One relationship, four views
Every relationship can be shown in four representations. Being able to move between them is one of the most important skills in algebra.
Example: A gym charges a $25 sign-up fee plus $10 per month.
| Representation | What it looks like |
|---|---|
| Verbal description | "$25 to join, then $10 a month" |
| Equation | C = 10m + 25 |
| Table of values | m: 0, 1, 2, 3 → C: 25, 35, 45, 55 |
| Graph | a straight line starting at (0, 25), rising 10 per month |
What each representation is best at
| Representation | Strength |
|---|---|
| Words | explains the context clearly |
| Table | shows exact values and patterns |
| Graph | shows the overall shape and trends at a glance |
| Equation | lets you calculate any value precisely |
Words → equation
Find the starting value and the rate of change.
"A pool has 200 L and is filling at 15 L/min." → V = 15t + 200
Table → equation
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 4 | 7 | 10 | 13 |
- Starting value (when x = 0): 4
- Rate of change: +3 each step
- Equation: y = 3x + 4
Equation → table → graph
y = −2x + 6
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 6 | 4 | 2 | 0 |
Plot the points and join them: a line falling from (0, 6) to (3, 0).
Graph → words
A graph of distance from home over time:
- rising line → walking away from home
- flat part → stopped
- falling line → walking back home
Describe each section: "She walked away for 10 min, stopped for 5 min, then returned in 15 min."
Linear or not?
| Clue | Linear? |
|---|---|
| Table: constant differences in y for equal steps in x | yes |
| Graph: straight line | yes |
| Equation: y = mx + b | yes |
| Table: y doubles each step | no (exponential) |
| Graph: curve | no |
Worked examples
Example 1: "A candle 30 cm tall burns 2 cm per hour." Equation? h = 30 − 2t.
Example 2: From the table x: 0, 2, 4; y: 1, 9, 17. Equation? Rate = 8/2 = 4 → y = 4x + 1.
Example 3: y = 5x graphed: describe it. A straight line through the origin, rising 5 for every 1 across.
Common mistakes
- Finding the rate using steps of x that aren’t 1 without dividing.
- Mixing up starting value and rate.
- Joining points in situations that are discrete (like whole numbers of people).
Try it yourself
- Write an equation: "$50 saved, plus $20 each week."
- Table: x: 0, 1, 2; y: 10, 7, 4. Equation?
- Make a table for y = x² for x = −2 to 2. Linear?
- What does a horizontal section mean on a distance–time graph?
Answers: 1) S = 20w + 50 2) y = −3x + 10 3) 4, 1, 0, 1, 4; not linear 4) no movement
Quick recap
- A relationship can be shown as words, a table of values, a graph or an equation.
- The starting value and the rate of change link them.
- Choose the representation that best suits the task.