Unit 5: Graphs of Linear Equations and Inequalities
We use graphs to help us visualize how one quantity relates to another. This unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how we can use lines to represent equations and relationships.
For example, we can graph how the location of a train depends on when it left the station. If the train is moving at constant speed, the line in the graph is straight. The slope or slant of the line depends on the speed: the greater the speed, the steeper the line. If the line is going up (from left to right), it tells you the distance is growing with time: the train is moving away from the station. If the line is going down, it tells you the distance is decreasing: the train is approaching the station. You can gather a lot of information about the train's journey from just one graph.
Completing this unit should take you approximately 5 hours.
5.1: Graphing Points in the Rectangular Coordinate Plane
First, we need to understand the coordinate plane, the space in which we produce graphs. In this section we will focus on finding and graphing points on the coordinate plane to become comfortable with it.
5.2: Ordered Pairs as Solutions of an Equation in Two Variables
Now we are ready to begin using graphs to determine if a pair of numbers (an ordered pair) is a solution to an equation.
5.3: Graphing Equations in Two Variables of the Form Ax + By = C
A common way equations can be written is: Ax + By = C, where A, B, and C are numbers.
5.4: Intercepts of a Straight Line
One of the properties of linear graphs is that they have intercepts on the x- and y-axis. The intercept is the point at which the line crosses the axis.
5.5: Definition of Slope and Slope Formula
Another important property of linear graphs is the slope of the graph. The slope tells us how steep the line is. When a linear equation is written in a specific form that we’ll discuss later, the slope helps us determine how to graph the line.
5.6: Slopes of Parallel and Perpendicular Lines
Here, we learn about how the slopes of parallel and perpendicular lines are related. Parallel lines have the same slope, while perpendicular lines have slopes that are reciprocals.
5.7: Graphing Equations in Two Variables of the Form y = mx + b
One of the most common types of graph is that of a line with the form y = mx + b. In this form, m is the slope of the line, and b is the y-intercept of the line. When an equation is in this form, it is easy to plot the linear graph, so it is important to be able to recognize when an equation is in this form.
5.8: Point-Slope Form
In the last section we discussed the slope-intercept form of a linear equation. We can also write linear equations in a form known as the point-slope form. This form works for when you want to make a line between two known points.
This form is: y − y1 = m(x − x1).
5.9: Graphing Linear Inequality of Two Variables on the Coordinate Plane
The last type of linear graphing we need to study is the graph of an inequality rather than an equation. When we graph inequalities, we must pay attention not only to the numbers and variables but also the inequality itself. That is, are we graphing a less-than, or greater-than inequality?