Quarry School

Build the equation from two points

Explain it like I am five

Suppose someone gives you two spots on a straight ramp but does not tell you its tilt. Those spots tell you how far forward and how far up the ramp goes between them. That gives the slope. Then you are back to the previous lesson: use one spot as an anchor. Two points do two jobs. Their changes tell you the tilt, and either point tells you where to place the line. Function notation can hide the same information in parentheses. Saying f(4) = 14 means the spot (4, 14), with input 4 and output 14. Translate the notation before calculating so the addresses stay clear.

123456789−10−8−6−4−2246810run 1rise 2(0, −9)(4.5, 0)firstsecondinitial value
Neither supplied output is the initial value unless its input is zero.
Reminder
  • Function notation. f(4) = 14 translates to (4, 14).
  • Slope. 7−18−5 = 63 = 2.
  • Subtracting negatives. 1 − (−2) = 3.
  • Signed division. −7−3 = 73.
  • Point-slope form. Slope 2 at (5, 1) gives y − 1 = 2(x − 5).
  • Distribution. 2(x − 5) = 2x − 10.
  • Solving and checking. 1 = 10 + b gives b = −9; plugging back gives 10 − 9 = 1.
  • Common denominators. −2 = −105, so −2 − 95 = −195.
  • Order of operations. 2(8) − 9 = 16 − 9 = 7; multiply first.
Why it works. Two points with different inputs fix one slope because their output change divided by input change has one value. Substituting either point into y = mx + b then fixes b. The other point also fits because the slope reproduces exactly the change between the two points. This determines one line through them. It does not prove that a real process follows that line between observations; using a linear model requires a stated or justified constant-change assumption.
RuleRule: for distinct inputs, find m = y2−y1x2−x1, then b = y1 − mx1. Write y = mx + b or f(x) = mx + b.
The same idea, five ways
Say it

Say: find the change per step, then the starting amount.

Write it

Write: two points with different inputs determine one line's slope and initial value.

In math
  • m = y2−y1x2−x1
  • b = y1 − mx1
  • y − y1 = m(x − x1)
  • y = mx + b
  • f(x) = mx + b
  • f(a) = c means (a, c)
Like

Two spots on a straight ramp tell you its tilt and position.

See it
2468−10−8−6−4−2246810run 1rise 2(0, −9)(4.5, 0)knownknown
Recover the line, then verify both marked points.
The same idea, other ways
As tilt plus location

The changes between the points give the tilt. Either point pins that tilt to a location. A different anchor with the same slope would usually give a different line.

2468−10−8−6−4−2246810run 1rise 2anchorsecond point
The two points locate the line as well as giving its tilt.
As a machine to recover

f(0) = 6 and f(4) = 14 mean the machine starts at 6 and adds eight across four steps. That is two per step, so f(x) = 2x + 6.

4multiply by 2, add 614inputoutput
Function values are input-output addresses.
.1Two coordinate pairs

A point with input zero is especially useful because its output is already the starting value. You still need both points to calculate slope. Listing the zero-input point second changes nothing. Reversing direction changes the sign of both differences, so the slope stays the same.

  • Rule: (0, b) lies on the y-axis because its horizontal coordinate is zero.
  • Rule: either point may come first if subtraction order matches in both differences.
  • Point-slope and slope-intercept forms describe the same line after correct distribution and equal addition.
24−6−4−22468run 1rise 2.33(0, −3)(1.29, 0)initialknown
Use the point on the vertical axis to read b.
Reminder
  • Signed division. −7−3 is positive 73.
  • Intercept. (0, −3) gives b = −3 because its input is zero.
The same idea, five ways
Say it

Say: input zero tells me the starting output.

Write it

Write: use both points for slope and the zero-input point for b.

In math
  • (3, 4), (0, −3)
  • m = −3−40−3 = 73
  • b = −3
  • y = 73x − 3
Like

A ramp marker at the start gives the starting height.

See it
24−6−4−22468run 1rise 2.33(0, −3)(1.29, 0)input zeroother point
Input zero identifies the initial value even when listed second.
Worked exampleThe verified line with a visible intercept

Write the line through (3, 4) and (0, −3) in both forms. This asks for a line whose initial value is supplied in one point.

