Quarry School

Start at the y-intercept and walk the slope

Explain it like I am five

Picture walking along a ramp drawn on a map. You need a starting address and instructions for each move. The y-intercept is the point where the line meets the upright y-axis. That axis has input 0, so the starting address is (0, b). The slope tells you how the height changes as you move across. Rise is vertical displacement, meaning change in height, negative when you go down. Run is horizontal displacement, meaning change across. A slope of two thirds tells you to go up 2 for every 3 right. A slope of negative two thirds sends you down 2 for every 3 right. Repeating the same move keeps you on the line.

−4−22468−2−1123456789run 2rise −1.33(0, 5)(7.5, 0)(0, 5)(3, 3)
The marked points give addresses on the same straight line.
Reminder
  • Substitution. At x = 0, 3x + 7 becomes 3 × 0 + 7 = 7.
  • Reducing fractions. 46 = 23 because both top and bottom divide by 2.
  • Subtracting a negative. 4 − (−2) = 6; removing a backward amount adds a forward amount.
  • Signs in division. −23 is negative, while −2−3 is positive.
  • Axis position. A point (0, 5) is on the y-axis because it has no horizontal move.
Why it works. Substitute x = 0 into y = mx + b. The product m × 0 disappears, leaving y = b. This output at input 0 is called the initial value; its graph point is the y-intercept. For two points, subtract their equations: y2 − y1 = m(x2 − x1). Divide by their nonzero horizontal change to obtain the slope formula. This explains why the walk is rise divided by run. You may reverse both directions because a negative divided by a negative gives the same slope. Reversing only one changes the slope.
RuleRule: In y = mx + b, the y-intercept is (0, b) and m = riserun = y2−y1x2−x1, with x2 ≠ x1.
The same idea, five ways
Say it

Say: slope is height change for each change across.

Write it

A linear function has a constant rate of change and starts at output b when the input is 0.

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

A ramp has a fixed climb for each distance forward.

See it
−4−2246−4−224run 2rise 1(0, −1)(2, 0)startright 2, up 1repeat
The marked points give addresses on the same straight line.
The same idea, other ways
As a walk

For m = −23, start at (0, 5), go right 3 and down 2 to (3, 3). Reverse both moves to go left 3 and up 2 to (−3, 7).

−4−22462468run 2rise −1.33(0, 5)(7.5, 0)(−3, 7)(0, 5)(3, 3)
The marked points give addresses on the same straight line.
As a rate

If you pay $2 per mile, three more miles add $6. The rate is 63 = 2. The fixed starting charge is separate from that rate.

3 more miles$2 per mile$6 moreinputoutput
The extra output divided by the extra input is the rate.
With small numbers

For y = 2x + 1, input 0 gives output 1. Moving from input 0 to input 1 changes the output from 1 to 3: a rise of 2 for a run of 1.

input xoutput f(x)011325
Every one-unit input increase raises the output by 2.
Why it must be true

The same b is added to both outputs. Subtracting them cancels b, leaving only m times the input change.

y2 − y1 = (mx2 + b) − (mx1 + b)
y2 − y1 = m(x2 − x1)
The starting value disappears when you compare changes.
Strategy: step by step
  1. 1. Read b, then substitute 0 to verify the point (0, b).
  2. 2. Read m. Write a whole-number slope over 1.
  3. 3. Choose signed rise and run whose quotient is m.
  4. 4. Start at (0, b), move by that run and rise, and repeat.
  5. 5. Draw the line through the points. Check one point by substitution.
Strategy
Strategy: graph using one point and the slope
1
Is m negative?
YesA positive rightward run must have a downward rise.
NoA positive rightward run has an upward rise when m > 0, and no rise when m = 0.
↓
2
Do you need a point to the left?
YesReverse both rise and run.
NoContinue the same rightward move.
  1. Identify the y-intercept (0, b).
  2. Turn m into a signed rise divided by a signed run.
  3. Walk that change at least twice.
  4. Check the plotted outputs in the original equation.
Worked exampleGraph by starting and walking

You need to draw f(x) = −23x + 5 using its starting point and its slant, instead of computing each point from scratch.

