Quarry School

Read a graph and write its equation

Explain it like I am five

Picture finding the instructions for a ramp after seeing only its drawing. First find where it crosses the upright axis. That gives its starting height. Next choose two marked addresses and count how far across and how far up you travel between them. Those changes tell you the slope. Put the starting height and slope into y = mx + b. If you are matching several drawings with equations, compare their starting heights first, then their directions and steepness. An increasing function climbs as you look from left to right. A decreasing function descends. A graph with a bigger slope size climbs or descends more for the same distance across.

−2−112345678−1123456789run 2rise −1.5(0, 7)(9.33, 0)(0, 7)(4, 4)
Compare the line's marked addresses and where it meets the axes.
Reminder
  • Slope. From (0, 1) to (2, 5), rise 4 divided by run 2 gives m = 2.
  • Substitution. If y = 2x + b and the point is (3, 8), write 8 = 2 × 3 + b.
  • Solving an equation. 8 = 6 + b gives b = 2 by subtracting 6 from both sides; 6 + 2 = 8 checks it.
  • Signed subtraction. 4 − 7 = −3, a downward change.
  • Reducing fractions. 84 = 2, because 8 ÷ 4 = 2.
Why it works. A line with a specified slope and y-intercept has only one possible graph: start at that intercept and repeat the required slope move. Therefore recovering m and b recovers its equation. If the intercept is not visible, any known point supplies the missing starting height through b = y − mx. When comparing steepness in a picture, the axes must use the same scales between graphs. Enlarging one axis alone can make a line appear steeper without changing its slope number.
RuleRule: Read b at (0, b), calculate m = y2−y1x2−x1, then write f(x) = mx + b. If b is hidden, use b = y1 − mx1.
The same idea, five ways
Say it

Say: read the starting height, then measure the climb per step across.

Write it

Two points on a graph let you recover the linear function that produced it.

In math
  • m = y2−y1x2−x1
  • b = y1 − mx1
  • f(x) = mx + b
  • m > 0: increasing
  • m < 0: decreasing
  • m = 0: constant
Like

You infer a road's starting elevation and grade from two map markers.

See it
−224682468run 2rise −1.5(0, 7)(9.33, 0)interceptsecond point
Compare the line's marked addresses and where it meets the axes.
The same idea, other ways
From a visible start

A graph through (0, 7) starts at 7. A second point (4, 4) tells you the output falls by 3 while the input rises by 4.

2462468run 2rise −1.5(0, 7)(4, 4)
Compare the line's marked addresses and where it meets the axes.
From a hidden start

If slope is 2 and a point is (3, 8), substitute 8 = 2 × 3 + b. Subtracting 6 gives b = 2, and putting it back gives 8 = 6 + 2.

8 = 2 × 3 + b
b = 8 − 6 = 2
Check: 8 = 6 + 2
One known point recovers an unseen starting height.
.1Increasing and decreasing directions

Think of walking right along a ramp. If your height grows, the function is increasing. If your height falls, it is decreasing. For a line, the sign of slope tells you which happens. These words always compare outputs while inputs increase.

  • Rule: m > 0 makes a linear function increasing; m < 0 makes it decreasing.
  • Rule: m = 0 gives constant outputs, so the function neither rises nor falls.
−22−8−6−4−2246810(0, −2)(0.667, 0)shared startincreasing outputdecreasing output
Both lines start at −2, but the solid line rises rightward while the dashed line falls.
Reminder
  • Signed subtraction. 1 − (−2) = 3 is an upward change.
The same idea, five ways
Say it

Say: when inputs grow, do the outputs grow or shrink?

Write it

A line's slope sign gives its direction from left to right.

In math
  • m > 0: increasing
  • m < 0: decreasing
  • m = 0: constant
Like

Walking right on a ramp can carry you upward, downward, or at one height.

See it
−22−8−6−4−2246810(0, −2)(0.667, 0)shared startincreasing outputdecreasing output
Both lines start at −2, but the solid line rises rightward while the dashed line falls.
Worked exampleRead direction from output changes

You need to decide whether y = 3x − 2 rises or falls as you move right, and compare it with y = −2x − 2.

