8/4/ Notes P. Real Numbers I. History of Numbers Number Sense? What is it? Natural/Counting Numbers:,,, 4, Whole Numbers:,,,, 4, Integers:, -, -,,,, Rationals:, ½, ¾, Irrationals:, π, 5, Together The set of all REAL NUMBERS!!! II. Describing Sets of Numbers A. Roster Notation: { abcd,,, } B. Rule Notation: a b C. Graphical Notation: { all integers } { } b or { } -5-4 - - - 4 5-5 -4 - - - 4 5 D. Interval Notation:.) Bounded Intervals 4 OPEN CLOSED HALF-OPEN [ 4, ] -5-4 - - - 4 5 4< < ( 4,) -5-4 - - - 4 5 4 < [ 4,) -5-4 - - - 4 5 4< ( 4,] -5-4 - - - 4 5.) Unbounded Intervals CLOSED OPEN III. Field Properties of Algebra PROPERTY ADDITION MULTIPLICATION 4 [ 4, ) > 4 < -5-4 - - - 4 5 [ ) ( 4, ) -5-4 - - - 4 5 (,] -5-4 - - - 4 5 (,) -5-4 - - - 4 5 Commutative Associative i Identity Inverse Distributive a+ b= b+ a ab = ba a+ ( b+ c) = ( ab) c = a( bc) ( a+ b) + c a+ = a a( ) = a a+ ( a) = a = a a b+ c = ab+ ac ( )
8/4/ IV. Laws of Eponents.).) ) a b a b = + a b =.) = 4.) a = a a b 5.) 6) 6.) 7.) ( y) a ( ) b a a a a = y = = y y ab a a V. Scientific Notation Used to facilitate mathematical operations involving very large or very small numbers. c m Where c < and m is an integer. VI. Quick Review Answer the following:.) List the positive integers between - and 7. {,,,,, 4,5,6,7,, }.) List the integers between - and 7. {,,,,,,, 4,5,6,7}.) List all the negative integers greater than -4. {,, } 4.) ( ) 4 ( 4) = 84 5.) 6.) 4 5 + = 4 4 + = 7 7
8/4/ I. Cartesian Plane Notes P. The Cartesian Coordinate System The y-aes Named after Reneé DesCartes y Used to plot data points and determine relationships between data. III. Absolute Value A. Magnitude or distance from zero a, a< a = a, a B. Properties a ab = a b a = a a a =, b b b II. Scatter Plots A. By Hand - Self eplanatory B. By TI-8+/84.) STAT EDIT SET UP EDIT.-ENTER.) STAT EDIT enter data in L and L. Year Population Enter the data on your 995, calculator and then 996,5 follow along with my 997,7 overhead 998,6 999 4, 5,7 7, 7, 8, C. Rewriting Absolute Value Equations: E. - f ( ) = +, < f( ) =,
8/4/ E. - f ( ) = + + First, find where each separate part would equal zero f = when =, Net, set up your intervals using only one and only one., < f( ) =, -, > Now, drop the absolute value signs and simplify the epression. This goes with the right-most part of the domain Negate the entire epression. This goes with the left-most part of the domain, < f( ) =, - +, > -, < f( ) =, - +, > Finally, subtract the second absolute value epression from the first, giving you the epression for the middle part of the domain. -, < f ( ) = + 4, - +, > V. Distance Formulas: A. Number Line: = a b B. Coordinate Plane: = ( ) + ( y y ) We need to be able to do this for CALCULUS!! VI. Midpoint Formulas: a+ b A. Number Line: = B. Coordinate + y+ y Plane: =, VII. Circles A. Def - All points in a plane equidistant from a particular point (h, k). B. Standard Form Equation: ( ) ( ) h + y k = r
8/4/ Notes P. Linear Equations and Inequalities I. Properties of Equality: Let u, v, w, and z be real numbers, variables, or algebraic epressions. Refleive u = u. Symmetric If u = v, then v = u. Transitive If u = v & v=w, then u = w 4. Addition If u = v & w=z, then u + w = v + z 5. Multiplication If u = v & w=z, then uw = vz II. Confirming Solutions A. Substitution NEVER PLUGGING IT IN!!!! III. Solving a +b = c A. By Hand Isolate the variable and coefficent B. Graphically B. By TI-8+/84.) Y = a+b.) Y = c.) GRAPH nd CALC INTERSECT ENTER ENTER - ENTER IV. Properties of Inequalities: Let u, v, w, and z be real numbers, variables, or algebraic epressions. Transitive If u < v & v < w, then u < w. Addition If u < v, then u + w < v + w If u < v & w < z, then u + w < v + z. Multiplication If u < v, & c >, then uc< vc If u < v, & c <, then uc> vc V. Solving Linear Inequalities A. By Hand Same as III B. By TI-8+/84 Same as III
