1.1 √Real Numbers Real – all numbers are real Rational – numbers that can be expressed as a ratio r/s and s≠0 - can be expressed as terminating decimals, or nonterminating, repeating decimals (ex. ¼, 6.37543754, -⅜) Irrational – numbers that can’t be expressed as a ratio - expressed as nonterminating, nonrepeating decimals where no blocks repeat - (ex. √5 , Π) Natural – positive, counting numbers (1,2,3,….) Whole – positive counting numbers, including 0 (0,1,2,3…) Integers – positive, negative and 0 counting numbers (-3,-2,-1,0,1,2,3) ** Do Monkey Worksheet ** Coordinate Plane (Cartesian Coordinate Plane) – a 2-dimensional coordinate system that corresponds ordered pairs of real numbers with locations in a coordinate plane. Made up of 4 quadrants formed by the intersection of 2 axes (x-axis and y-axis) - point of intersection is called the origin and has ordered pair of (0,0) - Ordered pairs are always (x,y) Scatter Plots – when data is plotted as points on the coordinate plane (usually Quad 1) Relations – when the x variable is related to the y variable, a group of ordered pairs - Domain = set of x values - Range = set of y values Functions – a set of ordered pairs in which the 1st coordinate(x) = input and the 2nd coordinate(y) = output. ** each input corresponds to 1 and only 1 output ** (can’t repeat x values) Function Notation- a convenient shorthand developed to make use and analysis easier f = denotes a given function a = denotes a number in the domain of f f(a) = denotes the output of the function f produced by input a ex. f(2) denotes the output of the function f that corresponds to input 2 y = the output produced by input x according to the rule of the function ** y = f(x) y = f of x ex. f(x)= √x² + 1 f(3)= √3² + 1 = √10 -------------------------------------------------------------------------------------------1.2 Mathematical Patterns Sequence – ordered list of numbers --term = each number in the list Infinite sequence = a sequence with an infinite number of terms Ex. -4,-1,2,5,8…. Point of ellipsis – indicates that same pattern continues on for infinite number of terms. Sequence Notation: u1 = 1st term of sequence u2 = 2nd term of sequence u3 = 3rd term of sequence un = the term in the nth position, called nth term un-1= the term before the nth term Graphs of sequences – is a function because each input corresponds to 1 output Domain – subset of integers Range – set of terms in the sequence Ex. {1,3,5,7,9} ordered pairs (1, first term) ,(2, second term)… (1,1),(2,3),(3,5)(4,7),(5,9) **graph will look like a scatterplot Recursively Defined Sequence – a sequence in which the 1st term is given and there is a method of determining the nth term by using the terms that precede it. Ex {¼, ½, 1, 2, 4… u1 = ¼ u2 = ½ u3 = 1 u4 = 2 u 5 = 4 x2 so. un = 2·un-1 Ex. x2 x2 x2 for n≥2 A basketball is dropped from a height of 8 feet. It hits the floor and bounces to a height of 6 feet. It continues to bounce and on each rebound it rises to ¾ the height of the previous bounce. a) write a recursive formula that represents the height of the ball on each bounce b) create a table a graph (calculator exercise) u0 = initial height = 8 feet u1 = 1st bounce height = 6 (which is ¾ of 8) so un = ¾un-1 for n≥1 Ex. Rick owns a car dealership. Last year he spent $16,000 on advertising. He plans to increase his advertising Expenditures by $1200 this year and in each subsequent year. What will be the amount he spends on advertising in the 6 year? Find a recursive function to represent this problem. th u0 = 16,000 so un = un-1 + 1200 for n≥1 (put table in calculator) ----------------------------------------------------------------------------------------------------------- 1.3 Arithmetic Sequences {un} – a sequence in which the difference between the preceding term is always constant. Ex…{3,5,7,9,11} -- constant increase of 2 Ex… {5,11,18,26} - not constant increase so not arithmetic RECURSIVE FORM OF AN ARITHMETIC SEQUENCE: un = un-1 + d for n≥2 d = common difference = the constant amount ex. If un is an arithmetic sequence and u1 = 2.5 and u2 = 6 are the 1st 2 terms find: a)common difference: u2 – u1 = 6-2.5 = 3.5 b) write sequence as recursive function: un = un-1 + 3.5 for n≥2 c) give