Transcript Chapter 2: Patterns and Relations

Chapter 2: Patterns and
Relations
This chapter begins on page 66
Chapter 2: Get Ready!

1.
2.
3.
4.
Before starting Chapter 2, we need to
review the following Math skills:
Independent and Dependant variables
Substitution and Evaluation of
expressions
Plot ordered pairs on a Cartesian Grid
Interpreting the graph of a straight line
A variable
A
variable is a letter that represents an
unknown value.
 For example, in the algebraic
expression 4x – 1, the variable is x
Independent variables
 In
a relation, the independent variable
is the variable that determines the
value of the other variable.
 For the most part in Mathematics, the
symbol used to represent the
independent variable is x
Dependent variables
 In
a relation, the dependent variable is
the variable that is determined by the
independent variable.
 For the most part in Mathematics, the
symbol used to represent the
dependent variable is y
Substitution and Evaluation of
algebraic expressions

In order to evaluate an algebraic
expression, equation or formula, you
should substitute the value of any variable
that you know, then simplify using the
order of operations.
Order of Operations
 Here
is the priority list for any order of
operations question:
1. Do all operations in the brackets.
2. Do all your exponents.
3. Multiply and Divide terms from left to
right.
4. Add and Subtract terms from left to right.
The Cartesian grid
The Cartesian grid is a two dimensional or
(x,y) plane.
 A Cartesian grid is also called a
« coordinate grid »

Plotting ordered pairs on a
Cartesian Grid
To plot an ordered pair (x,y):
1. Start at the origin, which is where the xaxis and y-axis intersect each other. The
origin has coordinates (0,0)
2. Move right if x is positive; move left if x is
negative.
3. Then, move up if y is positive; move down
if y is negative.

The vocabulary of a straight line
The vertical change of a straight line is the
distance up or down between the 2 points
on a graph. On the x-y plane, the
difference between y1 et y2
 The horizontal change of a straight line is
the distance right or left between 2 points
on a graph. On the Cartesian plane, the
distance between x1 et x2

Interpreting the graph of a Straight
Line
To find the vertical change or the
horizontal change, identify two points that
are easy to read on the line.
 To obtain the vertical change, measure the
distance up or down between the 2 points.
 To obtain the horizontal change, measure
the distance right or left between the 2
points.

Extrapolation
Extrapolation means to estimate values
lying outside the given data.
 You can extrapolate or estimate values
beyond the graphed data by extending the
line and reading new ordered pairs.

Interpolation
Interpolation means to estimate values
lying between given data.
 You can interpolate data by several
methods; however, I suggest using the
method of inspection when you are
interpolating data from a graph.

2.1: Represent patterns in a variety
of formats
Patterns and relations can be represented
in many different ways.
 Data tables, equations, diagrams/graphs
and word descriptions can all represent a
pattern and/or relation.
 To create a pattern, we need to have data.

The types of data

There a 2 types of
data:


Continuous data
Discrete data
Continuous data
 Continuous
data is a set of data where
a variable can be any real number.
 This type of data represents the
measure of a quantity that allows for
continuous change, such as speed or
temperature.
Discrete Data
 Discrete
data is a set of data where a
variable must be a whole number.
 This type of data represents a fixed
quantity, such as the number of pages
in a book or the number of students in
a class.
Interpreting a pattern
 To
interpret a pattern, we should
choose a relation that represents
specifically and accurately the collected
data.
 These relations are discussed in 2.2
2.2: Interpreting linear and nonlinear relationships
In this section, we are
going go learn about
three types of
relations:
 Each relation is easily
identified by its graph
(see page 82 of book)

A linear relation
 An exponential
relation
 A quadratic relation
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A linear relation
A
linear relation is a relation between 2
variables that appears as a straight line
when graphed.
 For example, y=2x+1 is a linear relation.
An exponential relation
 An
exponential relation is a relation
between 2 variables where one of the
variables is an exponent.
 For example, y=2x and y=4x are
exponential relations.
A quadratic relation
A
quadratic relation is a relation between
2 variables that appears as a parabola
when graphed.
 For example, y=x2 and y=-x2+8 are
quadratic relations.
 A parabola is a U-shaped curve and is
the graph of a quadratic relation.
Interpreting relations
To compare and interpret different
relations, it is necessary to complete a
table of values for each different relation.
 In this table of values, you will have
several ordered pairs for each different
relation.
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The table of values

