Download Module 5 notes Graphing -Graphs are used often to demonstrate the

Module 5 notes
Graphing
-Graphs are used often to demonstrate the relationship between two variables.
Graphs are plotted on the coordinate plane also known as the Cartesian plane. The
horizontal axis is the x-axis; the vertical axis is the y-axis. The point where the x-axis
and y-axis intersect is the origin and is denoted by the letter O.
-The position of any point on the coordinate/Cartesian plane is described by using
the ordered pair: (x, y). The first number in the ordered pair, x, is the horizontal
position of the point from the origin. It is called the x-coordinate. The second
number, y, is the vertical position of the point from the origin. It is called the ycoordinate.
To plot a point:
-1 Always start at the origin to find a
point. (The origin is the point (0,0)).
-2 Move in the x direction first - right for a
positive value, left for a negative value.
-3 From that position, move up or down
from the y direction. Move up for positive
and down for negative.
Independent and Dependent Variables
-Graphs show how the independent and dependent variable are related.
-The independent variable is what you, the experimenter, changes to do your
experiment. It appears on the x-axis/horizontal axis.
-The dependent variable is the result of the change in the independent variable. It
appears on the y-axis/vertical axis.
Graphs that compare two variables should always have the following components:
1) A title provides some information about what the graph displays.
2) The horizontal or x-axis is labeled including units. This axis indicates the
independent variable.
3) The vertical or y-axis is labeled including units. This axis indicates the
dependent variable. The dependent variable is affected by changes in the
independent variable.
4) The axes must increase by standard increments. In the example, the x-axis
increases by 1, and the y-axis increases by 20.
5) Depending on the data in the graph, you will
Use a line graph when the data is continuous.
Use no line when the data is discrete.
Continuous and Discrete Data
-Data is continuous if the data can be measured and broken into smaller parts and
still have meaning. There are real values between points and, therefore, the data is
continuous.
The following graph shows how the quantity of black forest ham affects the total
cost. The dots are connected because one could purchase 500 grams or 0.5 kg of
Black Forest ham. The cost for 500 grams of Black Forest ham according to the
graph is $10.
In this example, the data is continuous because the weight of the ham purchased
could be any value between the points chosen.
-Data is discrete if the data can be measured but not broken into smaller parts and
still have meaning. There are no real values between points and, therefore, the data
is discrete.
No line is used when the data is said to be discrete. In this next example, the data is
discrete.
In the graph, 0.5 or 1.5 or 2.5 of a dime is not possible. This demonstrates that, in
discrete data, values do not exist between points.
Graphing
-The way points are connected on a graph is highly dependent on the story the
graph is telling. For example, the following graph shows how a cell phone company
bills for air time:
-The relationship of the graph shows that the cell phone company charges $0.50 per
minute or portion thereof. (Obviously, this particular company does not have per
second billing.) A 20-second cell phone call is billed for one full minute. (It’s a rip-off,
but accept it for this example!) The vertical points on the graph are not connected
because it is impossible to have a phone call billed $0.75. The cost is either $0.50 or
$1.
-The open dot indicates that the value is not included in that portion of the graph.
For example, the open dot at x = 1 means that any phone call that it is more than one
minute is $1.00.
-The closed dot indicates that the value is included in the line. The closed dot at x =
1 means that the value is included in that portion of the graph. In other words, a 1minute call costs $0.50.
Domain and Range
-The domain is all possible input values. These values are often the x-values, and
they represent the horizontal values on a graph, and the first value in ordered pairs.
-The range is all possible output values. These values are often the y-values. They
represent the vertical values on a graph, and the second value in ordered pairs.
Example:
For the following relations, identify the domain and range.
A {(1, 5), (4, 3), (4, 9), (5, 5)}
B {(-3, -3), (-1, -1), (0, 4), (5, 9)}
C
Solutions
Domain: (1, 4, 5)
Range: (3, 5, 9)
Domain: (-3, -1, 0, 5)
Range: (-3, -1, 4, 9)
Domain: (-3, 1, 4)
Range: (-2, 1, 3, 4)
Using Graphs to Identify Domain and Range
Example:
-The x-values for this partial line begin at -1 and end at 2. Therefore, the domain or
input values are all values for x such that x is greater than or equal to -1 and less
than or equal to 2. This may be written as: -1< x < 2, and may be read as "x is
between -1 and 2, inclusive." The range of output values for the domain defined are
all y values that are greater than or equal to -1 and less than or equal to 5. This may
be written as: -1< y <5 and may be read as "y is between -1 and 5, inclusive."
