Math 9 · Linear relations
One pattern, four ways to show it
A linear relation is a pattern that changes by the same amount at every step. It can be shown as a picture pattern, a table, an equation or a graph, and this unit is mostly about moving between them — then using the graph to read values the table never gave you.
- 1. The words, first
- 2. From a pattern to an equation
- 3. Graphing a linear relation
- 4. Interpolation and extrapolation
- 5. What costs marks
The words, first
The idea: The last two are about reading between points and beyond them. The rest are about getting the points.
| Word | What it means |
|---|---|
| Linear relation | A relation between two quantities that goes up, or down, by the same amount at every step. Its graph is a straight line. |
| Variable | A letter standing for a quantity that changes: n for a figure number, t for time. |
| Table of values | Matching pairs of values, set out in two rows or two columns. |
| Ordered pair | One pair written as a point. (2, 7) means 2 across and 7 up; the horizontal value always comes first. |
| Constant term | The number in an equation with no variable attached. In t = 3n + 1 it is 1. It is the value when the variable is 0. |
| Discrete / continuous | Separate values only (there is no figure 2.5), or every value in between (time can be 2.5 hours). Discrete data is plotted as dots; continuous data as a joined line. |
| Interpolation | Reading a value that lies between the points you already have. |
| Extrapolation | Extending the pattern to read a value beyond them. |
From a pattern to an equation
The idea: How much it goes up at each step is what multiplies the variable. The constant term is whatever has to be added to make the first figure come out right.
Worked example. A row of squares is made from toothpicks.
| Squares n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Toothpicks t | 4 | 7 | 10 | 13 |
- Find the step. The count goes up 3 each time, because each new square shares a side with the one before. So the equation contains 3n.
- Fix the start. For n = 1, 3n is 3, but the figure has 4. Add 1: t = 3n + 1.
- Check another figure. n = 3 gives 3(3) + 1 = 10 ✓.
Use it. 20 squares need 3(20) + 1 = 61 toothpicks, without drawing them. And 100 toothpicks make the figure where 3n + 1 = 100, so n = 33.
The constant term means something. Run the table back to n = 0: 4 − 3 = 1, the single toothpick that starts the row. In a real situation it is usually a starting amount: a flat fee, an opening balance, a candle's height before it is lit.
Graphing a linear relation
The idea: Make a table, plot the ordered pairs, and check they fall in a straight line. If they do not, either the relation is not linear or a point is wrong.
Worked example: y = 2x − 3. Choose a few x-values, including a negative one, and substitute:
| x | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | −5 | −3 | −1 | 1 | 3 |
The points line up: y goes up 2 for every 1 across, and the line crosses the vertical axis at −3, the constant term.
Is a table linear? Look at the steps. With x going up by 1, y going 1, 4, 9, 16 is not linear — it goes up 3, then 5, then 7. y going 5, 8, 11, 14 is: up 3 every time.
Horizontal and vertical lines. y = 3 says y is 3 whatever x is — (−2, 3), (0, 3), (5, 3) — so the points make a horizontal line. x = −2 stacks the points into a vertical line. The equation names the one coordinate that never changes.
Interpolation and extrapolation
The idea: Interpolating reads between points you have, and is usually safe. Extrapolating assumes the pattern keeps going, and is only as good as that assumption.
Worked example. A 20 cm candle burns down 2 cm every hour, so its height is h = 20 − 2t. The table covers the first 4 hours.
- Interpolate: after 2.5 hours, inside the table: 20 − 2(2.5) = 15 cm.
- Extrapolate: after 7 hours, beyond the table: 20 − 2(7) = 6 cm.
- Work backwards: when is it 9 cm tall? 20 − 2t = 9, so 2t = 11 and t = 5.5 hours. On the graph, go across from 9 to the line, then down.
Where extrapolation breaks. The equation says that after 12 hours the candle is 20 − 24 = −4 cm tall. The arithmetic is fine; the candle is not, because it burnt out at 10 hours. Always ask whether the pattern could really continue that far.
Math 10C comes back to these straight lines and gives the step size — “down 2 every hour” — a name and a formula of its own.
What costs marks
The idea: Test every equation on the first figure, and most of these disappear.
- Forgetting the constant term, writing t = 3n instead of 3n + 1.
- Mixing up y = 3 and x = 3. y = 3 is horizontal.
- Joining dots that should stay separate, or leaving a continuous relation as dots.
- Swapping the coordinates. (2, 7) is 2 across, then 7 up.
- Extrapolating past where the situation makes sense.