24−6−4−22468run 1rise 2.33(0, −3)(1.29, 0)startthree right, seven up
Run three makes the fractional slope's rise a whole number.
  1. m = −3−40−3 = −7−3 = 73.Reversing travel makes both differences negative.
  2. Using (0, −3), write y − (−3) = 73(x − 0), or y + 3 = 73x.Subtract matching anchor coordinates.
  3. Subtract 3: y = 73x − 3.This isolates y and reveals b = −3.
  4. At x = 0, y = −3. At x = 3, y = 7 − 3 = 4.Both points must fit the solved equation.
Answer
  • Point-slope: y + 3 = 73x.
  • Slope-intercept: y = 73x − 3.
Check Run three times slope 73 gives rise seven. −3 + 7 = 4 reproduces the second output.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: b = 4 because (3, 4) is listed first.
Initial value is tied to input zero, not listing order.
✓ Instead: Use b = −3 from (0, −3).
Tips and tricks
  • Tip: an input-zero point is the quickest anchor.
.2Two values written in function notation

Function notation reports what comes out when an input goes in. The parentheses do not multiply the letter f. f(0) = 6 means input zero gives output six. f(4) = 14 means input four gives output fourteen. These are two points for the same method. Read each statement aloud before writing its point address.

  • Rule: f(a) = c means point (a, c), because a is input and c is output.
  • Rule: when f is given to be linear, two distinct input-output pairs determine its equation.
  • Use the supplied function name in the answer.
04614function
Parenthesized inputs belong beside their stated outputs.
Reminder
  • Evaluation. For f(x) = 2x + 6, f(4) = 2(4) + 6 = 14.
The same idea, five ways
Say it

Say: f of four is fourteen; input four gives output fourteen.

Write it

Write: the function gives six at zero and fourteen at four.

In math
  • f(0) = 6
  • f(4) = 14
  • (0, 6), (4, 14)
  • f(x) = 2x + 6
Like

Two receipts show a machine's result at two settings.

See it
04614function
Each input stays connected to its own output.
Worked exampleThe verified function-value exercise

A linear function has f(0) = 6 and f(4) = 14. Find f(x). This asks for the whole rule from two known input-output pairs.

244681012141618run 1rise 2(0, 6)(−3, 0)f(0) = 6f(4) = 14
Function notation and coordinate notation locate the same points.
  1. Write (0, 6) and (4, 14).Inputs occur in parentheses and outputs after equals signs.
  2. m = 14−64−0 = 84 = 2.Eight output units across four input units means two per unit.
  3. b = 6, so f(x) = 2x + 6.Input zero directly gives the starting output.
  4. f(0) = 2(0) + 6 = 6; f(4) = 2(4) + 6 = 14.The final equation must reproduce both values.
Answer
f(x) = 2x + 6.
Check Four steps of two add eight to starting output six, reaching fourteen.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: f(4) = 14 means 4f = 14.
These parentheses label a function input rather than a multiplication.
✓ Instead: It means input 4 gives output 14, the point (4, 14).
Tips and tricks
  • Tip: translate into points before calculating slope.
Strategy: step by step
  1. 1. Convert function values into coordinate pairs: f(a) = c becomes (a, c).
  2. 2. Check that the inputs differ, then find slope in matching subtraction order.
  3. 3. Use an input-zero point to read b directly, or use an available point in point-slope form.
  4. 4. Rewrite in the requested form.
  5. 5. Substitute both given inputs into the final rule; each output must match.
Strategy
Strategy: recover a line from two points
1
Are the inputs equal?
YesIf outputs differ, there is no function line through both. If outputs also match, you have only one distinct point and need more information.
NoFind the slope.
↓
2
Does either input equal zero?
YesThat output is b.
NoUse point-slope form or solve for b by substitution.
↓
3
Is function notation requested?
YesUse the supplied function name, such as f(x).
NoWrite the requested line form, often y = mx + b.
  1. 1. Turn the data into input-output pairs.
  2. 2. Compute matching changes and divide to get m.
  3. 3. Read b at input zero or solve for it using one point.
  4. 4. Write the equation and verify both points.
Worked exampleA first line from two small points

Find the line through (0, 1) and (2, 5). This asks for a rule returning both listed outputs at their corresponding inputs.

2−22468run 1rise 2(0, 1)(−0.5, 0)firstsecond
The input-zero point gives b while the two-point change gives m.
  1. m = 5−12−0 = 42 = 2.The output grows four while input grows two.
  2. b = 1 from (0, 1).The output at input zero is the initial value.
  3. Write y = 2x + 1.Slope-intercept form combines the rate and starting value.
  4. At x = 0, y = 1; at x = 2, y = 2(2) + 1 = 5.Both given points must fit the recovered line.
Answer
y = 2x + 1.
Check The other anchor gives y − 5 = 2(x − 2). Distribute and add 5 to isolate y: y = 2x + 1 again. Substituting input 2 gives 5.
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: one point gives b

Find the line through (3, 4) and (0, −3). This asks for slope and starting value when the starting point is supplied.