1.
2.
This table always has 2 columns:
A column for the independent
variable (x)
A second column for the dependent
variable (y)
Evaluating ordered pairs
 To
evaluate the value of y:
 Substitute each value of x directly into
the relation that you are using then
evaluate the expression to find the
exact value of y
 These 2 coordinates, x and y, are
going to create an ordered pair that
you can place of a Cartesian grid.
An example of a table of values for
a linear relation

For example, consider
the linear relation,
y=2x
X
Y=2X
(X,Y)
0
0
(0,0)
1
2
(1,2)
2
4
(2,4)
3
6
(3,6)
4
8
(4,8)
5
10
(5,10)
2.3: Finding the slope of a straight
line
To determine the slope of a straight line,
you need to find the line of best fit for
your scatter plot.
 A scatter plot is a diagram that contains
several ordered pairs situated on a
Cartesian grid.
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The Line of Best Fit
The line of best fit is the straight line that
passes through or near as possible to the
points on a scatter plot.
 After finding the line of best fit, we can
find the slope of the straight line.
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The Incline of a Line
The incline of a straight line is the
measure of the steepness of a straight
line.
 We can measure the steepness of a
straight line by finding the slope of the
straight line.
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The Slope of a Straight Line
The slope of a straight line is the ratio of
the vertical change to the horizontal
change.
 The vertical change is the vertical distance
between 2 points (aka the rise)
 The horizontal changes is the horizontal
distance between 2 points (aka the run)
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How to calculate the slope
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1.
2.
We can calculate the slope of the line:
Directly by inspecting the diagram of the
straight line.
Mathematically by using 2 ordered pairs
that are found directly on the straight
line.
Finding the slope of a straight line
mathematically

The equation of the slope of a straight line is:

Rise/Run

= (y2-y1)/ (x2-x1)

(x1,y1) is the start destination and (x2,y2) is the
final destination.
An example of the slope of a
straight line
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1.
2.
For example, a slope of ½ indicates
that:
The Rise is 1 unit upwards from its
Reference Point.
The Run is 2 units to the right from its
Reference Point.
The types of slopes
A
straight line that has a positive slope
slopes upward, from left to right.
 A straight line that has a negative slope
slopes downward, from left to right.
2.4: The equation of a Straight Line
You can write the
equation of a line (a
linear relation) in this
form:
 y = mx + b
 The symbols in this
equation mean the
following:

y is the dependent
variable
 m is the slope of the
straight line
 x is the independent
variable
 b is the y-intercept
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How to create a straight (linear)
line
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1.
2.
To prepare any straight, linear line, one
needs 2 things:
A start point (typically the y-intercept, b)
A slope, m
The y-intercept
The y-intercept is y-coordinate of the point
where a line or curve crosses the y-axis.
 The x-coordinate of the y-intercept is
always zero.
 The y-intercept is represented in the
equation of a straight line by the letter b

The x-intercept
The x-intercept is the x-coordinate of the
point where a line or curve crosses the xaxis.
 The y-coordinate of the x-intercept is
always zero.
 The x-intercept is represented by the
letter a.
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2.5: Diagrams of horizontal and
vertical lines
 Up
to this point, we have only
discussed diagonal straight lines (i.e.
straight lines with a positive or
negative slope)
 However, there are 2 other types of
straight lines: Horizontal lines and
Vertical lines
The Diagram of a Horizontal Line
A
slope of zero (i.e. a slope that has a
vertical change of zero) always
indicates a horizontal line.
 For example, the equation of the
horizontal line y = 4 should be
thought of as such:
 Y = 0/1 + 4
The Diagram of a Vertical Line
An undefined slope (i.e. a slope that has a
horizontal change of zero) always
indicates a vertical line.
 For example, the equation of the vertical
line x = 4 should be thought of as such:
 A vertical line starting at (4,0) and rising
up one unit and moving horizontally zero
units.
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Summary of Chapter 2

What did we learn in Chapter 2? What
concepts?