Quarry School

Read the starting amount and the amount per step

Explain it like I am five

Imagine a taxi meter begins at five dollars before the taxi moves, then adds two dollars for each block. The starting amount and the amount for each block do different jobs. A linear formula records both. The amount multiplying the input is the slope, the output change for one input unit. The amount left at input zero is the initial value. On a graph, that starting point is the y-intercept, also called the vertical intercept. You choose or measure the input, called the independent variable. The resulting output is the dependent variable because it depends on that input.

48121620242832364048121620242832(0, 14.7)(−33.9, 0)surface33 feet
The input runs horizontally and the output runs vertically.
Reminder
  • Substitution. Replace the input everywhere: P(33) = 0.434 × 33 + 14.696.
  • Decimal addition. Line up decimal places: 14.322 + 14.696 = 29.018.
  • Function notation. P(33) is the pressure output when the depth input is 33.
  • Coordinate order. An intercept has input zero: (0, 14.696), not (14.696, 0).
  • Multiplying by zero. 0.434 × 0 = 0, so P(0) leaves the starting pressure 14.696.
  • Decimal multiplication. 0.434 × 33 = 14.322; restore the three decimal places after multiplying 434 × 33.
  • Units. PSI per foot times feet gives PSI; add the starting pressure in PSI.
Why it works. In f(x) = mx + b, putting in zero makes m × 0 equal zero, leaving f(0) = b. That explains why b is the initial value and why the graph crosses the vertical axis at (0, b). Putting in one adds one group of m, so f(1) = m + b. Removing the starting b leaves m. The numbers therefore describe different quantities: an amount and an amount per input unit.
RuleRule: In f(x) = mx + b, slope m is output change per input unit and initial value b = f(0). The y-intercept or vertical intercept is the point (0, b).
The same idea, five ways
Say it

Say: m tells the change per input unit; b tells the output at zero.

Write it

The independent variable is the input; the dependent variable is the output determined by that input.

In math
  • f(x) = mx + b
  • m: output units per input unit
  • b = f(0)
  • y-intercept = (0, b)
  • P(d) = 0.434d + 14.696
  • d: independent variable
  • P(d): dependent variable
Like

A taxi has a starting charge and a separate price per block.

See it
224681012run 1rise 2(0, 5)(−2.5, 0)initial valueone block
The input runs horizontally and the output runs vertically.
The same idea, other ways
As a taxi meter

The meter's five-dollar starting charge stays once. Each block adds two dollars. Five dollars is an amount; two dollars per block is a rate.

1 block2 × 1 + 5$7inputoutput
The total combines a per-block charge with the amount already on the meter.
As a graph

Input zero places you on the vertical axis. Its output is the start. One step right changes the output by the slope.

224681012run 1rise 2(0, 5)(−2.5, 0)initial valueone block
The input runs horizontally and the output runs vertically.
As units

The diver's output pressure uses PSI, meaning pounds per square inch. PSI measures push spread over each square inch. The slope uses PSI per foot, because each foot of depth adds pressure.

Input: depth in feet
Output: pressure in PSI
Slope: PSI per foot
Initial value: PSI at zero feet
A rate and an initial amount require different unit labels.
.1Slope as an amount per input unit

A rate answers how much changes for each step. If a taxi charge adds two dollars for one more block, its rate is two dollars per block. You need both units because two dollars alone does not say how often it is added. In a linear function this rate is the slope. For the diver, the input is depth in feet and the output is pressure in PSI. The slope 0.434 means that one more foot of depth adds 0.434 PSI to the pressure.

  • Rule: In mx + b, slope m is the output change per input unit, because one more input adds one more group of m.
  • Rule: Slope units are output units per input unit, because the rate relates output change to input change.
  • Rule: In P(d) = 0.434d + 14.696, the slope is 0.434 PSI per foot, because d measures feet and P measures pressure in PSI.
Input: depth in feet
Output: pressure in PSI
Slope: 0.434 PSI per foot
33 feet adds 14.322 PSI
The slope has output-per-input units.
Reminder
  • Decimal multiplication. 0.434 × 3 = 1.302 because three groups of 434 thousandths are 1302 thousandths.
The same idea, five ways
Say it

Say: four hundred thirty-four thousandths PSI for each extra foot.

Write it

Each extra foot of depth increases the diver's pressure by 0.434 PSI.

