Quarry School

Build and check a line from a table

Explain it like I am five

Think of a table as several photographs of the same growing plant. Each column matches a time with its height. To see whether one straight-line rule fits, compare the change in height with the time between photographs. Different time gaps can produce different height changes even when the growth per month stays the same. A photograph at time zero shows the starting height. If that photograph is missing, work backward from another one. Check every neighboring pair of columns. A few photographs can fit a line without proving that the plant grows steadily between the photographs or afterward. Put the photographs in time order first. Repeating the same photograph adds no information. Two different heights recorded for the same time cannot be outputs of one function.

input t monthsoutput H(t) feet012.5213.5414.5816.51218.5
Use every column and every time gap before deciding that one line fits the measurements.
Reminder
  • Decimal subtraction. Line up decimal places: 16.5 − 14.5 = 2.0, while 8 − 4 = 4, so the rate is 2.0 ÷ 4 = 0.5.
  • Function notation. H(8) means find the height at input 8 months: 12.5 + 0.5(8) = 16.5 feet.
  • Slope. An output gain 80 over two weeks gives 802 = 40 rats per week.
  • Fractions. 12 = 24 = 0.5; different-looking gains can have the same rate.
  • Solving for b. 7 = 3(2) + b gives b = 1 by subtracting 6; check 6 + 1 = 7.
  • Ordered pairs. A column with input 2 and output 7 supplies point (2, 7).
  • Domain. Elapsed time can be 2.5 months when the model allows continuous time; a count of whole objects cannot be 2.5.
  • Zero run. A repeated input gives a gap of 7 − 7 = 0. Do not divide by that gap. Identical pairs repeat information; different outputs at that input fail the function definition.
Why it works. For a line, output change equals m times input change, so dividing the two changes must give the same m on every interval. Matching all adjacent rates shows that the listed points fit a single line. The output under input zero gives b directly. If zero is missing, substituting a listed pair into y = mx + b finds b by subtraction. Unequal raw output changes are harmless when the input gaps also differ. Samples establish agreement at the sampled inputs, not at every possible input.
RuleRule: With at least two distinct inputs and one output per input, compare outputchangeinputchange on every adjacent interval of increasing inputs. Equal rates mean the listed points fit f(x) = mx + b. Read b under zero, or use b = y − mx.
The same idea, five ways
Say it

For every pair of neighboring columns, compare change in output with change in input.

Write it

A table fits one line when each interval has the same output change per input unit.

In math
  • m = y2−y1x2−x1
  • f(0) = b
  • b = y − mx
  • f(x) = mx + b
Like

Plant photographs may be taken at different time gaps; compare growth per month rather than growth per photograph.

See it
input xoutput f(x)011223
Both one-unit intervals add one output unit, and the output under zero is 1.
The same idea, other ways
By reading the columns

In the small table, the output goes up 1 when the input goes up 1, on both intervals. That gives m = 1. The output under zero is 1, so b = 1 and f(x) = x + 1.

input xoutput f(x)011223↓ evaluate: input given, read the output below it
The highlighted zero-input column identifies the starting output.
As growth measured per month

A tree gaining 1 foot in 2 months and 2 feet in 4 months has the same rate in both intervals: half a foot per month. Compare the rate, because the photographs may be taken at different time gaps.

1 foot ÷ 2 months = 0.5 foot per month
2 feet ÷ 4 months = 0.5 foot per month
Different gaps, same rate
Unequal gains can still represent equal change per input unit.
.1Input zero is visible

A column under zero is a photograph taken at the start. Its output is the initial value. For the rat population, that column shows 1,000 rats before any weeks have passed. The other columns show how the population changes. Divide each population gain by the weeks in its interval to find the rate per week.

  • Rule: P(w) = 1000 + 40w fits the given population table, because each two-week interval adds 80 rats and the starting count is 1,000.
  • The slope is 40 rats per week, because 80 rats ÷ 2 weeks = 40 rats per week.
  • The initial value is 1,000 rats, because the output under w = 0 is 1,000.
  • Elapsed weeks can be nonnegative real numbers. Actual rat counts are whole numbers, so the drawn population line can estimate change between observations. The table alone does not prove constant growth at every time.
input w weeksoutput P(w) rats01000210804116061240
The two-week gaps belong in the denominator of the slope calculation.
Reminder
  • Rate units. 80 rats ÷ 2 weeks = 40 rats per week. The input unit belongs under the output unit.
The same idea, five ways
Say it

Start with 1,000 rats and use a modeled gain of 40 rats per week.