−22−8−6−4−2246810(0, −2)(0.667, 0)shared startincreasing outputdecreasing output
Both lines start at −2, but the solid line rises rightward while the dashed line falls.
  1. At inputs 0 and 1, the first rule gives −2 and 1: 3 × 0 − 2 = −2 and 3 × 1 − 2 = 1.Evaluate by substituting the inputs, multiplying, then subtracting.
  2. The output change is 1 − (−2) = 3, for input change 1 − 0 = 1. The first function is increasing.Its output grows when input grows, so its positive slope describes a rightward climb.
  3. At inputs 0 and 1, the second rule gives −2 and −4. Its output change is −4 − (−2) = −2.A negative change means a lower ending height at a larger input.
  4. The second function is decreasing. Its slope is −2 for this one-unit run.Decreasing means outputs get smaller when inputs get larger.
Answer
  • y = 3x − 2: increasing.
  • y = −2x − 2: decreasing.
Check The slope signs agree with the measured changes: 3 is positive and −2 is negative.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A negative output means the function is decreasing.
Direction concerns a change, not a height. For y = 3x − 2 the outputs rise from −2 to 1.
✓ Instead: Compare outputs at increasing inputs, or inspect the slope sign.
Tips and tricks
  • Tip: Read direction from left to right, and distinguish output sign from slope sign.
.2Match equations to labeled graphs

Think of identifying roads from their starting elevation and grade. First compare starting heights. When two roads start together, compare whether they climb or descend. If both climb, measure how much each climbs for the same trip across. A graph match needs all these clues to agree.

  • Rule: Match y-intercept, slope sign, then slope size.
  • Rule: Use matching axis scales when comparing steepness by appearance; calculating rise divided by run avoids scale guessing.
−224−9−6−3369121518(0, −2)(0.667, 0)A startA second
Line A rises 3 for each 1 right.
Reminder
  • Subtracting a negative. −1 − (−2) = 1 finds D's rise.
The same idea, five ways
Say it

Say: match the start, direction and climb per step.

Write it

The matching equation must reproduce both marked graph points.

In math
  • y = mx + b
  • b = f(0)
  • m = riserun
Like

Roads may start at one elevation but have different grades.

See it
−224−9−6−3369121518(0, −2)(0.667, 0)A startA second
Line A rises 3 for each 1 right.
Worked exampleMatch lines that can share an intercept

You need to attach an equation to each labeled line. Line A passes through (0, −2) and (1, 1). Line B passes through (0, 4) and (1, 7). Line C passes through (0, −2) and (1, −4). Line D passes through (0, −2) and (3, −1). Match A, B, C and D to y = 3x − 2, y = 3x + 4, y = −2x − 2, and y = 13x − 2.

−224−9−6−3369121518(0, −2)(0.667, 0)A startA second
Line A rises 3 for each 1 right.
−224−9−6−3369121518(0, 4)(−1.33, 0)B startB second
Line B has slope 3 and starting height 4.
−224−9−6−3369121518(0, −2)(−1, 0)C startC second
Line C falls to the right from starting height −2.
−224−9−6−3369121518(0, −2)(6, 0)D startD second
Line D shares A's intercept but rises only 1 for a run of 3; every panel uses the same scales.
  1. Line B starts at height 4, so it matches y = 3x + 4. Its second point checks the slope: 7−41−0 = 3.Only that equation has intercept 4. The slope calculation checks that the remaining feature agrees too.
  2. A, C and D all start at −2. Intercept alone cannot distinguish them.The same starting height may belong to several different line directions.
  3. Line C falls to the right: m = −4−(−2)1−0 = −2. Match C to y = −2x − 2.The negative slope is the one falling option.
  4. Both A and D rise to the right. For A, m = 1−(−2)1−0 = 3. For D, m = −1−(−2)3−0 = 13.Both have positive slope and intercept −2. Counting rise and run distinguishes their steepness without guessing from the drawing.
  5. Match A to y = 3x − 2 and D to y = 13x − 2.Their different slope numbers select different equations even though their intercept and direction match.
Answer
  • A: y = 3x − 2.
  • B: y = 3x + 4.
  • C: y = −2x − 2.
  • D: y = 13x − 2.
Check Check the nonintercept point of each match: 3 × 1 − 2 = 1; 3 × 1 + 4 = 7; −2 × 1 − 2 = −4; 13 × 3 − 2 = −1. Each formula returns its labeled graph's second height.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Two rising graphs with the same intercept must match the same equation.
Their positive slopes can differ. Slopes 3 and 13 both rise but give different heights at the same input.
✓ Instead: Use two marked points to calculate slope, then match the coefficient too.
Tips and tricks
  • Tip: State each graph label with its full equation; direction alone is not a complete match.
Strategy: step by step
  1. 1. Read the y-intercept if it is shown. Confirm that the first coordinate is 0.
  2. 2. Choose two clearly marked points. Keep their order the same in both subtractions.
  3. 3. Compute rise divided by run and reduce the fraction.
  4. 4. Write the slope and intercept in f(x) = mx + b.
  5. 5. Substitute both points to verify the recovered equation.
Strategy
Strategy: recover or match a line equation
1
Is the y-intercept visible and labeled?
YesRead b directly.
NoUse a known point in b = y − mx, then plug b back into that point's equation.
↓
2
Do candidate lines have the same intercept?
YesUse slope sign and size to distinguish them.
NoMatch the intercept before comparing slopes.
  1. Find two readable points and compute slope.
  2. Find b from a visible y-intercept or from b = y − mx.
  3. For matching, compare b, slope sign, then slope size.
  4. Verify the chosen equation at a point other than the intercept.
Worked exampleThe line through two graph markers

