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Foundations · Analytic geometry

Coordinates and reading graphs

Plotting points, the four quadrants, reading values off a graph, and drawing a straight line from a table of values.

First, the rules

Along the corridor, then up the stairs

A point on the plane is two numbers: how far right (x), then how far up (y). Negative means left or down. That is the whole alphabet of graphs.

★ Coordinates
Example 1
(x, y) — x first, y second — (3, −2) is 3 right and 2 down; (0, 0) is the origin.
★ Quadrants
Example 1
I: (+, +) · II: (−, +) · III: (−, −) · IV: (+, −), counted anticlockwise from the top right.
On the axes
Example 1
Points with y = 0 lie on the x-axis; points with x = 0 lie on the y-axis.
★ Table of values
Example 2
Pick x values, calculate y for each, plot, join. For y = 2x − 1: x = −1, 0, 1, 2 → y = −3, −1, 1, 3.
★ Reading a graph
Example 3
Go up from the x value to the line, then across to the y-axis (or the other way round). Say what the axes measure before you read.
Distances along a grid line
Example 1
Same x: distance = difference of the y values; same y: difference of the x values. Diagonal distances come later (Pythagoras).

See it

Drag a point

Move the point or the sliders and watch its coordinates, quadrant and distances; switch on the reflections to see how a sign change moves it.

Step by step

Worked examples

1

Plot and describe

coordinates · quadrants

Plot A(−3, 4), B(2, 4), C(2, −1). Which quadrant is each in, and how long are AB and BC?

Ask yourself: Same y for A and B; same x for B and C.

Show the solution
  1. A: left 3, up 4 → quadrant II. B: right 2, up 4 → quadrant I. C: right 2, down 1 → quadrant IV.
  2. AB is horizontal: 2 − (−3) = 5. BC is vertical: 4 − (−1) = 5.
  3. The fourth corner of the square would be D(−3, −1).

Answer II, I, IV · |AB| = |BC| = 5

2

Draw a line from a table

table of values

Draw y = 3 − x for x from −2 to 4, and say where it crosses the axes.

Ask yourself: Make the table, plot, join with a ruler — then read the crossings.

Show the solution
  1. x: −2, −1, 0, 1, 2, 3, 4 → y: 5, 4, 3, 2, 1, 0, −1.
  2. The points lie on a straight line sloping down. It crosses the y-axis at (0, 3) and the x-axis at (3, 0).
  3. Every point on the line satisfies y = 3 − x — for example (1, 2): 3 − 1 = 2 ✓.

Answer (0, 3) and (3, 0)

3

Reading a real graph

reading values

A distance–time graph for a walk goes from (0, 0) to (20, 1.5) in a straight line, is flat until (30, 1.5), then goes to (50, 0). (Time in minutes, distance from home in km.) What happened, and how fast was the return?

Ask yourself: A rising line is moving away, a flat line is standing still, a falling line is coming back.

Show the solution
  1. 0–20 min: walked 1.5 km away. 20–30 min: stopped. 30–50 min: walked 1.5 km back.
  2. Return speed: 1.5 km in 20 min = 0.075 km/min = 4.5 km/h.
  3. Outward speed was 1.5 km in 20 min too — the same; the flat part is the only difference.

Answer out 20 min, rest 10 min, back 20 min · 4.5 km/h

Practice: coordinates

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Quadrants, reading coordinates, points on a line, distances along a grid line and reflections. Your best score is kept in this browser only.

About this lesson

Coordinates and reading graphs: the x- and y-axes, plotting a point from its coordinates, the four quadrants and the signs in each, reading a value from a line graph or a table, and drawing y = mx + c from a table of values. A point you can drag, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.