Digital SAT · Linear Functions
How to Solve Matching a Table to a Rule on the Digital SAT
A table lists a function's values, and you find the equation behind them. Two quick routes work. Read the pattern, since outputs that change by a constant amount signal a linear rule whose slope is that change over the input step. Or test each candidate against a row and keep the one that fits every row. Desmos can plot the points and show which equation passes through them all.
- per test
- less than 1 per test per test
- typical difficulty
- Medium typical difficulty
- practice questions
- 90 practice questions
Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.
What the question bank shows
A wrong rule can hit one point by luck, so a match must hold across rows.
Plot the points and graph a candidate to see if it fits them all.
Equal input steps giving equal output changes means a linear rule.
How to recognize matching a table to a rule questions
- A table lists inputs and outputs of a function.
- The question asks which equation or rule matches.
- Outputs may change by a constant amount, hinting at a linear rule.
- Choices are equations or expressions.
Why students miss these
The step-by-step method
- 1
Look for a pattern
See whether outputs change by a constant amount as the input steps, signaling a linear rule.
- 2
Find the rate and start
For a linear table, the slope is the output change over the input change; the start is the output at input zero.
- 3
Test candidates on a row
Plug a table input into each equation and keep those that return the matching output.
- 4
Confirm on a second row
Check the survivor against another row, since one match can be coincidence.
Solving it on Desmos
- Plot the points. Enter the table's pairs as points in Desmos to see the shape.
- Graph a candidate. Type an answer choice and check that its curve passes through every plotted point.
- Keep the fit. The equation whose graph hits all the points is the match; wrong rules miss at least one.
Full Desmos walkthrough for matching a table to a rule→
Worked examples
Easy example
| x | f(x) |
|---|---|
| 0 | -3 |
| 1 | 0 |
| 2 | 3 |
A: Incorrect. increases, so the slope is positive.
B: Incorrect. The intercept is , not .
C: Correct. gives intercept , and rises 3 per step, so .
D: Incorrect. The slope is 3, not 1.
Explanation
Intercept -3, slope 3: f(x) = 3x - 3.
Medium example
| x | f(x) |
|---|---|
| 0 | -5 |
| 2 | 3 |
| 4 | 11 |
A: Correct. gives intercept , and rises 8 over a run of 2, a slope of 4, so .
B: Incorrect. This swaps the slope and the intercept.
C: Incorrect. The intercept is , not .
D: Incorrect. The slope is 4, not 2.
Explanation
Intercept -5, slope (3-(-5))/2 = 4: f(x) = 4x - 5.
Hard example
| x | f(x) |
|---|---|
| 0 | -1 |
| 5 | 1 |
| 10 | 3 |
A: Correct. gives intercept , and rises 2 over a run of 5, a slope of , so .
B: Incorrect. The slope is , not .
C: Incorrect. The intercept is , not .
D: Incorrect. increases, so the slope is positive.
Explanation
Intercept -1, slope (1-(-1))/5 = 2/5: f(x) = 2/5 x - 1.
The common traps
| Pattern | What it does | The tell |
|---|---|---|
| Matched one row only | Kept an equation that fits a single point by coincidence. | A correct rule fits every row; test at least two. |
| Wrong slope from the table | Computed the rate with mismatched rows. | Use consistent input and output changes between the same two rows. |
| Ignored the starting value | Got the rate right but the intercept wrong. | Check the output at input zero, or back it out from a row. |
Try it: two real questions
| x | f(x) |
|---|---|
| 0 | 5 |
| 1 | 8 |
| 2 | 11 |
| 3 | 14 |
The table gives several values of the linear function . Which equation could define ?
The graph of a line in the -plane is shown. Which equation could define the line?
Question 3 is ready when you are
Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.
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Related reading
Common questions
How do you find the equation from a table on the SAT?
Check whether the outputs change by a constant amount; if so, the rule is linear with that slope. Then test candidate equations against a row, confirming with a second so a lucky single match does not fool you.
How do I know if a table is linear?
If equal steps in the input produce equal changes in the output, it is linear. The constant change is the slope, and the output at input zero is the intercept.
How do I test which equation matches a table?
Plug a table input into each choice and keep the ones that return the listed output. Then check a second row, since a wrong equation can match one point by chance.
Can Desmos match a table to a rule?
Yes. Plot the table's points, then graph each candidate; the one whose curve passes through them all is the match. Wrong rules miss at least one point.
Practice matching a table to a rule the way it is tested
Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.
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