while b is varied and the variables a and c are held constant. So let's start with the graph where all variables, a, b and c, are equal to 1.
The graph opens upward because a is postive. If a was negative the graph would open in the negative direcction. It is not symmetrical around the yaxis because c = 1. Because c = 1, when x = 0, the parabola passes through the point (0, 1). If c equaled 0, then the parabola would be symmetrical around the yaxis.
Now let's look at what happens when we change b, while a and c remain 1.
For these positive values of b, the graph always intersects the yaxis at the point (0, 1). The vertex is always to the left of the graph when b > 0. When b = 0, the vertex is on the yaxis. We know that where the parabola intersects or hits the xaxis is where the real roots of that particular equation occur. For those graphs that do not intersect or hit the xaxis, then they do not have any real roots. For example, when b = 1, the graph does not hit the xaxis, therefore it does not have any real roots. Notice that when tb is positive, the real roots occur on the negative side of the yaxis.
Now let's see what happens when b is negative.
For these negative values of b, the graph always intersects the yaxis at the point (0, 1). The vertex is always to the right of the graph when b > 0. When b = 0, the vertex is on the yaxis. We know that where the parabola intersects or hits the xaxis is where the real roots of that particular equation occur. For those graphs that do not intersect or hit the xaxis, then they do not have any real roots. For example, when b = 1, the...
... 1. PIE CHART
This pie chart shows Mark’s monthly budget. The highest designation of his budget will go to his foods with 45% of his total allowance. Next is for lodging with 30% followed by the projects and fare which will have 10%. The least designation for his budget will be for his savings which has 5% only.
2. BAR GRAPH
The bar graph shows the yearly tourist count for the provinces of region V. the province of Albay got the highest number of tourist with 450 000. It is followed by the provinces of Camarines Sur and Camarines Norte with 400 000 and 350 000 respectively. Sorsogon got 300 000 and Catanduanes with 250 000. The province of Masbate got the lowest number of tourist with 200 000.
3. LINE CHART
Here is a line chart for the number of absentees in class of Mr. Lozada for the 1st semester in 4 of her subjects. English has the most number of absents with 5 meetings. It is then followed by Math and Science with 4 and 3 meeting respectively while Filipino has the least absentees with only 2 meetings.
4. TABLES
KLINE DORMITORY SPORTS EQUIPMENT SPORT  NUMBER OF EQUIPMENT 
VOLLEYBALL  7 
BADMINTON  7 
SOCCER  4 
BASEBALL  12 

This table shows the number of sport equipment for each of the favorite sport of the KLINE scholars. The dormitory has the most sufficient sport equipment with 12. And Soccer is the sport with less number of equipment with only 4 sport equipment.
5.
PICTOGRAPH
MEMBERS’ SAVINGS...
... 
Male 57 59% 
Female 40 41% 
Total 97 100% 
Table 1 reveals the sex profile of the respondents. As reflected on the table, the male has the larger percentage than the female. Out of 97 respondents, 57 or 59% are male while 40 or 41% are female.
To illustrate visually the sex profile, the graph is presented below.
20
Graph 1
[pic]
Gender Profile of the Respondents
Table 2
Analytical Skills
Respondents  S N Computed t Tabular t Decision Remark 
Male 10.84 2.95 57     
    0.33 1.9852 Accept Ho Significant 
5% level of significant and 26 degrees of freedom
Table 2 reveals the level of significant and the degrees of freedom. As reflected on...
...The Pursue of Happiness
"Dimensions" by Alice Munro is a tragic story that talks about self discovery and the courage to start all over again. Doree is a woman who has been broken in every way, but refuses to give her right to continue to live. The story describes Doree's psychological and emotional metamorphosis from an innocent young girl who has to face many difficulties to become a woman. All the circumstances that she goes through helps her mature, think more critically, and find the strength to pursue the happiness that she yearned.
Doree was sixteen when her mother died of an embolism; and was sheltered, to a certain point, by Lloyd. Doree was a girl who had to leave her adolescence behind at an early age to become a wife/mother/woman and due to her lack of experience in all aspects, she had to depend on her husband, Lloyd. During their whole marriage Doree was isolated from others; having no social skills due to her lack of interaction, she could not establish a bond with any other person strong enough to overcome her need for Lloyd: “It was Lloyd and Doree and their family that mattered…the bond was not something that anybody else could understand”(Munro,6). At that exact moment in her life, she found in Lloyd the love that she desperately needed; especially after the lost of her mother she felt helpless. Lloyd represented a father figure for Doree; he replaced the family that she had lost to become part of a new one.
Throughout the story...
...CHAPTER 4 : FUNCTIONS AND THEIR GRAPHS
4.1 Definition of Function
A function from one set X to another set Y is a rule that assigns each element in X to one element in Y.
4.1.1 Notation
If f denotes a function from X to Y, we write
4.1.2 Domain and range
X is known as the domain of f and Y the range of f. (Note that domain and range are sets.)
