About Graphing Lines

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Anis Goodman
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1 About Graphing Lines

2 TABLE OF CONTENTS About Graphing Lines... 1 What is a LINE SEGMENT?... 1 Ordered Pairs... 1 Cartesian Co-ordinate System... 1 Ordered Pairs... 2 Line Segments... 2 Slope of a Line Segment... 2 Slope Properties... 3 Types of Line Segments... 4 Midpoint Formula... 5 Midpoint Formula... 5 Distance Formula... 5 Right Triangle... 5 Pythagorean Theorem... 6 Distance Formula... 6 Glossary... 7 References... 9

3 What is a LINE SEGMENT? About Graphing Lines A line segment is composed of two points on a line and all the points between those two points. Line segments are generally used in mathematics to calculate the slope of the line between two points. The two points are typically called an ordered pair (or co-ordinates), with the first number being a value on the horizontal (or x) axis, and the second number being a value on the vertical (or y) axis. Cartesian Co-ordinate System Ordered Pairs With a graph consisting of x and y axes, intersecting at what is known as the origin (0,0), the system is designed to uniquely classify individual points within the plane. Within the ordered pairs, the first value can be read from the horizontal (or x) axis; whereas, the second value can be read from the vertical (or y) axis. The plane is broken into four quadrants by the intersection of these axes, resulting in the possible signs (i.e. positive or negative) of x and y, as shown on this diagram. 1

4 Ordered Pairs Ordered pairs refer to groups of two numbers, representing unique points on the Cartesian plane, in which the first number represents a value on the horizontal (or x) axis and the second number represents a value on the vertical (or y) axis. Ordered pairs can be placed anywhere on the Cartesian plane, including on both the x and y axes. The Cartesian plane can also be drawn to eliminate unused quadrants. Note, each point on the plane has a unique corresponding ordered pair of real numbers (x, y). Equivalently, each ordered pair has only one point on the plane. Slope of a Line Segment Line Segments Given two ordered pairs, one can draw a line to connect them, which is known as a line segment. In the given diagram, the section of the line between A and B is what would be identified as a line segment. 2

5 To find the slope of the line, simply divide the rise by the run. Try to be consistent in your choice of formula, as you may run into some confusion otherwise! Slope Properties Provided is a comprehensive list of the properties of slope. 3

6 Types of Line Segments There are a number of types of line segments. The most common is known as a co-linear line segment: Collinearity refers to three or more points lying on the same straight line (or line segment). In the diagram given above, D, E, and F are colinear. Another common type of line segment is a parallel line segment: A parallel line segment refers to one line segment parallel to another. Both line segments have equivalent slope, as in the above. Perpendicular line segments are also common. A perpendicular line segment refers to the intersection of one line segment with another, forming a right (or 90 o ) angle. The slopes of these segments are negative reciprocals of one another, as in the diagram below. 4

7 Midpoint Formula Midpoint Formula A midpoint is the point equidistant from the endpoints in a line segment. This ordered pair is found using the midpoint formula. Given A ( x1, y1 ) and B ( x2, y2 ): This formula may seem intuitive, but it comes up often, making it advantageous to know. Right Triangle Distance Formula Before considering the distance formula, first consider the basics behind the formula. Begin by recalling the Cartesian plane and how you found the slope of a line segment. It is apparent that the rise and run lines intersect to form a right-angled triangle. 5

8 Pythagorean Theorem By using the triangle formed by the rise and run lines, one can find the distance between the two corresponding points on the line segment, using what is known as the Pythagorean theorem. By the Pythagorean theorem, if you have the length of (or distance between) 2 of the 3 sides of the triangle, you can calculate the length of the remaining side. Distance Formula The distance formula follows from the Pythagorean Theorem. This equation, finding the distance between A and B, simply represents a rearrangement of the equation corresponding to the Pythagorean theorem. 6

9 Glossary Cartesian plane: Co-linear Line Segment: Co-ordinate: Distance: Infinity: Line segment: Midpoint: Ordered Pair: Origin: Parallel Line Segment: A graphical representation of all points, each uniquely represented by ordered pairs of the form (x, y). a line segment containing 3 or more points from the original line. is a point on the Cartesian plane that has both a horizontal (x) and a vertical (y) value. the length between two points or ordered pairs. the maximum or minimum (if negative) limit for all real numbers. a portion of a line, drawn between two points on the line. the half way point between two ordered pairs or points in a line segment. a group of two numbers, where the first number represents a value on the horizontal (or x) axis, and the second number represents a value on the vertical (or y) axis in the plane. the midpoint of the co-ordinate plane, usually being (0, 0). formed when there are two line segments parallel to one another. 7

10 Perpendicular Line Segment: formed when two line segments intersect to form a right-angled (or 90 degree) angle. Plane: Pythagorean Theorem: Quadrant: Rise: Run: Slope: x-axis: y-axis: made up of the vertical and horizontal number lines that intersect at (0, 0). an equation allowing the calculation of the length of the sides of a right-angled triangle, given the length of two of the three sides. one of the four quarters of the Cartesian plane. the distance between the y co-ordinates of two ordered pairs on a line segment. the distance between the x co-ordinates of two ordered pairs on a line segment. calculated using two numbers on the line, demonstrates the steepness (or incline) of the line. known as the horizontal number line that runs left to right on the Cartesian plane. known as the vertical number line that runs up and down the Cartesian plane. 8

11 References 9

The Rectangular Coordinate System and Equations of Lines. College Algebra

The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis