Download Chapter 1 Organizing Data Review File

FINAL EXAM REVIEW
SEMESTER 1
TECHNICAL ALGEBRA 3
Chapter 1: Modeling Data
Topics Covered:
• Matrix
• Slopes and Lines
• Scatterplots and Classifications
• Regression lines
• Relations and Functions
• Translations
Matrix
Matrix is an array of numbers organized by rows
columns
(the dimensions).
Matrices contain only numbers!
1
-3
2
10
dimensions
2x2
-1
0
1X3
5
and
Organizing Data in a Matix
9th grade
Male
has pt job
5
no pt job
95
Female
15
90
Same data represented as a matrix:
 5 95
A= 

15 90
Each number in a matrix is an element.
a21 = 95
a12 = 15
What is a scatterplot?

A graph that relates data from two different data
sets by plotting data as an ordered pair.
Slopes and Lines

What is the slope of the line?
 Rate
of change between two points.
 The pitch or steepness of the line. Formula is below:
y2  y1
m
x2  x1

There are 2 types of slopes.
 Positive
slope and negative slope.
Slope and Linear Equation

The slope in the linear equation
y = mx + b
Tells us the direction of the line and how steep it is.
- If m is positive, line goes up to right.
- If m is negative, line goes down to right.
- If m is large, the line gets steeper (more vertical)
Classification of Lines by Correlation


We classify lines as strong or weak and positive or
negative.
It is based on the slope (positive or negative) and
how close the points are to forming a straight line.
Correlation Classifications
Positive, strong
Positive, weak
negative, strong
Lines of Best Fit or Trend Lines or
Regression Lines



Trend Line, best fit line and regression line are
mean the same thing.
It means that we can draw a line through our data
that best represents our data.
From this we can find the equation for the line and
make predictions about our data.
Modeling Functions
1. Turn plot on: Y= highlight
PLOT1.
1. To place data in calculator
use: Stat – Edit.
2. To calculate the regression:
Stat – Calc – 4:LinReg (for
linear regression) or
5:QuadReg (for quadratic
regressions) – Vars – Y
Vars – 1:Function – 1:Y1.
3. Choose Zoom – 9:Zoomstat
to actually view the graph.
Function Vocabulary



Relation
 Set of ordered pairs (x, y)
Domain
 Set of x-values or 1st number in an ordered pair.
Range
 Set of y-values or the 2nd number in an ordered pair.
Ex.
State the domain and range for the relation:
{(1, 2)(0, 6)(4, 7)(8, -12)}
Domain = 1, 0, 4, 8
Range = 2, 6, 7, -12
Function


A function is a set of ordered pairs where no xvalue can be repeated with a different y-value.
Are these relations functions?
 (1,
2)(-2, 6)(9, -5)(0, 0) - YES no x value repeats
 (1,
2)(1, 2)(10, 3)(-5, -8)(0, -2) – YES-no x value
repeats with a different y value.
 (4,
3)(-8, 9)(-8, 3) – NO because of (-8, 9)(-8, 3)
Vertical Line Test
If you can draw a vertical line through the graph and have it cross
the graph more than once, then the graph is not a function.
Are these graphs functions? Why or why not?
Is not a function –
fails VLT
Is a function – passes
Vertical line test
Is not a function
– fails VLT
Function Notation
y = 3x + 2 written this way (with a “y”) means it could be a
function.
f(x) = 3x + 2 read “f of x equals 3x + 2” means it is
definitely a function.
We are simply replacing y with f(x) in this situation.
Evaluating Functions

Evaluating functions means we are finding a point on
the graph. You need to think substitution.
Ex. Given f(x) = 3x + 5, find f(2) then find f(-2).
f(2) = 3*2 + 5
=6+5
= 11
Found pt (2, 11)
f(-2) = 3*-2 + 5
= -6 + 5
= -1
Found pt (-2, -1)
Composition of Functions


Combining one or more functions to create a new
function.
Given 2 functions: f(x) = x + 2 and g(x) = 2x – 5
Find f(g(0)). This is read “f of g of 0.” It is evaluated from the inside out.
First do g(0) = 2*0 – 5
=0–5
= -5
Then place this answer into outside function.
f(-5) = -5 + 2
= -3
So, f(g(0)) = -3
Translations of graphs - Parabolas
y = a(x –
2
h)
+k
+ k moves graph up
-k moves graph
down
Width (wide or steep)
+a Opens up
-a Opens Down
x – h moves graph right
x + h moves graph left
Absolute Value Graph
y = a|x – h| + k
+ k moves graph up
-k moves graph
down
Width (wide or steep)
+a Opens up
-a Opens Down
x – h moves graph right
x + h moves graph left