Lesson 3 · from §79, 80 of the 1910 text
Coördinates and Plotting Points
The graph of an equation
A graph can also show a relation between numbers that have nothing to do with temperature or time. "What two numbers added give five?" is the equation x + y = 5, where x and y are any two numbers whose sum is 5.
If x is 1, y is 4; if x is 2, y is 3; if x = 3\frac12, then 3\frac12 + y = 5 and y = 1\frac12. Proceeding this way we can tabulate as many pairs as we like:
| A | B | C | D | E | F | G | J | H | I | K | R | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| x | 1 | \frac{3}{2} | 2 | 3 | \frac{19}{5} | 4 | 5 | 0 | 6 | 7 | -1 | -2 |
| y | 4 | \frac{7}{2} | 3 | 2 | \frac{6}{5} | 1 | 0 | 5 | -1 | -2 | 6 | 7 |
Lay off equal spaces along OX for values of x and along OY for values of y. The point A is 1 space to the right and 4 spaces up; B is 1\frac12 right and 3\frac12 up; and so on. Negative values of x go to the left, negative values of y go down.
All the points lie on one straight line, even B and E, whose coördinates are fractions. That line, extended without end in both directions, is the complete graph of x + y = 5: every pair of numbers that satisfies the equation is a point on it.
Coördinates
It is agreed that:
- Two lines at right angles, X'OX (the x-axis) and Y'OY (the y-axis), are drawn.
- A definite length is the unit of distance.
- The distance of a point from the y-axis (measured parallel to the x-axis) is its x-distance or abscissa; its distance from the x-axis is its y-distance or ordinate.
- Distances measured from the axes to the right and upward are positive; to the left and downward, negative.
- Every point of the paper corresponds to a pair of numbers, and the first number is always the x-distance. The point (2, 3) has x-distance 2 and y-distance 3.
- The point where the axes cross is the origin, (0, 0).
The two distances are the coördinates of the point. The axes divide the plane into four quadrants, numbered counterclockwise from the upper right:
| Quadrant | I | II | III | IV |
|---|---|---|---|---|
| signs of (x, y) | (+, +) | (-, +) | (-, -) | (+, -) |
A point with a zero coördinate is on an axis, not in any quadrant. Locating points this way is called plotting them. Cross-ruled paper is a great convenience, but a ruler marked in inches or centimeters works too.
Try one:
Quick check
In which quadrant is the point (-4, 7)? (Type 1, 2, 3, or 4.)
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Plot each point on paper (one unit = ½ inch or 1 cm). Then give the quadrant it lies in: 1, 2, 3, or 4, or 0 if the point is on an axis.
(3, 5); (-3, 5); (-3, -5); (+3, -5)
(4, -2); (-6, 4); (-1, -2); (+2, -4)
(0, 4); (0, -4); (4, 0); (-4, 0)
(2, 2); (0, 2); (-2, 6); (2, 0)
(0, -5); (-5, 0); (0, 0)
If one coördinate of a point is zero, where is it located? Where, if both are zero?