−4−22468−22468run 2rise −1.33(0, 5)(7.5, 0)startright 3, down 2repeat
The marked points give addresses on the same straight line.
  1. Read b = 5. Check f(0) = −23 × 0 + 5 = 5, and plot (0, 5).The y-axis is where the input is 0.
  2. Read m = −23, so choose rise −2 and run 3.The quotient of these signed changes is the slope.
  3. From (0, 5), go right 3 and down 2 to (3, 3). Repeat to (6, 1).Add 3 to each input and subtract 2 from each output; this repeats the same rate.
  4. Draw the line through the points, extending it both ways.The rule accepts inputs on both sides of the starting point.
Answer
The line goes through (0, 5), (3, 3) and (6, 1).
Check Substitute the new input 6: f(6) = −4 + 5 = 1. The walk and the formula agree.
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: a whole-number slope

You need to draw y = x + 2 from one starting point and a one-unit move.

−224−2246run 1rise 1(0, 2)(−2, 0)startfirstsecond
The marked points give addresses on the same straight line.
  1. At input 0, y = 0 + 2 = 2, so plot (0, 2).This verifies the y-intercept.
  2. Write the slope 1 as 11. Go right 1 and up 1 to (1, 3), then to (2, 4).Each move has rise divided by run equal to 1.
Answer
The line through (0, 2), (1, 3) and (2, 4).
Check At input 2, 2 + 2 = 4, matching the final point.
Rung 2Rung 2: a positive fractional slope

You need to draw y = 12x − 1 using whole-number movements.

−2246−22run 2rise 1(0, −1)(2, 0)startfirstsecond
The marked points give addresses on the same straight line.
  1. At input 0, y = −1, so plot (0, −1).The product with 0 vanishes.
  2. Use rise 1 and run 2. Move to (2, 0), then (4, 1).The fraction measures the two-unit run, so no fractional coordinate is needed.
Answer
The line through (0, −1), (2, 0) and (4, 1).
Check At x = 4, 12 × 4 − 1 = 2 − 1 = 1.
Rung 3Rung 3: a negative slope

You need to draw y = −32x + 4 and keep the downward direction correct.

24−4−2246run 1rise −1.5(0, 4)(2.67, 0)startfirstsecond
The marked points give addresses on the same straight line.
  1. Input 0 gives output 4. Plot (0, 4).This finds the starting point on the y-axis.
  2. Move right 2 and down 3 to (2, 1), then (4, −2).The signed quotient −32 is the given slope.
Answer
The line through (0, 4), (2, 1) and (4, −2).
Check Substitution at 4 gives −6 + 4 = −2, confirming the final point.
Rung 4Rung 4: a negative input

You need a point to the left of the y-axis on f(x) = −23x + 5.

−4−2242468run 2rise −1.33(0, 5)(7.5, 0)negative inputstart
The marked points give addresses on the same straight line.
  1. Start at (0, 5). Reverse the rightward move: go left 3 and up 2.Changing both directions preserves 2−3 = −23.
  2. The new point is (−3, 7). Substitute −3: f(−3) = −23 × (−3) + 5 = 2 + 5 = 7.Plugging the found input back in checks that the point is on the original graph.
Answer
(−3, 7)
Check Between (−3, 7) and (0, 5), the rise is −2 and the run is 3, so the slope is −23.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: a slope of −23 means down 2 and left 3.
Those changes give −2−3 = 23, a positive slope.
✓ Instead: Use down 2 and right 3, or up 2 and left 3.
✗ Not this: Counterexample: y = 2x + 5 starts at (5, 0).
That point is on the horizontal axis. The starting value 5 occurs when x = 0.
✓ Instead: Start at (0, 5). Substitution gives 2 × 0 + 5 = 5.
Tips and tricks
  • Tip: Memory cue: rise over run. Say the top movement before the bottom movement.
  • Tip: A downward line from left to right must have negative slope; compare the picture with your sign.
  • Tip: Each unrestricted linear function has a y-intercept because input 0 is allowed. A context may restrict inputs, so check its stated domain.
Trap. Reading −23 as down 3 and right 2. The top is rise, the bottom is run. Write the two movements beside the fraction before plotting.