Topic 1.13

Function Model Selection and Assumption Articulation

10 MIN READ7 IDEAS23 PROBLEMS8 flashcards

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01
This restriction is a modeling assumption, not a mathematical fact about the equation. $P(x) = -50x^{2}+500x-800$ is perfectly well-defined for every real $x$ — the parabola keeps going in both directions. We are choosing to only trust the model where it matches reality (profit ≥ 0), and that choice is exactly what "assumption articulation" means: stating out loud which part of the math you're relying on, and why.

Practice 23 problems Next: Chapter 1.14 Function Model Construction and Application

Selecting Models from Real Contexts

Look at the context, the data, or a graph, and select the function type that best models the situation. Remember: models often have a restricted domain and range once you bring real-world context into play.

Maya's Stickers

day	1	2	3	4	5	6
stickers	10	14	18	22	26	30

1st differences: 4, 4, 4, 4, 4 — constant → LINEAR

Theo's Followers

day	1	2	3	4	5	6
followers	5	9	15	23	33	45

1st: 4,6,8,10,12 (not constant). 2nd: 2,2,2,2 — constant → QUADRATIC

Priya's Views

day	1	2	3	4	5	6
views	2	10	30	68	130	222

1st: 8,20,38,62,92. 2nd: 12,18,24,30. 3rd: 6,6,6 — constant → CUBIC

Owen's Savings — Watch the Spacing!

day	1	2	4	5	7	8
dollars	50	53	59	62	68	71

CONCEPT

Careful — the days aren't evenly spaced!

You can't just subtract consecutive values here, because the day-gaps aren't all the same size.

Instead, compute the RATE per day for each gap: $(53-50)/(2-1)=3$ , $(59-53)/(4-2)=3$ , $(62-59)/(5-4)=3$ , $(68-62)/(7-5)=3$ , $(71-68)/(8-7)=3$ .

Every rate is 3 — constant → LINEAR (slope 3), even though the days skip around.

Selecting a Model Type from Context Alone

Sometimes you won't have a data table at all — just a description of a situation. Here's how to pick a model type from the CONTEXT itself.

CONCEPT

Quick Criteria

Linear: roughly constant rate of change (steady growth/decline).

Quadratic: roughly LINEAR rate of change, OR roughly symmetric with a single max/min, OR context involving AREA.

Cubic: context involving VOLUME.

Rational: a quantity that's naturally a RATIO or DIVISION of two other quantities (like dose ÷ time, or cost ÷ units), OR a situation that LEVELS OFF near a fixed value instead of growing forever (concentration diluting toward 0, a rate approaching a saturation point).

Piecewise: the rule genuinely CHANGES at specific input values — different pricing tiers, distinct phases of a process (e.g., climbing, then cruising, then descending).

Worked example

(Calculator Active) A patient receives a dose of painkiller. The function $p(t) = (4t^{2}+12t)/(t^{3}+3)$ models the amount in the bloodstream over time $t$ (hours). Which model TYPE is this, and how can you tell from the equation's structure alone?

01
The right-hand side is one polynomial DIVIDED BY another polynomial — that structure itself signals a RATIONAL function, regardless of the specific numbers.
02
This also matches the context clue: medicine amount typically rises then tapers off toward 0 as it's processed by the body — a leveling-off shape, not one that grows or falls forever like a polynomial would.

Worked example

For each situation, decide: linear, quadratic, or cubic model?

CONCEPT

a) Stacking cube-shaped storage boxes

Each box is a cube. Comparing the SIDE LENGTH to how much it can hold (VOLUME) → CUBIC (volume context).

CONCEPT

b) Buying movie tickets

Each ticket costs the same amount. Comparing number of tickets to total cost → LINEAR (constant rate per ticket).

CONCEPT

c) A circular pool cover

Comparing the pool's RADIUS to the AREA of material needed to cover it → QUADRATIC (area context).

Geometry Naturally Suggests a Degree

CONCEPT

Perimeter, Area, Volume

Perimeter — adding up side lengths → LINEAR in terms of a single side variable.

Area — multiplying two length dimensions together → QUADRATIC.

Volume — multiplying three length dimensions together → CUBIC.

This is a quick sanity check: if a problem describes a length-times-length-times-length quantity, expect a cubic model, even before you see any data.