In math
  • m = 0.434 PSI per foot
  • pressure change = 0.434 × depth change
  • m has output units per input unit
Like

A taxi meter adds the same charge each time another block is traveled.

See it
2414151617run 2rise 0.868surfaceone foot
The input runs horizontally and the output runs vertically.
Worked exampleAdded amount versus total amount

Find the added amount means count only the change; find the total means include the start. First find the added and total taxi charge for one block in C(n) = 2n + 5. Then find the added and total pressure at 33 feet in P(d) = 0.434d + 14.696.

33 feet0.434 × 33 + 14.69629.018 PSIinputoutput
The rate produces the added pressure; the start makes it a total.
  1. For one block, the added charge is 2 × 1 = 2 dollars and the total is 2 + 5 = 7 dollars.The slope contributes the new charge, while the constant contributes the starting charge.
  2. Read the diver's rate as 0.434 PSI per foot.The output is pressure in PSI and the input is depth in feet.
  3. Calculate the added pressure: 0.434 × 33 = 14.322 PSI.Thirty-three feet contributes thirty-three copies of the one-foot pressure change.
  4. Add the surface pressure: 14.322 + 14.696 = 29.018 PSI.The total includes the pressure already present at depth zero.
Answer
  • Small model: added charge $2.
  • Total charge $7.
  • Diver slope: 0.434 PSI per foot.
  • At 33 feet: added pressure 14.322 PSI.
  • Total pressure 29.018 PSI.
Check Subtracting the surface pressure from the total gives 29.018 − 14.696 = 14.322 PSI, agreeing with rate times depth.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: The slope is 0.434 feet per PSI.
That reverses which quantity is the input and which is the output.
✓ Instead: The slope is 0.434 PSI per foot.
✗ Not this: Counterexample: At 33 feet, total pressure is 14.322 PSI.
That counts only added pressure and leaves out the surface pressure.
✓ Instead: Total pressure is 14.322 + 14.696 = 29.018 PSI.
Tips and tricks
  • Tip: Underline 'per' in the rate units and write output units before it.
.2Initial value as the output at input zero

The initial value is what your output says before any input steps happen. A taxi can already charge five dollars while it has traveled zero blocks. A diver already experiences pressure at the water's surface, where her depth is zero feet. Putting in zero clears away the changing part and leaves the starting part. On the graph this becomes a point on the vertical axis, because that axis contains points whose horizontal coordinate is zero. That point is the y-intercept or vertical intercept.

  • Rule: Initial value means f(0) = b, because the input term m × 0 vanishes.
  • Rule: The y-intercept or vertical intercept is the point (0, b), because its input coordinate is zero.
  • Rule: P(0) = 14.696 PSI is the diver's surface pressure, because zero depth is the water's surface.
0 feet0.434 × 0 + 14.69614.696 PSIinputoutput
At zero depth, only the initial pressure remains.
Reminder
  • Zero multiplication and coordinates. 2 × 0 = 0. In (0, 5), zero is the input and five is the output.
The same idea, five ways
Say it

Say: the initial value is the output when the input is zero.

Write it

The diver's surface pressure is 14.696 PSI and the vertical intercept is (0, 14.696).

In math
  • f(0) = b
  • y-intercept = (0, b)
  • P(0) = 14.696
  • vertical intercept = (0, 14.696)
Like

A taxi meter shows its starting charge before the wheels move.

See it
48121620242832364048121620242832(0, 14.7)(−33.9, 0)surface33 feet
The input runs horizontally and the output runs vertically.
Worked exampleThe initial charge and the surface pressure

Find the initial value means evaluate at zero; find the intercept means write the matching point. Do both for C(n) = 2n + 5, then for P(d) = 0.434d + 14.696. Compare the diver's surface pressure with her pressure at 33 feet.