24−6−4−22468run 1rise 2.33(0, −3)(1.29, 0)initialknown
Reading b directly removes a solving step.
  1. m = 4−(−3)3−0 = 73.Seven output units are gained across three input units.
  2. b = −3; write y = 73x − 3.The point with input zero gives the initial value.
  3. Check x = 0 gives −3, and x = 3 gives 7 − 3 = 4.Both given points must fit.
Answer
y = 73x − 3.
Check Slope 73 times run three gives rise seven, matching the two outputs.
Rung 2Rung 2: recover a hidden starting value

Find both forms through (5, 1) and (8, 7). This asks for an equation when neither known input is zero.

2468−10−8−6−4−2246810run 1rise 2(0, −9)(4.5, 0)bfirstsecond
Extend the line back to input zero to see its initial value.
  1. m = 7−18−5 = 63 = 2.Six output units across three input units means two per unit.
  2. y − 1 = 2(x − 5).Use the first point as anchor.
  3. Distribute: y − 1 = 2x − 10. Add 1: y = 2x − 9.These steps isolate y and reveal b.
  4. 2(5) − 9 = 1 and 2(8) − 9 = 7.Verify both points in the solved form.
Answer
  • Point-slope: y − 1 = 2(x − 5).
  • Slope-intercept: y = 2x − 9.
Check Directly solving 1 = 2(5) + b gives b = 1 − 10 = −9; plugging back gives 10 − 9 = 1.
Rung 3Rung 3: fractions while isolating y

Find the line through (3, −2) and (8, 1). This asks for an exact equation without rounding slope or intercept.

246810−4−22run 2rise 1.2(0, −3.8)(6.33, 0)b = −nineteen fifthsfirstsecond
Retain exact fifths in the written equation.
  1. m = 1−(−2)8−3 = 35.Rise is three and run is five.
  2. y − (−2) = 35(x − 3), so y + 2 = 35x − 95.Subtracting a negative adds, and the slope distributes to both terms.
  3. Subtract 2 = 105: y = 35x − 195.Isolate y and combine equal-sized fifths.
  4. At x = 3: 95 − 195 = −2. At x = 8: 245 − 195 = 1.Both original points must fit the solved equation.
Answer
  • m = 35.
  • Point-slope: y + 2 = 35(x − 3).
  • Slope-intercept: y = 35x − 195.
Check The other anchor gives y − 1 = 35(x − 8). Expanding and adding 1 = 55 gives the same intercept −195; input 8 still returns 1.
Rung 4Rung 4: negative coordinates and slope

Find the line through (−2, 5) and (4, −4). This asks you to combine signed coordinates, a fractional slope and a hidden intercept.

−224−6−4−22468run 1rise −1.5(0, 2)(1.33, 0)firstinitialsecond
Positive initial value and negative slope can occur together.
  1. m = −4−54−(−2) = −96 = −32.Output falls nine while input increases six; reduce by three.
  2. y − 5 = −32(x + 2).Subtracting anchor input −2 gives addition.
  3. Distribute: y − 5 = −32x − 3. Add 5: y = −32x + 2.Distribution and addition isolate y.
  4. At x = −2, y = 3 + 2 = 5; at x = 4, y = −6 + 2 = −4.Substitution verifies both points.
Answer
  • Point-slope: y − 5 = −32(x + 2).
  • Slope-intercept: y = −32x + 2.
Check Slope −32 times run six gives rise −9; 5 − 9 = −4.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: through (5, 1) and (8, 7), use y = 2x + 1.
Output 1 belongs to input 5, not input zero. The proposed rule returns 11 at input 5.
✓ Instead: 1 = 2(5) + b gives b = −9; check 10 − 9 = 1. The line is y = 2x − 9.
✗ Not this: Counterexample: f(0) = 6 and f(4) = 14 give points (6, 0) and (14, 4).
The parenthesized number is input and the number after the equals sign is output.
✓ Instead: Use (0, 6) and (4, 14), which give f(x) = 2x + 6.
✗ Not this: Counterexample: input 2 gives outputs 1 and 6, so use a horizontal line.
Horizontal means equal outputs, whereas these inputs are equal.
✓ Instead: The points lie on vertical x = 2, which is not a function of x and has undefined slope.
Tips and tricks
  • Tip: circle an input-zero point to identify b immediately.
  • Tip: when neither input is zero, choose the point with simpler coordinates as the anchor.
  • Tip: check both points. One successful substitution cannot catch every slope error.
Trap. Trap: taking the first listed output as b. It gives b only when that point's input is zero; otherwise recover b by substitution.