Math 1 · Unit 2 · Lesson 2 · part 2 of 6

Fill in the gaps

Lesson 1 ended with a promise.

Fill in the gaps

Lesson 1 ended with a promise. You filled in a table for y = 2x − 1, plotted the rows, and the dots lined up. Here they are again. Same rule, same six points.

Now fill in the gaps. Nothing in the rule says x has to be a whole number. Try an in-between input, x = 0.5. Feed it in: y = 2 × 0.5 − 1 = 1 − 1 = 0. That is the point (0.5, 0), and it sits exactly between the dots at (0, −1) and (1, 1). Try x = 1.5: y = 2 × 1.5 − 1 = 2, the point (1.5, 2). Try x = 2.25: y = 2 × 2.25 − 1 = 3.5, the point (2.25, 3.5). Every one of them lands on the same straight path through the dots.

The leap. Any input at all, a decimal, a fraction, a negative, has exactly one output, because that is what a function (a rule with one output for each input) promises. One input and its output is one point. Every one of those points sits on the same straight path. So the line is not a new thing: it is all of the points of the rule at once, every input you could ever choose, drawn together.

That is why we can draw a line through the dots. We are not guessing what happens between them. We are drawing every in-between point, all at the same time.

Where to go next

Next: Draw the line, then test itTwo rules to graph. All the partsConnect the Dots: From Points to a Line