Algebra 1
Unit 4 · Lesson 4.1

What is a graph

You already have one number line from Lesson 1.2, a single line where every number has its spot. One number is enough to pin a place along that line. But a flat page is bigger than a line. To point at a seat in a row, or a place on a map, one number can't say where; you need two, how far across and then how far up or down.

Two number lines crossing at the center, with a single dot reached by going across and then up.

Every point on the grid gets an address written (x, y). The first number is how far across: right is positive, left is negative. The second is how far up or down: up is positive, down is negative. You always read the across number first. That order is the whole idea, and it's why (3, 2) and (2, 3) are two different points, not one point written two ways. 4.1.f1

x y (3, 2)
Figure 4.1.f1

New words

  • 4.1.d1 Coordinate plane: a flat grid made by two number lines crossing at right angles.
  • 4.1.d2 x-axis / y-axis: the across (horizontal) and up-and-down (vertical) number lines. They cross at the origin, (0, 0).
  • 4.1.d3 Ordered pair (x, y) / coordinates: a point's address: x first (how far across, right is +, left is −), then y (how far up or down, up is +, down is −). Order matters: (3, 2) ≠ (2, 3).
  • 4.1.d4 Plot: to mark the point for an ordered pair by starting at the origin and stepping across, then up or down.

Work through these one at a time, and picture each step on the grid before you read the next line. The move to get comfortable with is starting at the origin every single time.

Worked example

4.1.w1 Example 1: plot (3, 2). Start at the origin. The first number is 3, so step 3 to the right. The second number is 2, so step 2 up. Mark the point there. Across first, then up. 4.1.f2

x y (3, 2)
Figure 4.1.f2

4.1.w2 Example 2: plot (−2, −1). Start at the origin again. The first number is −2, so step 2 to the left (left because it's negative). The second number is −1, so step 1 down (down because it's negative). The point lands below and to the left of the origin. Same routine as before; only the directions change with the signs. Here it is on the plane 4.1.f3.

x y (-2, -1)
Figure 4.1.f3

4.1.w3 Example 3: read a dot back into an address. A dot sits 1 step left of the origin and 3 steps up. Left means the across number is negative, so x is −1; up means the second number is positive, so y is 3. Its address is (−1, 3). Reading a point is just plotting in reverse: see how far across, then how far up or down. 4.1.f4

x y (-1, 3)
Figure 4.1.f4

4.1.w4 Example 4: a list of pairs becomes a picture. Take three pairs: (1, 2), (2, 4), (3, 6). Plot each one from the origin: 1 right and 2 up, then 2 right and 4 up, then 3 right and 6 up. Three dots. That is the real point of the grid: a list of number-pairs turns into something you can see. These three happen to fall in a row, but that's not a rule. Sometimes plotted pairs line up and sometimes they stay scattered; you'll see which, and when, later. 4.1.f5

x y (1, 2) (2, 4) (3, 6)
Figure 4.1.f5

You'll notice the grid splits into four corners around the origin. Those corners get names in Unit 5; here you don't need them. Naming a direction, across-then-up, is all it takes to find any point.

Check yourself

  1. 4.1.c1 Is (4, 1) the same point as (1, 4)? (No. (4, 1) is 4 right then 1 up; (1, 4) is 1 right then 4 up. The two land in different spots, because the across number comes first.)
  2. 4.1.c2 A dot sits 2 steps left and 2 steps down of the origin. Write its ordered pair. ((−2, −2): left makes the across number −2, down makes the second number −2.)
  3. 4.1.c3 Where do you start to plot any point, and which number do you use first? (Start at the origin, and use x, the across number, first.)

The problems below mix plotting points and reading them back, which takes a little more thought than doing one kind over and over. Every answer is at the end of the lesson if you want to check yourself.

Practice

A. Plot the point (start at the origin; across first, then up or down).

4.1.1 (5, 1)
Reveal answerHide to problem 1From the origin, 5 right and 1 up.
4.1.2 (0, 3)
Reveal answerHide to problem 2From the origin, 0 across and 3 up (it sits on the y-axis).
4.1.3 (−3, 2)
Reveal answerHide to problem 3From the origin, 3 left and 2 up.
4.1.4 (2, −4)
Reveal answerHide to problem 4From the origin, 2 right and 4 down.

B. Write the ordered pair.

4.1.5 3 across (right), 0 up or down
Reveal answerHide to problem 5(3, 0).
4.1.6 0 across, 2 down
Reveal answerHide to problem 6(0, −2).
4.1.7 A dot 4 steps left of the origin and 1 step up.
Reveal answerHide to problem 7(−4, 1).

C. True or false?

4.1.8 (2, 5) is the same point as (5, 2).
Reveal answerHide to problem 8False — (2, 5) is 2 right then 5 up; (5, 2) is 5 right then 2 up, a different point.