4: Graphs

  • 4.1: Use the Rectangular Coordinate System
    Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.
    • 4.1E: Exercises
  • 4.2: Graph Linear Equations in Two Variables
    • 4.2E: Exercises
  • 4.3: Graph with Intercepts
    When graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, but if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.
    • 4.3E: Exercises
  • 4.4: Understanding the Slope of a Line
    When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat? In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world: the pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples.
    • 4.4E: Exercises
  • 4.5: Use the Slope–Intercept Form of an Equation of a Line
    We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.
    • 4.5E: Exercises
  • 4.6: Find the Equation of a Line
    The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line.
    • 4.6E: Exercises
  • 4.7: Graphs of Linear Inequalities
    • 4.7E: Exercises
  • Chapter 4 Review Exercises

4: Graphs

A graph is a data structure that consists of the following two components:
1. A finite set of vertices also called as nodes.
2. A finite set of ordered pair of the form (u, v) called as edge. The pair is ordered because (u, v) is not the same as (v, u) in case of a directed graph(di-graph). The pair of the form (u, v) indicates that there is an edge from vertex u to vertex v. The edges may contain weight/value/cost.
Graphs are used to represent many real-life applications: Graphs are used to represent networks. The networks may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like linkedIn, Facebook. For example, in Facebook, each person is represented with a vertex(or node). Each node is a structure and contains information like person id, name, gender, and locale. See this for more applications of graph.
Following is an example of an undirected graph with 5 vertices.

The following two are the most commonly used representations of a graph.
1. Adjacency Matrix
2. Adjacency List
There are other representations also like, Incidence Matrix and Incidence List. The choice of graph representation is situation-specific. It totally depends on the type of operations to be performed and ease of use.
Adjacency Matrix:
Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w.

The adjacency matrix for the above example graph is:

Pros: Representation is easier to implement and follow. Removing an edge takes O(1) time. Queries like whether there is an edge from vertex ‘u’ to vertex ‘v’ are efficient and can be done O(1).
Cons: Consumes more space O(V^2). Even if the graph is sparse(contains less number of edges), it consumes the same space. Adding a vertex is O(V^2) time.
Please see this for a sample Python implementation of adjacency matrix.
Adjacency List:
An array of lists is used. The size of the array is equal to the number of vertices. Let the array be an array[]. An entry array[i] represents the list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is the adjacency list representation of the above graph.

Note that in the below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of the linked list. The vector implementation has advantages of cache friendliness.

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The file is a self-extracting installation program. You just execute the file and the installation will ask in which directory you want to install Graph. The installation will add a shortcut to Start->Programs->Graph, which may be used to run Graph.

If an older version of Graph is already installed, you can just install the new version over it. There is no need to uninstall the old version first, but make sure Graph is not running during the installation process.

Under the installation you can choose between different languages for the user interface. This may be changed later from inside the program.

If later you wish to remove Graph, you can use the shortcut in Start→Programs→Graph.

4. Representations of graphs

A good graph representation will allow us to answer one or both of these questions quickly. There are generally two standard representations of graphs that are used in graph algorithms, depending on which question is more important.

For both representations, we simplify the representation task by insisting that vertices be labeled 0, 1, 2, . n-1, where n is the number of vertices in the graph. If we have a graph with different vertex labels (say, airport codes), we can enforce an integer labeling by a preprocessing step where we assign integer labels, and then translate the integer labels back into more useful user labels afterwards. The preprocessing step can usually be done in O(n) time, which is likely to be smaller than the cost of whatever algorithm we are running on our graph, and the savings in code complexity and running time from working with just integer labels will pay this cost back many times over.

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Graph Paper Template

Other Versions

Template Details

"No installation, no macros - just a simple spreadsheet" - by Jon Wittwer


This Excel (.xls) file contains square grids of different sizes, triangular, and isometric graph paper.

Excel can provide a useful grid-based drawing canvas for landscape plans and simple architectural floor plans. Many people may not realize that the drawing tools in Office are basically the same in Excel, Word, and Powerpoint. In Excel, you can snap to the grid by holding the Alt key as you insert shapes and lines. You can also use another worksheet within the same workbook to do calculations.

Note: If you are just looking for printable graph paper, the Word (.docx) templates available on the new printable graph paper page provide more accurate measurements and aspect ratios.

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Resizing the chart in Excel

To resize the Excel graph, click on it, and then drag the sizing handles to the size you want.

Alternatively, you can enter the desired chart height and width in the Shape Height and Shape Width boxes on the Format tab, in the Size group:

For more options, click the Dialog Box Launcher next to Size and configure the needed parameters on the pane.

Watch the video: Stata Tutorial: Combining Graphs (December 2021).