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
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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)
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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