You need an equation for the graph that crosses the y-axis at (0, 7) and passes through (4, 4).

−224682468run 2rise −1.5(0, 7)(9.33, 0)(0, 7)(4, 4)
Compare the line's marked addresses and where it meets the axes.
  1. Read b = 7 from the point (0, 7).The output at input 0 is the y-intercept.
  2. Use the same point order: m = 4−74−0 = −34 = −34.Slope is the output change divided by the input change, from the first point to the second.
  3. Write f(x) = −34x + 7.The recovered slope and starting output determine the line.
  4. Put the found values back in: f(0) = 7 and f(4) = −3 + 7 = 4.Both original graph markers must satisfy the new equation.
Answer
f(x) = −34x + 7.
Check Walk right 4 and down 3 from (0, 7); you land at (4, 4), so the slope interpretation reproduces the graph.
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: graph through the origin

You need the equation through (0, 0) and (1, 1).

−22−2−1123(0, 0)(−0, 0)startsecond
Compare the line's marked addresses and where it meets the axes.
  1. b = 0 and m = 1−01−0 = 1.The first point is the intercept and the second gives one up for one right.
  2. Write y = x.Slope 1 and intercept 0 give y = 1x + 0.
Answer
y = x.
Check At input 1 the output is 1; at input 0 it is 0.
Rung 2Rung 2: a visible nonzero intercept

You need the equation through (0, 2) and (2, 6).

−224−22468(0, 2)(−1, 0)startsecond
Compare the line's marked addresses and where it meets the axes.
  1. b = 2 and m = 6−22−0 = 42 = 2.Read the starting value and calculate the common rate.
  2. Write y = 2x + 2.These are the slope and intercept.
  3. At input 2, 2 × 2 + 2 = 6.Plugging the recovered values back in verifies the second point.
Answer
y = 2x + 2.
Check A run of 2 with slope 2 gives a rise of 4, taking output 2 to 6.
Rung 3Rung 3: a negative fraction

You need the equation through (0, 7) and (4, 4).

2462468(0, 7)(9.33, 0)startsecond
Compare the line's marked addresses and where it meets the axes.
  1. m = 4−74−0 = −34, and b = 7.The line drops 3 while moving right 4.
  2. Write f(x) = −34x + 7 and check f(4) = −3 + 7 = 4.The equation must reproduce its known point.
Answer
f(x) = −34x + 7.
Check A down-3, right-4 walk returns to the second marked address.
Rung 4Rung 4: the intercept is hidden

You need the equation through (−1, 1) and (3, 9), even though neither given point is on the y-axis.

−224−2246810(0, 3)(−1.5, 0)firstsecondfound intercept
Compare the line's marked addresses and where it meets the axes.
  1. m = 9−13−(−1) = 84 = 2.The input change is 4 because subtracting −1 adds 1.
  2. Use (−1, 1): 1 = 2(−1) + b = −2 + b. Add 2 to both sides to find b = 3.A point fixes the missing starting value.
  3. Plug b back in: 2(−1) + 3 = 1. Write y = 2x + 3.The recovered intercept must make the point true.
Answer
y = 2x + 3.
Check At the other point, 2 × 3 + 3 = 9, so both given addresses fit.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: from (0, 7) and (4, 4), m = 7−44−0 = 34.
The output subtraction runs backward while the input subtraction runs forward. The graph descends to the right, so a positive slope cannot fit.
✓ Instead: Use 4−74−0 = −34, or 7−40−4 = −34.
Tips and tricks
  • Tip: When two graphs share an intercept, the larger |m| is steeper if their axes use matching scales.
  • Tip: A graph that rises to the right is increasing; one that falls is decreasing. Check the sign before accepting your equation.
Trap. Subtracting the output coordinates in one direction and the input coordinates in the opposite direction. Write arrows between the two points, then use that same direction for both differences.