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 y-axis 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 y-axis.

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 y-axis 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 y-axis. We know that where the parabola intersects or hits the x-axis is where the real roots of that particular equation occur. For those graphs that do not intersect or hit the x-axis, then they do not have any real roots. For example, when b = 1, the graph does not hit the x-axis, therefore it does not have any real roots. Notice that when tb is positive, the real roots occur on the negative side of the y-axis.

Now let's see what happens when b is negative.

For these negative values of b, the graph always intersects the y-axis 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 y-axis. We know that where the parabola intersects or hits the x-axis is where the real roots of that particular equation occur. For those graphs that do not intersect or hit the x-axis, then they do not have any real roots. For example, when b = -1, the...

...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...

...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...

...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...

...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...

...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 real-world problems that can be represented and manipulated in a computer. Sometimes the process of abstraction is simple. For example, we use a logic to...

...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...

...
Graphs
1 Introduction
We have studied one non-linear data structure so far i.e Trees. A graph is another non-linear data structure that is widely used to solve many real-life computing problems. For example, we need to use a graph to find out whether two places on a road-map 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...

...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...

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