Chapter 5
Graphing Lines
A bike rental charges a flat fee plus a set amount for each hour. You can write down the cost of one hour, then two, then three, but the list never ends and it never shows the shape of the deal. Plot those pairs on a grid instead and they do not scatter. They fall in a perfectly straight row. A line and an equation turn out to be two views of the same relationship, and that raises a question: what can you read off one that the other keeps hidden?
What You'll Explore
5 lessons.
- Graphing Linear Equations
An equation with two letters does not have a single answer; endlessly many pairs of numbers make it true. Plotting them all would take forever, yet the picture they make is the same simple shape every time. Here you will find out why it comes out straight, and how little of it you need to draw.
- Slope
Two roads can both climb, but only one makes you get off your bike and walk. Steepness is obvious to your legs and vague in words, so the work here is to pin it down with a single number, and to ask whether it matters which stretch of the line you measure.
- Finding the Equation of a Line
Suppose you know one point a line passes through and how steeply it climbs. Is that enough to say exactly which line it is, or could many lines fit that description? The task here runs backwards, from a couple of facts about a line to an equation that names it exactly.
- Slope and Intercepts
Written one way, a line's equation tells you almost nothing about how it looks. Written another way, its steepness and the height it starts from sit right there in the symbols. That second arrangement is what you will explore, along with how much of a graph those few numbers already decide.
- Parallel and Perpendicular Lines
Railway tracks never meet, and a street corner turns exactly square. Both facts are easy to see in a picture and far less obvious in a pair of equations. What has to be true of two equations for either relationship to hold is the question here, along with the lines that break the pattern.