of variations of sign in P ( ), or (b) Less than the number of variations of sign in P ( ) b a positive even integer. A zero of multiplicit m must be counted m times. Test 5 ( ) (1) Definition of log: log a = a = () Product rule: log M + log N = log (MN) (3) Quotient rule: log M log N = log ( ) M N (4) Power rule: log M p = p log M (5) Change of base formula: log b a = log M a log M b

4 4 (6) General log facts log () means log 10 log a 1 = 0 log a a = 1 log a a = a log a = Last Section: 5.6 (1) Compound interest formual: A(t) = A 0 (1 + 1 n )nt () Continuousl compounded interest formula: P (t) = P 0 e kt (3) Eponential growth formula: P (t) = P 0 e kt (4) Eponential deca formula: P (t) = P 0 e kt A Librar of Graphs Linear Graphs: f() = m + b f() = 3 Vertical Line = 3 Horizontal Line/ Constant Function f() = Quadratics: f() = a + b + c or a( h) + k, a 0 f() = f() = 3 1 f() = ( + ) +

5 5 Cubics: f() = a 3 + b + c + d, a 0 f() = 3 f() = 3 + f() = Eponential: f() = a, a > 0 and Logarithmic: f() = log a () f() = e f() = 3 f() = 1 f() = 4 f() = log 10 () f() = log e () Miscellaneous: Rational Function f() = Square Root f() = Function, 0