4.1.3 Object and image
If and , then x and y are known respectively as the objects and images of f. We can write
, , .
We can represent a function in its general form, that is
f(x) = y.
Example 4.1
a. Given that , find f(0), f(1) and f(2).
Example 4.2
a. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
Solution
a. Given that , find f(0), f(1) and f(2).
b. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
(a)
(b)
4.2 Graphs of Functions
An equation in x and y defines a function y = f(x) if for each value of x there is only one value
of y.
Example:
y = 3x +1, , .
The graph of a function in the xy plane is the set of all points (x, y) where x is the
domain of f and y is the range of f.
Example
Figure 1 below shows the graph of a linear function, the square root function and a general function.
y = f(x)
y = x
(a) (b) (c)
Figure 1
It is easy to read the domain...
...Graphs are often used to deliver a visual and compelling case in many applications and businesses. Graphs are the information delivery vehicle of choice for many numerical data applications. With graphs a lot of information can be condensed into a visually descriptive object. They reduce the amount of time that would have been expended in reading or parsing through a lot of information.
On the other hand, they can also be easily used to misrepresent or skew interpretation towards a favorable outcome. With modern tools like Microsoft Excel, graphs can be made to look more visually appealing and creative. Such appeal could often be used to mask incorrect or misleading information. Robyn Raschke and Paul John Steinbart argued that training users on graph design and making them aware of proper design would often reduce the problem of poor or bad graphs but would not eliminate it altogether (Robyn Raschke & Paul John Steinbart, 23).
There are far too many examples of poor graph in use in the media and other less formal online sources as well. The prevalence of advanced graphing capabilities like those found in Microsoft Excel can lead poorly trained people into creating these graphs (Levine et al. 57). The availability of these advanced graphing tools make it a lot easier to produce bad graphs with a visual appeal. People may be less inclined...
...
Graphs
1 Introduction
We have studied one nonlinear data structure so far i.e Trees. A graph is another nonlinear data structure that is widely used to solve many reallife computing problems. For example, we need to use a graph to find out whether two places on a roadmap are connected and what is the shortest distance between them. Graphs are used in simulating electrical circuits to find out current flows and voltage drops at various points in the circuit. Graphs are widely used in telephone and computer networks.
Graphs have great historical significance too. In 1736, the famous mathematician Euler used the concept of a graph to solve the “Koenigsberg problem”. In the small town of Koenisberg in Prussia, the river Pregal flows around the island of Kneiphof and then divides into two. The four land areas ( A, B, C, D) bordering the river are connected by seven bridges ( a,b,c,d,e,f,g). The problem is to find out whether it is possible to start walking from some area, cross each bridge exactly once and return to the starting land area. Euler used graphs to prove that this would not be possible. A walk which achieves this is called an “Eulerian Walk”.
{{{ Diagram }}}
In this chapter, we will study this data structure, its implementation and its applications. Before that, we will study some definitions and terminology.
Definitions and Terminology...
...V. Adamchik
1
Graph Theory
Victor Adamchik Fall of 2005
Plan
1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs
Introduction
A.Aho and J.Ulman acknowledge that “Fundamentally, computer science is a science of abstraction.” Computer scientists must create abstractions of realworld problems that can be represented and manipulated in a computer. Sometimes the process of abstraction is simple. For example, we use a logic to design a computer circuits. Another example  scheduling final exams. For successful scheduling we have to take into account associations between courses, students and rooms. Such set of connections between items is modeled by graphs. Let me reiterate, in our model the set of items (courses, students and rooms) won't be much helpful. We also have to have a set of connections between pairs of items, because we need to study the relationships between connections. The basic idea of graphs were introduced in 18th century by the great Swiss mathematician Leonhard Euler. He used graphs to solve the famous Königsberg bridge problem. Here is a picture (taken from the internet)
V. Adamchik
21127: Concepts of Mathematics
German city of Königsberg (now it is Russian Kaliningrad) was situated on the river Pregel. It had a park situated on the banks of the river and two islands. Mainland and islands were joined by seven bridges. A problem was whether it...
...Graph Theory
GraphsGraph: A graph consists of a nonempty set of points or vertices, and a set of edges that link together the vertices. A simple real world example of a graph would be your house and the corner store. Where the house and the store are the vertices and the road between them is the edge connecting the two vertices.
Or a graph is a network consisting of vertices (or nodes) and edges (V,E)
Simple Graph
A graph can take on many forms: directed or undirected.
Directed Graph: A directed graph is one in which the direction of any given edge is defined.
Or A graph with directed edges = directed graph (digraph)
Directed edges = arcs
Directed Graph
Undirected Graph: An undirected graph is one in which the direction of any given edge is not defined. Conversely, in an undirected graph you can move in both directions between vertices. Or a graph with undirected edges is called undirected graph.
Undirected graph
Mixed Graph: A graph is one in which contains both directed...
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