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David Terr
Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> 1. Graphs, Functions, and Models

>> 1.2. Lines and Slopes

 

1.1. Graphs and Graphing Utilities

Graphing is an indispensable mathematical tool. They say a picture is worth a thousand words, and a graph provides a useful pictorial representation of a mathematical formula or set of formulas. In this chapter, we explain how to graph points, lines, circles, and some simple functions.

Most graphs are two-dimensional, i.e. they represent two variables, usually denoted x and y. The variable x is represented as the horizontal distance from a central point in the graph, called the origin. Similarly, y is represented as the vertical distance from the origin. Units are marked on vertical and horizontal lines, known as axes, which cross at the origin. The horizontal axis is known as the x-axis and has units of the variable x marked along it. Similarly, the vertical axis is known as the y-axis and has units of the variable y marked along it. All this is shown in the following figure, known as the rectangular or Cartesian coordinate system. The name Cartesian comes from the French mathematician and philospher René Descartes (1596-1650), who invented this system in 1637.

rectangular coordinate system

Figure 1.1.1: Rectangular (Cartesian) Coordinate System

 

Graphing points is easy; one simply locates the coordinates of a given point along the x-axis and y-axis and draws it at this location. As an example, we graph the point P=(3,4). This is the point with x=3 and y=4 as shown. The dashed vertical and horizontal line segments going from the axis to P are not part of the graph; they are just shown here in order to illustrate how to locate P.

rectangular coordinate system with labeled point

 

Once we can graph points, it is easy to graph lines and line segments connecting pairs of points, as well as polygons. The following examples illustrate how to do so.

 

Example 1: Graph the line segment connecting the points P=(-3,2) and Q=(4,-1).

Solution: First we graph the points P and Q, then we draw the line segment connecting these points. The resulting graph is shown below:

xy graph with line segment

 

Example 2: Graph the triangle with vertices P=(-3,-3), Q=(0,3), and R=(3,-3).

Solution: First we graph the points P,Q, and R; then we draw the line segments PQ, QR, and RP. The resulting graph is shown below:

triangle graph

 

There are many utilities for rendering graphs. The simplest one is graph paper. There are several types of graph paper. The standard type is rectangular graph paper, which is what most people are most familiar with. Rectangular graph paper is marked with a square grid of vertical and horizontal lines, from which it is easy to draw and label the x-axis and y-axis and draw graphs as explained above. In addition, there are several specialized forms of graph paper, each of which is useful for drawing particular types of graphs. Other types include polar, log-log, and semilog graph paper. More sophistocated graphing utilities include computers, graphing software and graphing calculators.

 

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