Chapter 12
Graphing Functions
You already know how to turn a straight line's equation into a picture. Most rules are not straight, though. Square an input, or take its square root, and the pairs of numbers it produces stop lining up; a column of them tells you almost nothing about what the picture would look like. A graph gathers every input and its output into one shape you can take in at a glance. And a shape is something you can move and reshape, which is a strange thing to be able to do to a rule.
What You'll Explore
4 lessons.
- Graphs of Functions
Once a rule is drawn, what can you actually get back out of the picture? This lesson plots a function from a table of values, then works the other way, reading outputs off the finished curve and finding what its crossings with the axes, its outer edges, and its climbs and dips are telling you.
- Shifting Graphs
Change a rule by adding a number to it and the graph you get is the same shape, just parked somewhere else on the grid. Where does it land, and does adding that number inside the function do the same thing as adding it outside? You will follow how one small edit relocates an entire graph.
- Stretching and Reflecting Graphs
Sliding a graph leaves its shape untouched. Multiply inside or outside the rule instead and the shape itself gives way: a graph can stretch taller, pinch narrower, or turn over completely. You will sort out which multiplications produce which of those effects, and one of the answers is not the one most people expect.
- Graphs of Inverse Functions
Undoing a function means swapping every input with its output. Do that to every point of a graph and the new picture is not scattered at random; it stands in a strikingly regular relationship to the original. This lesson hunts for that relationship, and asks which functions can be undone this way at all.