A function can be shown in four ways: words, a rule (an equation), a table and a graph. They all describe the same input-output pairs, so you can move from any one to the others.
Why it works
A rule like $y = 3x + 2$ is a recipe for making pairs: choose an input, get its output. A table lists some of those pairs. A graph draws every pair as a point $(x, y)$. Words tell the story the pairs come from. Moving between forms is just making pairs or reading them.
Key idea
To go from a rule to a table or graph, choose inputs and compute outputs. To go from a table or graph to a rule, look for how the output changes as the input grows, and what the output is when the input is 0.
Example: a bike rental costs \$2 plus \$3 per hour. In words: the cost starts at 2 and goes up 3 each hour. As a rule: $y = 3x + 2$, where $x$ is hours and $y$ is dollars.
| hours $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| cost $y$ (\$) | 2 | 5 | 8 | 11 | 14 |
Worked examples
Worked example: Rule to outputs
For $y = -2x + 7$, find the outputs for $x = 3$ and $x = -1$.
- $x = 3$: $y = -2 \cdot 3 + 7 = -6 + 7 = 1$.
- $x = -1$: $y = -2 \cdot (-1) + 7 = 2 + 7 = 9$.
Answer: $1$ and $9$
Worked example: Table to rule
Write a rule for this table. $$\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline y & 5 & 9 & 13 & 17 \end{array}$$
- Each time $x$ goes up 1, $y$ goes up 4, so the rule has $4x$.
- At $x = 0$, $y = 5$, so 5 is added.
- Check $x = 3$: $4 \cdot 3 + 5 = 17$.
Answer: $y = 4x + 5$
Worked example: Output to input
For the bike rental, how many hours cost \$20?
- Set the rule equal to 20: $3x + 2 = 20$.
- Subtract 2: $3x = 18$, so $x = 6$.
Answer: 6 hours
Common mistake
Mixing up the input and the output, for example reading a table’s bottom row as the inputs. In a table the input row (or column) is usually labeled $x$; on a graph the input is read on the horizontal axis.