8/4/ VI. Solving Double Inequalities A. Work both simultaneously! B. Eample - 7 < 8 6 < 5 < 5 [,5)
8/4/ Notes P.4 Lines in the Plane I. Slope: Def. The slope of a nonvertical line through the points (, y ) and (, y ) is Δ y y y m = = Δ Δ If the line is vertical, the slope is undefined. II. Forms of Linear Equations: A. Point-Slope Form: y y = m( ) D. Vertical Line: = h B. Slope-Intercept Form: y = m + b E. Horizontal Line: y = k C.) General Form: A + By + C =, with A and B III. Linear Regression: A. Def. The process of using data to determine a linear model or relationship between two variable quantities. B. E. Determine the linear relationship between Fahrenheit and Celsius temperature. Then, find the Celsius equivalent of 9 F and the Fahrenheit equivalent of -5 C. Solution: First, we provide ourselves with a linear equation to model the relationship of F = mc + b. Now, use the fact that water boils at C and F and freezes at C and F to set up a system of two equations and two variables. = m+ b = m+ b
8/4/ Solution cont.: Solving the second equation leaves us with Using substitution, we find that Therefore, 9 m =. 5 9 5 F = C+ and C = ( F ). 5 9 b =. Solution cont.: 5 9. 9 C =. When F =9, C = ( ) 9 When C =-5, F = ( 5) +. 5 F =. IV. Linear Regression with TI-8+ A. Enter the following data on your calculator and follow along with the TI-8+ overhead. World Population in millions: Year Population 98 4454 985 485 99 585 995 5696 65 4 678 5 645
8/4/ Notes P.5 Solving Equations I. Graphically: E.- Solve + = 6 graphically, using two different methods. Solution See graphing Calculator Overhead II. Algebraically: A. Zero Property Let a and b be real numbers: If ab =, then a = or b =. B. E.- Solve = using the following methods:. Factoring: = = ( + )( ) = ( ) ( ) 7 + = or 7 = = or = 7. Quadratic Formula : b± b 4ac = a ( ) ( ) 4( )( ) ( ) ± = ± 7 = 4 ± + 68 = = or = 7 4 ± 89 = 4. Completing the Square : = = + = + 6 8 = 89 = 4 8 89 = 4 6 7 =± 4 4 = or = 7
8/4/ III. Approimate Solutions: A. = vs. B. Graphically: ALWAYS THREE DECIMAL PLACES C. Tables: Find where the sign changes from negative to positive or vice-versa.
8/4/ Notes P.6 Solving Inequalities I. Absolute Values: A. If u < a, then u is in the interval (-a, a). B. If u > a, then u is in the interval (-, -a)or (a, ). C. Solve the following both algebraically and graphically: < 4 4 < < 4 Graphically: See TI-8+ Overhead. < < 7 7 < < 7, II. Solving Quadratic Inequalities: A. E- Solving >. Factor to get ( )( ) 5 + > Algebraically - We will use a function sign test, i. e., we will check the sign of the function in the intervals between the zeroes of the function The epression on the left is equivalent to zero at = - and = 5. We will therefore check the SIGN of the epression in the intervals (-,-), (-, 5) and (5, ) for our solution.
8/4/ + _ + Graphically: See TI-8+ Overhead. - - - 4 5 6 7 (, ) ( 5, ) III. Cubics, Quartics, etc: A. Algebraically Factor and use the sign test. B. Graphically if unfactorable. IV. Projectile Motion: A. Vertical Straight up and/or down B. Units Initial Velocity Initial Position.) In feet s = 6t + vt+ s.) in meters - s = 4.9t + v t+ s V. Quick Review. ) Solve 5 + 4..) Solve 5 > 6 by using the sign test and confirming graphically..) Solve by using the sign test and confirming graphically. Quick Review cont. 4. ) Solve + + 6< 5.) A projectile is launched from a height of 6 ft. above ground with an initial velocity of 56 ft./sec. Determine when the projectile reaches a height of 8 ft.