Write it

The sampled population increases by 80 rats for every two-week interval shown.

In math
  • P(w) = 1000 + 40w
  • P(0) = 1000
  • m = 802 = 40 rats per week
  • w ≥ 0
Like

The first photograph gives the starting population; later photographs reveal the weekly rate.

See it
input w weeksoutput P(w) rats01000210804116061240↓ evaluate: input given, read the output below it
The complete source table shows the initial population under zero weeks.
Worked exampleUse the complete rat population table

Writing P(w) means finding a population rule that matches all four columns of the pictured source table. Find the rate per week and the starting population.

input w weeksoutput P(w) rats01000210804116061240
All three adjacent two-week intervals add 80 rats.
  1. Read P(0) = 1000 from the column under 0, so b = 1000 rats.The initial value is the output at zero elapsed weeks.
  2. First interval: 1080−10002−0 = 802 = 40 rats per week.Divide the population increase by its two-week time gap.
  3. Second interval: 1160−10804−2 = 40. Third interval: 1240−11606−4 = 40.Every adjacent interval must share the rate, not only the first one.
  4. Write P(w) = 1000 + 40w.Combine the starting population with the common modeled weekly gain.
Answer
  • P(w) = 1000 + 40w
  • Initial population: 1,000 rats.
  • Modeled rate: 40 rats per week.
Check P(0) = 1000, P(2) = 1080, P(4) = 1160 and P(6) = 1240. All columns fit; behavior between these sampled times still requires the constant-growth assumption.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: The population grows by 80 rats per week.
The gain of 80 takes two weeks. Omitting the input gap doubles the stated rate.
✓ Instead: Use 80 ÷ 2 = 40 rats per week.
Tips and tricks
  • Tip: Write the time gap below each population change before reducing the fraction.
.2Input zero is missing

A table may begin after the process has already started. Its first output is then a later photograph, not the starting amount. First find the rate from the listed columns. Then use one column to work backward: remove the change accumulated since input zero. What remains is the starting output, even though the zero-input column was never printed.

  • Rule: If zero is missing, find b = y − mx from a listed pair, because subtracting accumulated change removes what happened after the start.
  • The first listed output is b only if its input is zero, because b describes f(0).
  • The original example's common rate is 3 and its recovered initial value is 1, because 7 − 3(2) = 1.
76 + b=do the same thing to both sides
Subtract the accumulated change 6 to uncover the starting output 1.
Reminder
  • Solving for an added constant. To isolate b in 7 = 6 + b, subtract 6 from both sides: b = 1. Plug back: 6 + 1 = 7.
The same idea, five ways
Say it

Use a later total and take away the change since the start.

Write it

A missing zero-input column does not stop you from finding the initial value.

In math
  • y = mx + b
  • b = y − mx
  • 7 = 3(2) + b
  • b = 1
Like

A later savings balance minus all later deposits reveals the money present before those deposits.

See it
input xoutput f(x)27413619
No column under zero appears in this original table, so the initial value must be recovered.
Worked exampleFind the absent starting output

Finding f(x) means reproducing the outputs in this original table, including recovering its missing output at input zero. Use the pictured columns.

input xoutput f(x)27413619↓ evaluate: input given, read the output below it
The highlighted known pair supplies the equation used to recover the absent initial value.
  1. First rate: 13−74−2 = 62 = 3. Second rate: 19−136−4 = 62 = 3.Both intervals must agree before one line can describe the listed values.
  2. Use the column under 2: 7 = 3(2) + b = 6 + b.This equation makes the missing starting output the only unknown.
  3. Subtract 6 to find b = 1. Plug back: 3(2) + 1 = 7.Removing the change since zero isolates the start, and substitution verifies it.
  4. Write f(x) = 3x + 1.The common slope is 3 and the recovered starting output is 1.
Answer
  • f(x) = 3x + 1
  • The missing initial value is f(0) = 1.
Check f(4) = 12 + 1 = 13 and f(6) = 18 + 1 = 19. The other columns agree with the recovered equation.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: The first output is 7, so b = 7.
Its input is 2. It includes two steps of accumulated change and therefore is not the output at zero.
✓ Instead: Remove 3 × 2 = 6 from 7 to find b = 1. Check: 3(2) + 1 = 7.
Tips and tricks
  • Tip: Circle the input attached to any candidate b. It must be zero.
.3Unequal input gaps

If plant photographs are taken two months apart at first and four months apart later, the later gains should be larger under steady growth. That does not mean the rate has changed. Divide each height gain by its own time gap. A one-foot gain over two months and a two-foot gain over four months both mean half a foot per month.