48121620242832364048121620242832(0, 14.7)(−33.9, 0)surface33 feet
The input runs horizontally and the output runs vertically.
  1. C(0) = 2 × 0 + 5 = 5 dollars, so the intercept is (0, 5).Zero blocks removes the per-block contribution and places the point on the vertical axis.
  2. P(0) = 0.434 × 0 + 14.696 = 14.696 PSI.Setting depth to zero finds the surface pressure.
  3. Write the diver intercept as (0, 14.696).A point needs its input first and output second.
  4. P(33) = 0.434 × 33 + 14.696 = 14.322 + 14.696 = 29.018 PSI; the increase is 29.018 − 14.696 = 14.322 PSI.Below-surface pressure contains the same initial amount plus the change caused by depth.
Answer
  • Taxi: initial value $5.
  • Intercept (0, 5).
  • Diver: initial value 14.696 PSI.
  • Intercept (0, 14.696).
  • At 33 feet, pressure is 14.322 PSI above the initial pressure.
Check Substitute the diver intercept coordinates: 14.696 = 0.434 × 0 + 14.696. Both sides equal 14.696, so the point lies on the graph.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: Any pressure on the graph can be b, so b = 29.018.
The initial value belongs specifically to input zero. The pressure 29.018 belongs to input 33.
✓ Instead: b = P(0) = 14.696 and the intercept is (0, 14.696).
Tips and tricks
  • Tip: Check a claimed initial value by putting zero into the original formula.
Strategy: step by step
  1. 1. Identify input, output, and their units.
  2. 2. Read the input coefficient as slope, with output-per-input units.
  3. 3. Substitute zero to find the initial value and vertical intercept.
  4. 4. Describe the start and rate in words, then evaluate any requested input.
Strategy
Strategy: Read and explain a slope-intercept model
  1. 1. Match the formula with mx + b.
  2. 2. Give m output-per-input units.
  3. 3. Give b output units and verify it at input zero.
  4. 4. Say the starting amount and the change per step.
  5. 5. Substitute the requested input, multiply, and add.
Worked exampleA small charge, then pressure on a diver

Read a model means explain its starting output and its change per input unit. First read C(n) = 2n + 5 for dollars after n blocks. Then restate P(d) = 0.434d + 14.696 in words, with d in feet and P in PSI, and evaluate it at 33 feet.

1 block2 × 1 + 5$7inputoutput
The small model separates the starting charge from the rate.
48121620242832364048121620242832(0, 14.7)(−33.9, 0)surface33 feet
The input runs horizontally and the output runs vertically.
  1. C(0) = 2 × 0 + 5 = 5 dollars and C(1) = 2 × 1 + 5 = 7 dollars.Zero blocks leaves the start, and one block adds one two-dollar group.
  2. Say: the taxi charge begins at five dollars and adds two dollars per block.The coefficient is the rate and the added constant is the start.
  3. Identify depth d in feet as the independent variable and pressure P(d) in PSI as the dependent variable. PSI means pounds per square inch.Pressure depends on depth, so the output uses pressure units rather than length units.
  4. Say: pressure equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths. It starts at 14.696 PSI and adds 0.434 PSI per foot.0.434 multiplies depth, while input zero leaves 14.696.
  5. P(0) = 0.434 × 0 + 14.696 = 14.696 PSI; the intercept is (0, 14.696).Depth zero finds the surface and the graph's vertical-axis point.
  6. P(33) = 0.434 × 33 + 14.696 = 14.322 + 14.696 = 29.018 PSI.Thirty-three feet adds thirty-three pressure changes to the pressure already present at the surface.
Answer
  • Taxi: start $5.
  • Rate $2 per block.
  • Diver input: d, depth in feet.
  • Diver output: P(d), pressure in PSI.
  • Diver initial value: 14.696 PSI at the surface.
  • Diver slope: 0.434 PSI per foot.
  • Diver vertical intercept: (0, 14.696).
  • At 33 feet: P(33) = 29.018 PSI.
Check Split 33 feet into 30 and 3: 0.434 × 30 = 13.020 and 0.434 × 3 = 1.302. Then 14.696 + 13.020 + 1.302 = 29.018 PSI. These decimals are exact for the stated model.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: 14.696 is the slope because it is the larger number.
A number's role depends on its position in the formula, not its size.
✓ Instead: Slope 0.434 multiplies depth. Initial value 14.696 remains at depth zero.
✗ Not this: Counterexample: The diver intercept is (14.696, 0).
That reverses input and output. The vertical intercept has input zero.
✓ Instead: The intercept is (0, 14.696).
Tips and tricks
  • Tip: Say 'start' for b and 'per' for m, with units for both.
  • Tip: PSI is pressure; PSI per foot is the change in pressure for one extra foot.
Trap. Tip: At 33 feet, 14.322 PSI is the added pressure and 29.018 PSI is the total. Include the initial pressure when the question asks for pressure.