Title: The Basics: Graphs
Course: Open MAT107 Contextualized > Section: Graphs

The Basics:

Graphing Lines

The Rectangular Coordinate System (or Cartesian Coordinate System, named after French mathematician and philosopher René Descartes) is a coordinate system that is used to graph lines in the plane. The Cartesian plane consists of two intersecting axes; the x-axis or horizontal axis, and the y-axis or vertical axis. The axes intersect in a point called the origin, in which it is assigned the ordered pair (0, 0). An ordered pair (or coordinate, or point) is a pair of numbers that are ordered so that the x-coordinate (or abscissa) is listed first, and the y-coordinate (or ordinate) is listed second to form the point (x, y). To plot ordered pairs in the plane we start at the origin, and then move a horizontal distance of x units (positive is to the right, and negative is to the left) and a vertical distance of y units (positive is up, and negative is down). There are four quadrants in the rectangular coordinate system. Quadrant I starts in the upper right portion, and then the following quadrants are listed in a counterclockwise manner.

Slope-Intercept Form of a Line: Any linear equation of the form y = mx + b is in slope-intercept form where m is the slope and (0, b) is the y-intercept.

x-intercept: The point on the line that touches the x-axis. To find the x-intercept, Let y = 0, then solve for x. The x-intercept written as a coordinate is of the form (a, 0).

y-intercept: The point on the line that touches the y-axis. To find the y-intercept, Let x = 0, then find y. The y-intercept written as a coordinate is of the form (0, b).

Slope: Slope is a rate of increase (positive slope) or a rate of decrease (negative slope) in which the rate remains constant.

Special Forms of Lines:
1. A line with zero slope (m = 0) yields a horizontal line parallel to the x-axis. The equation of this line is of the form y = b, for some constant b. The line y = b must pass through the point (0, b).
2. A line with no slope (or undefined slope) yields a vertical line parallel to the y-axis. The equation of this line is of the form x = a, for some constant a. The line x = a must pass through the point (a, 0).

Graphing Lines in the Plane:
Two points determine a line. If we can find two distinct points that satisfy the line we can then plot the points and draw a straight line through them. To find two distinct points, you may use the x- and y-intercept, or just randomly find two points by substituting an x-value into the linear equation and then finding its corresponding y-value. You may repeat this process to find as many points as you like. If the equation of the line is given to us in slope-intercept form, we may then plot the y-intercept and then use the slope to find another point. Remember that slope is “rise over run”, so a slope of 2, for example, would tell us to go up 2 and to the right 1 from the lines y-intercept to get to another point on the line.

Watch these helpful Khan Academy videos for more information:

The Coordinate Plane

X and Y intercepts

Slope of a line

Graphing a line in slope intercept form

Linear Equations in Slope Intercept Form

Now move on to the energy technology video demonstrations. When you are ready, take the practice quiz to show what you know. If you think you need more practice, ask your instructor if the optional practice assignments would help you.