first 6 terms: 2.5, 6, 9.5, 13, 16.5, 20 d) graph Explicit form of Arithmetic sequence – when terms of a sequence can be found based on position in sequence. un = u1 + (n-1)d for every n≥1 ** if initial term is denoted u0 – substitute that for u1 in the formula. Ex. Find the nth term of arithmetic sequence with 1st term = -3 and d = 4 un = -3+(n-1)4 un = -3 + 4n -4 un = 4n – 7 Ex. Find the 38th term of the arithmetic sequence with 1st 3 terms as: 15,10,5 u1=15 d = -5 un = u1 (n-1)d so u38 = u1 + (38-1)-5 u38 = 15 + (37)(-5) = -170 Finding Explicit Recursive Functions (when u1 is not given) Ex. If un is an arithmetic sequence w/ u5 = 22 and u11 = 64 find u1, a recursive formula, and an explicit formula. a) first find d ---- 64 -22 = 42 b) then divide by the # of missing spots between u5 and u11 = 6 so 42/6 = 7 c) so.. u5 = u1 + (n-1)d = 22 = u1 + (5-1)7 = -6 d) recursive formula: un=un-1 + 7 for n≥2 e) explicit formula : un = -6 + (n-1)7 = -6 + 7n -7 un = 7n -13 for n≥1 Summation Notation – used when want to find sum of various terms in sequence m ∑ ck = c1 + c2 + …cn k=1 4 ex. ∑(5-4n) = (5-4(1)) + (5-4(2)) + (5-4(3)) + (5-4(4)) = n=1 1 + -3 + -7 + -11 = -20 ** Do Calculator example p. 26 SUM(SEQ(expression,variable,begin,end)) mo Partial Sums: if un is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by: k Either: a) ∑ un = k/2 (u1 + uk) n=1 ex. Find the 14th partial sum of the arithmetic sequence 21, 15, 9, 3… a) d = -6 and u1=21 so =21 + (14-1)-6 = -57 14 ∑ = 14/2(21 + -57) n=1 b) = 14(21) + 14(13)/2 ·-6 = -252 = -252 ex. Find the sum of all multiples of 4 from 4 to 404 a) 404/4 = 101th term b) = 101/2(4+404) = 20,604 1.4 Lines Graph – a set of points in a plane Solution of Equations – a pair of numbers such that the substitution of the 1st number for x and the 2nd number for y produces a true statement. Graph of an Equation – set of points in a plane whose coordinates are solutions of the equation. ** graph of a sequence = has an infinite number of discrete points b/c the value of the sequence depends on the term of the sequence which must be a counting number. Slope (m) = change in y (vertical distance) change in x (horizontal distance) **measures steepness of line = y2 – y1 x2 – x1 ex. Find the slope that passes through (1,-2) and (4,1) 1- -2 3 4–1 3 =1 Properties of Slope (m) if m>0, positive slope, the line rises from left to right, larger the m= steeper line if m<0, negative slope, the line falls from left to right, larger │m│ = more steep the line falls. If m = 0 , then horizontal line If m is undefined, then vertical line Slope Intercept Form y = mx + b m=slope, b=y-intercept Point Slope Form : y – y1 = m(x – x1) Ex. Sketch the graph and find the equation of the line that passes through pt(-2,5) with m = -3. Write in slope intercept form. y – y1 = -3(x – x1) y = -3x-1 y – 5 = -3(x- -2) y-5 = 3x -6 Equation of Horizontal Lines : y = 0x + b Equation of Vertical Lines: x + 0y = b Parallel and Perpendicular Lines ** parallel are 2 nonvertical lines that have exactly the same slope ** perpendicular are 2 nonvertical lines when the product of the slope is -1 (are negative reciprocals of each other) Ex. Given line M whose equation is 5x-4y+8=0 find the equation of lines through pt. (-4,2) a)parallel to M 5x-4y+8=0 = y = 5/4x + 2 now using (-4,2) use y – 2 = 5/4(x + 4) = y = 5/4x + 7 **same slope used b) perpendicular to M y = 5/4x + 2 ** must use negative reciprocal y – 2 = -4/5(x + 4) = y= -4/5x -6/5 Standard Form of a Line: Ax + By = C where A,B,C are integers with A≥0 Ax + By = C y = mx + b y – y1 = m(x – x1) Horizontal line Vertical line Standard Form Slope Intercept Form Point Slope Form Has slope=o and equation in form y = b Has undefined slope and equation in form x = c Connection between Arithmetic Sequences and Lines: un = u1 + (n-1)d and y = mx + b a) slope of line corresponds to common difference m = d b) y-intercept represents the value of the 1st term of the sequence minus the difference b = u1 – d Linear Depreciation Ex. An office buys a new color copier for $8000. 4 years later its value is $2320. Assume the copier depreciates linearly. a) Write the equation that represents the value as a function of years b) Find the value 3 yrs. After it was purchased, the y value when x=3 c) Graph d) Find how many years before the system is worthless, the x when y =0 a ) b = 8000 bc when x=0 the system is worth 8000 so y = mx + 8000 – bc copier worth 2320 after 4 yrs 2320 = 4·m + 8000 m = -1420 so y = -1420x + 8000 b) y = -1420(3) + 8000 = $3740 c) use calculator to graph d) use trace function on calc to see what x = when y=0 Mathematical Models - demonstrates the desired relationship or predicts the likely outcomes in cases not included in the data. 1.5 ** When given a set of data points: 1 : a) determine whether a straight line would be a good model for the data b) if line is a good fit w/o graphing by using Finite Differences: if the differences between each y entry and the preceding one is the same = good model of data. st Year 1993 1994 1995 1996 1997 Exp. Per share 1.24 1.45 1.66 1.88 2.09 **determine if line a good fit Ex. a)calculate the finite differences for data points: 1.45-1.24 = .21 ** approx = so is a good model 1.66 – 1.45 = .21 1.88 – 1.66 = .22 2.09 – 1.88 = .21 Modeling Terminology Suppose (x,r) is a data point and that the corresponding point on the model is (x,y): Residual = the difference r – y **area measure of the error between the actual value of the data, r, and the value y, given by the model ** graphically = the residual is the vertical distance between the data point (x,r) and the model point (x,y) ** when the sum of the residuals = 0, which indicates that the + and – error cancel each other out ---- model is probably a reasonable one Do example #2 p. 45 To find which model fit the data the best --- use Sum of the Squares –of the residuals Because this sum has no negative terms and no canceling Sum of Squares: a) emphasizes large error – those w/absolute values >1 b/c the square is greater than the residual. b) Minimizes small error – those w/absolute values <1 b/c the square is less than the residual Least Squares Regression Lines – for any set of data there is 1 and only 1 line for which the sum of the squares of the residuals is as small as possible. **Linear Regression – computational process for finding least squares regression line **Correlation Coefficient (r) – a statistical measure of how well the least squares regression line fit the data points *value between 1 and -1 a)closer absolute value of r is to 1 --- better the fit b) when │r│ = 1 the fit is perfect – all data points on regression line c) when │r│= 0 the fit is poor **Coefficient of Determination (r²) – the proportion of variation in y that can be attributed to a linear relationship between x and y in the data Do Ex. #3 p. 47 - technology hints for linear regression General Rule: 1. Use a linear model when the scatterplot of the residual shows no obvious pattern 2. Use a nonlinear model when the scatterplot of the residuals has a pattern Do Ex. #4 p. 49 1.6 Geometric Sequences - a sequence in which terms are found by multiplying a preceding term by a nonzero constant. A quotient of consecutive terms = r = common ratio Un/un-1 = r or un = run-1 = Recursive form of Geometric Sequence Ex. Find the common ratio of geometric sequence w/ u1= 3 and u2 = 6, list 1st 5 terms. 6/3 = 2 = r so un = 2un-1 3,6,12,24,48 Ex. Is it a geometric sequence? {5,10,20,40,…} – find the r --- 10/5 = 2 so un = 2un-1 for n ≥ 2 20/10 = 2 40/20 = 2 Explicit Form of Geometric Sequence – value of a sequence determined by position of term. If un is a geometric sequence with common ratio, r, then for all n≥1 un=u1r n-1 Ex. Write the explicit form of a geometric sequence where the 1st 2 terms are 3 and -¾. Find the first 5 terms R = -¾/3 = -¼ so u1 = 3 u2 = -¾ u3 = 3(-¼)² = 3/16 u4 = 3(-¼)3 = -3/64 U5 = 3(-¼)4 = 3/256 Ex. The 3rd and 6th terms of a geometric sequence are 80 and -5120. Find the explicit form of the sequence. 3rd term: u3 = u1rn-1 = u1r2 = 80 6th term: u6 = u1rn-1 = u1r5 = -5120 ** find r by dividing terms u6/u3 = -5120/80 = -64 √r3 = -√64 then un = u1rn-1 = un = 5(-4)n-1 = -4 Partial Sums of a Geometric Sequence - the kth partial sum of the geometric sequence with common ratio r ≠ 1 is: k ∑ un = u1( 1- rk/ 1- r) n=1 ex. Find the sum: -3/2 + ¾ - 3/8 + 3/16 – 3/32 + 3/64 – 3/128 + 3/256 – 3/512 un = -3/2 (1-(-½)9 )/(1- (-½)) = -513/512 r = ¾/-3/2 = -½