  • Rule: H(t) = 12.5 + 0.5t fits the source tree measurements, because all four interval rates are 0.5 foot per month.
  • The starting height is 12.5 feet, because the column under t = 0 gives that height.
  • The two later intervals span four months each, because the input gaps are 8 − 4 and 12 − 8. Use those actual gaps rather than assuming every gap is two months.
  • Time can be measured continuously, so t can be a nonnegative real number. The observations support the line at the listed times from 0 to 12 months. Predictions between or beyond them rely on an added constant-growth assumption.
1/22/4
One over two and two over four represent the same rate.
Reminder
  • Equivalent fractions. Dividing numerator and denominator by 2 gives 24 = 12 = 0.5.
The same idea, five ways
Say it

Divide each gain by the time that gain took.

Write it

Unequal time gaps can have equal rates even when their height gains differ.

In math
  • H(t) = 12.5 + 0.5t
  • 12 = 24 = 0.5 foot per month
  • H(0) = 12.5
  • 0 ≤ t ≤ 12 for the observed time span
Like

Two hours of steady walking cover twice the distance of one hour without changing walking speed.

See it
input t monthsoutput H(t) feet012.5213.5414.5816.51218.5
The complete source table includes both two-month and four-month gaps.
Worked exampleUse every interval in the tree-height table

Writing H(t) means finding one height rule for all the source table's sampled months. Use the pictured five columns and compare growth per month, including the wider time gaps.

input t monthsoutput H(t) feet012.5213.5414.5816.51218.5
The gains double when the time gaps double, leaving the same monthly rate.
  1. Read b = 12.5 feet from the column under zero months.That is the height when the measurements begin.
  2. First rate: 13.5−12.52−0 = 12 = 0.5. Second rate: 14.5−13.54−2 = 12 = 0.5.Each of these gains takes two months.
  3. Third rate: 16.5−14.58−4 = 24 = 0.5. Fourth rate: 18.5−16.512−8 = 24 = 0.5.Each of the larger gains takes four months, so its rate remains half a foot per month.
  4. Write H(t) = 12.5 + 0.5t.Every interval has the same rate, and the starting height is known.
Answer
  • H(t) = 12.5 + 0.5t
  • Initial height: 12.5 feet.
  • Modeled growth rate: 0.5 foot per month.
Check The formula gives H(0) = 12.5, H(2) = 13.5, H(4) = 14.5, H(8) = 16.5 and H(12) = 18.5 feet. Every listed measurement matches.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: The tree is nonlinear because the later output gains are 2 feet instead of 1 foot.
The later input gaps are also twice as large. Output gains must be compared per input unit.
✓ Instead: Compute 1 ÷ 2 = 0.5 and 2 ÷ 4 = 0.5 foot per month. All listed rates match.
Tips and tricks
  • Tip: Write a separate denominator for each interval. Never carry the first gap across the whole table.
.4A table that cannot fit one line

The first two photographs can look consistent with steady growth, while a later one breaks the pattern. Check every interval before choosing a linear rule. If one input step adds one output unit but the next equal step adds three, the rate changed. You can still describe those observations, but a single constant-rate line cannot pass through all of them exactly.

  • Rule: If any adjacent rates differ, no single linear function fits all listed points exactly, because a line has one constant rate.
  • Two points with different inputs always determine a candidate line. A third point can fail that rule, because agreement at two inputs does not establish a constant rate elsewhere.
  • Even equal rates in a finite table do not prove that the complete process is linear, because unmeasured inputs may behave differently.
input xoutput g(x)001124
Check every neighboring pair rather than assuming the first rate continues.
Reminder
  • Testing a formula. For a candidate y = x, input 2 gives output 2. A table output of 4 at that input disproves the candidate.
The same idea, five ways
Say it

One changed rate is enough to rule out a single exact line for the full table.

Write it

The listed points are not all on one line when their interval rates differ.

In math
  • 1−01−0 = 1
  • 4−12−1 = 3
  • 1 ≠ 3
Like

A car can go slowly during one hour and faster during the next; one fixed speed does not describe both hours.