Expressing Domain and Range
The notation used to express domain and range can be expressed in various ways:
- words
- a number line
ex.
In this case, the open circle around -1 means that -1 is not included. The solid dots
means that the 8 is included in the set of numbers. So the domain here is greater
than -1 and less than or equal to 8.
- a list
ex.
the domain is {0, 2, 4, 6}
the range is {0, 5, 9, 13}
- interval notation
ex.
Domain [0, ∞)
Range [0, ∞)
Interval notation uses different brackets to indicate an interval.
This style of
bracket "]" is used if the end number is included.
This style of bracket ")" is used
if the end number is not included.
- set notation/algebraic form
ex.
Domain {x| 3 ≤ x ≤ 20, x  R}
Range {y| y > 0, y  R}
Relation and Functions:
Relation – a rule that associates the elements of one set with the elements of
another set (a set of ordered pairs).
Set – a collection of distinct objects
Element – One distinct object in a set.
ie.) {2, 4, 6, 8} = a set of natural numbers with elements 2, 4, 6, and 8.
When we represent relations using numbers, a relation is a set of ordered pairs. The
elements in the relation are the numbers that represent specific coordinate points
on a Cartesian plane.
ie.) {(2,4), (4,8), (6,12)} is a relation.
Relations can be represented in a number of ways:
1.
2.
3.
4.
5.
Table of Values
Graphs
Arrow Diagrams (mapping diagram)
Equations
In Words (description)
Example #1 – Represent the following relation in 5 different ways:
{(1,3), (2,4), (3,5)}
Recall: (x, y)
Table of Values
x
1
2
3
y
3
4
5
Graph
Equation
Words
Y
6
y=x+2
5
4
3
two.
2
1
X
0
1
2
3
4
5
6
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
y is always equal to
the value of x plus
-Another way to represent a relation is by drawing an “arrow diagram” or “mapping
diagram”. Each element represented by an x (domain discussed in L2) in the
coordinate points is matched with an element represented by a y (range discussed
in L2) coordinate.
eg.) Draw an arrow diagram for the following relation:
{(3,7), (0,4), (-1,0), (-5,-1)}
x
3
0
-1
-5
y
4
0
7
-1
Relation vs. Function
-Recall: A relation is a set of ordered pairs. A relation produces one or more output
numbers for every valid input number.
eg.) {(-2, 3), (3, 4), (4, 7), (-2, 6)}
Notice that we get 2 different output numbers for an input of -2. (3 and 6)
-A function is also a set of ordered pairs, however, for every valid input number,
there is only ONE output number.
eg.) {(0, 2), (2, 6), (3, 8), (4, 10)
-In order to determine if a graph is a relation that is NOT a function, or whether it is
in fact a function, we can use what we call the “Vertical Line Test”. This means that
if we draw a vertical line anywhere on the graph, and 2 or more points on the graph
are touching (lie on) that line, then the graph is NOT a function.
Function Notation
-To use a generic form is much easier than always using y = for defining functions.
Instead, we can use function notation that begins with f(x) = .
-f(x) is read as “f of x” and means the same thing as y. f(x) is used because it gives
more information than y. On your graphing calculator, you will define functions for
graphing as y1, y2, and so on. In most math courses, however, you will see f(x), g(x),
h(x), and so on.
-Functions give us opportunities to analyze what we see around us and come up
with some rules that help to explain what we experience.
-The domain is the input and the range is the output. In a function, each member of
the domain is paired with exactly one member of the range. Each member of the
range may be paired with one or more members of the domain.
-Note that in function notation, f(x) = x + 1 is the same as y = x + 1.
Evaluating functions
Consider the function f(x) = 2x - 3, and ask what is its value when x is 2?
f(x) = 2x – 3
f(2) = 2(2) – 3
f(2) = 4 – 3
f(2) = 1
Generating a Table of Values
-Given the graph of a line, you can generate the values of a function or a table of
values. For the following line, name the points in blue:
First, identify the points: {(-4, 7), (-2, 6), (0, 5), (2, 4), (4, 3)}
This is also the following table of values:
Graphing a Line
The equation of a line may look like the following: y = 2x + 5
In function notation, it would look like: f(x) = 2x + 5
To generate the points and complete a table of values, you would use the
substitution method that you used for evaluating functions above. In the table, y
becomes f(x) but you still graph the ordered pairs in the same way.
Generate the table of values for the function f(x) = 2x + 5 and graph the line.