See it
input xoutput g(x)001124
Equal input gaps produce different output gains in this original counterexample.
Worked exampleTest a tempting line against the last column

Checking whether the table fits a line means comparing all interval rates, then seeing whether one equation gives every output. Test the pictured original table.

input xoutput g(x)001124↓ evaluate: input given, read the output below it
The last output defeats the line suggested by the first two columns.
  1. First rate: 1−01−0 = 1.The first input step adds 1 to the output.
  2. Second rate: 4−12−1 = 3.The next equal input step adds 3 instead.
  3. The rates 1 and 3 differ, so no single line fits all three columns.A linear function has the same output change per input unit everywhere.
  4. The first two columns suggest g(x) = x, but at x = 2 that rule gives 2 rather than 4.Checking the last column exposes why fitting only the first two points is insufficient.
Answer
No single linear function fits all three listed pairs exactly.
Check The endpoints alone suggest slope 4−02−0 = 2 and the line y = 2x. At x = 1 it gives 2 instead of 1, confirming the failure another way.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: Any table is linear because you can connect its first two points with a line.
The remaining points may not be on that line. In this example the next rate changes from 1 to 3.
✓ Instead: Check every interval and every listed point. Report that no single line fits when a rate differs.
Tips and tricks
  • Tip: Use the final column as a quick check after finding a candidate equation from the first two.
Strategy: step by step
  1. 1. Arrange columns in increasing input order. Keep one copy of repeated identical pairs. Different outputs at the same input rule out a function, and one distinct pair alone does not determine a slope.
  2. 2. Read each column as one input-output pair, because values in the same column belong together.
  3. 3. For every neighboring pair, subtract the outputs and divide by the corresponding input difference, because unequal input gaps must be accounted for.
  4. 4. Compare all the rates. If any differ, no single linear function fits the whole table exactly.
  5. 5. If the rates match, read b in the column under zero. If zero is missing, substitute any listed pair and solve for b to recover the missing starting output.
  6. 6. Substitute every listed input into the candidate equation, because fitting the first pair alone does not establish agreement with the whole table.
Strategy
Strategy: Check a table and recover its equation
1
Do repeated inputs have different outputs?
YesStop. One input would have more than one output, so no function fits this data.
NoKeep one copy of each repeated identical pair and put the remaining inputs in increasing order.
↓
2
Are at least two distinct inputs left?
YesTheir input gaps are nonzero, so calculate the interval rates.
NoOne distinct pair cannot determine an unknown slope. Obtain another distinct pair or a stated slope.
↓
3
Are all adjacent input gaps the same?
YesEqual output changes give equal rates, but still calculate the rate in output units per input unit.
NoDivide each output change by its own input gap before comparing.
↓
4
Are all the calculated rates equal?
YesUse that common rate as m and find b.
NoNo single line fits the whole table exactly. Do not force a linear equation onto it.
↓
5
Is there a column with input zero?
YesRead its output as b.
NoUse y = mx + b with a listed pair, subtract mx to isolate b, and plug the result back into that pair.
  1. 1. Arrange columns in increasing input order. Repeated input with different outputs means the data are not a function. Keep one copy of each identical pair; at least two distinct inputs are needed to recover an unknown slope.
  2. 2. Read the input unit and output unit.
  3. 3. Compute a rate for every adjacent interval using that interval's actual input gap.
  4. 4. If rates differ, report that one line does not fit the listed values exactly.
  5. 5. If rates agree, find the starting output from the zero-input column or by substitution.
  6. 6. Check the equation in every column and state any limits on the model's use.
Worked exampleStart with a three-column table

Finding f(x) means making a rule that reproduces the output in every shown column. Use the pictured original table.

input xoutput f(x)011223
Read one input and output from each column and check both intervals.
  1. From the column under 0 to the column under 1, m = 2−11−0 = 1.The output rises 1 over an input change of 1.
  2. From the column under 1 to the column under 2, m = 3−22−1 = 1.Every adjacent interval must have the same rate for one line to fit.
  3. Read b = 1 in the output cell under input 0. Write f(x) = x + 1.The common rate is 1 and the zero-input output is 1.
Answer
f(x) = x + 1 fits all three listed pairs.
Check The formula gives f(0) = 1, f(1) = 2 and f(2) = 3, matching every column. These samples alone do not establish the rule at unlisted inputs.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: Unit input gaps

Finding f(x) means making a rule for every output in this original table. Use the pictured columns with one-unit input gaps.

input xoutput f(x)02142638
Each one-unit input gap corresponds to an output gain of 2.
  1. Rates: 4−21−0 = 2, 6−42−1 = 2 and 8−63−2 = 2.Every adjacent one-unit gap must have the same output gain.
  2. Read b = 2 from the output under input zero.That column shows the starting value.
  3. Write f(x) = 2x + 2.Use the common rate 2 and the starting output 2.
Answer
f(x) = 2x + 2
Check f(0) = 2(0) + 2 = 2.
f(1) = 2(1) + 2 = 4.
f(2) = 2(2) + 2 = 6.
f(3) = 2(3) + 2 = 8.
Every computed output matches its cell in the picture.
Rung 2Rung 2: Rat population with two-week gaps

Writing P(w) means finding the rat population rule that fits all four columns of the complete source table. Find the weekly rate even though the samples are two weeks apart.

input w weeksoutput P(w) rats01000210804116061240
The rate is 40 per week because each gain of 80 spans two weeks.
  1. Read b = 1000 rats from the column under zero weeks.The zero-input output is the initial population.
  2. Check all rates: 1080−10002−0 = 40, 1160−10804−2 = 40 and 1240−11606−4 = 40 rats per week.Every two-week gain must be divided by its input gap, and all intervals must agree.
  3. Write P(w) = 40w + 1000.The listed population values share the same modeled weekly rate and starting count.
Answer
P(w) = 40w + 1000
Check Substitution gives P(0) = 1000, P(2) = 1080, P(4) = 1160 and P(6) = 1240. The line fits the observations; steady growth between them remains a model assumption.
Rung 3Rung 3: Recover a missing starting value

Finding f(x) means recovering a rule for the listed original pairs and its missing output at zero. Use the pictured table, which begins at input 2.

input xoutput f(x)210516822↓ evaluate: input given, read the output below it
Use a known pair to recover the missing output at zero.
  1. First rate: 16−105−2 = 63 = 2. Second rate: 22−168−5 = 63 = 2.Both three-unit intervals must have the same rate.
  2. Use the column under 2: 10 = 2(2) + b = 4 + b.A known output and the common slope give an equation for the missing start.
  3. Subtract 4 to find b = 6. Plug back: 2(2) + 6 = 10.Removing the accumulated change isolates the starting output, and the known pair verifies it.
  4. Write f(x) = 2x + 6.The common slope is 2 and the recovered initial value is 6.
Answer
  • f(x) = 2x + 6
  • f(0) = 6
Check At x = 5, 10 + 6 = 16. At x = 8, 16 + 6 = 22. Both remaining columns fit.
Rung 4Rung 4: Tree height with unequal time gaps

Writing H(t) means finding a height rule that fits every source measurement. Use the complete pictured tree table and account for its unequal month gaps.

input t monthsoutput H(t) feet012.5213.5414.5816.51218.5
Check all four interval rates, including both of the wider four-month gaps.
  1. Read b = 12.5 feet under t = 0.This is the starting height of the experiment.
  2. First two rates: 13.5−12.52−0 = 12 = 0.5 and 14.5−13.54−2 = 12 = 0.5 foot per month.Each of these height gains took two months.
  3. Last two rates: 16.5−14.58−4 = 24 = 0.5 and 18.5−16.512−8 = 24 = 0.5 foot per month.Each later gain took four months, so a larger gain does not mean a larger rate.
  4. Write H(t) = 0.5t + 12.5.All four rates agree and the zero-input height is known.
Answer
  • H(t) = 0.5t + 12.5
  • The modeled rate is 0.5 foot per month.
Check H(0) = 12.5, H(2) = 13.5, H(4) = 14.5, H(8) = 16.5 and H(12) = 18.5 feet. The equation matches all five columns. Predictions between the measured months or later depend on the steady-growth assumption.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: Equal raw output gains are required in every linear table.
Output gains are equal only when the corresponding input gaps are equal. Unequal gaps can produce unequal gains at the same rate.
✓ Instead: Compare output change divided by input change on every interval.
✗ Not this: Counterexample: A table whose samples fit a line proves that the complete process is linear forever.
The table says nothing directly about unmeasured times or future changes in the process.
✓ Instead: Say the listed points fit a line. State the constant-rate assumption before predicting other outputs.
✗ Not this: Counterexample: Every repeated input can be ignored, even when its output changes.
A function gives one output for an allowed input. Deleting a conflicting output hides a failure of that definition.
✓ Instead: Ignore only repeated identical input-output pairs. Different outputs at the same input rule out a function.
Tips and tricks
  • Tip: Rate, rate, rate, then starting value. Write one rate for every adjacent interval.
  • Tip: If zero is absent, recover b by removing mx from a known output and plug it back in.
Trap. Looking only at the output differences when the input gaps change. Always divide by the actual input gap for that interval.