Quick check

A can's height is always twice its radius $r$ . What type of function models its volume in terms of $r$ ?

Restricted Domain and Range in Context

Worked example

A candle shop's weekly profit is $P(x) = -50x^{2} + 500x - 800$ dollars, where $x$ is the price charged per candle (in dollars). Find $P(4)$ in context, the restricted domain, the restricted range, and the average rate of change on $[5,7]$ .

01
$P(4)$ : plug in $x=4$ :
$P(4) = -50(16) + 500(4) - 800 = 400$
02
In context: charging $4 per candle, the shop makes $400 profit that week.
03
Restricted domain — profit can't realistically be negative, so find where $P(x)=0$ :
$-50x^{2} + 500x - 800 = 0 \implies x = 2,\; x = 8$
04
Restricted domain: $[2, 8]$ dollars — charging less than $2 or more than $8 would lose money, which the shop wouldn't sustain.

KEY RULE

This restriction is a MODELING ASSUMPTION, not a mathematical fact about the equation. $P(x) = -50x^{2}+500x-800$ is perfectly well-defined for EVERY real $x$ — the parabola keeps going in both directions. We are CHOOSING to only trust the model where it matches reality (profit ≥ 0), and that choice is exactly what "assumption articulation" means: stating out loud which part of the math you're relying on, and why.

05
Restricted range — the vertex (at $x=5$ ) gives the maximum: $P(5) = 450$ .

06
Range: $[0, 450]$ dollars.

07
Average rate of change on $[5,7]$ :

\frac{P(7) - P(5)}{7 - 5} = \frac{250 - 450}{2} = -100

08
In context: raising the price from $5 to $7 DECREASES weekly profit at an average rate of $100 per dollar of price increase — you've priced past the sweet spot.

CONCEPT

Articulating the Assumption (the Other Half of This Topic's Name)

Whenever you restrict a domain or range "because it wouldn't make sense otherwise," you're making an assumption — the equation itself doesn't know or care about real-world limits.

Good practice: state the assumption explicitly, e.g. "We assume the shop only operates at prices where profit is non-negative," rather than treating $[2,8]$ as if it fell out of the algebra alone.

Quick check

A box is cut from a $20$ cm by $30$ cm sheet with corner squares of side $x$ . What is the restricted domain of the volume model?

Piecewise Functions in Context

Worked example

The graph shows a regional flight's altitude over 35 minutes. Find the domain, range, and describe what happens on each piece.

01
Domain in context: $[0, 35]$ minutes — from takeoff to landing.
02
Range in context: $[0, 8000]$ feet — from the ground up to cruising altitude.
03
From $t=0$ to 10: the plane climbs steadily (linear piece, rate +800 ft/min). From $t=10$ to 25: it cruises at a constant 8000 ft. From $t=25$ to 35: it descends steadily (linear piece, rate −800 ft/min).

COMMON MISTAKE

The domain and range "in context" should reflect the REAL-WORLD limits of the scenario, not just "all real numbers" — profit can't sustain being negative, a flight doesn't climb forever.

When x-values in a data table are unevenly spaced, always convert to a RATE (change ÷ time) before comparing — raw differences will mislead you.

Practice Problems

Worked example

Problem 1. Select linear, quadratic, or cubic, and justify with differences.

x	1	2	3	4	5	6
y	6	4	4	6	10	16

01
1st diffs: −2, 0, 2, 4, 6 — not constant.
02
2nd diffs: 2, 2, 2, 2 — constant. Model: QUADRATIC.

Worked example

Problem 2 (Calculator Active). The strength of a radio signal (in relative units) at distance $x$ km from the tower is modeled by $S(x) = 5x/(x^{2}+4)$ . Find $S(3)$ , the AROC from $x=2$ to $x=3$ , and the maximum signal strength.

01
$S(3) = 15/13 \approx 1.15$ — at 3 km from the tower, the signal strength is about 1.15 units.
02
AROC[2,3] ≈ $(1.15 - 1.25)/1 \approx -0.096$ — signal strength is DECREASING on average over this stretch.
03
Using a graph or table (calculator active), the maximum is exactly 1.25, reached at $x=2$ km — signal actually peaks a couple km out before fading, not right at the tower.

Worked example

Problem 3 (Calculator Active). A gardener has 40 m of fencing to enclose a rectangular garden against the side of a house (so only 3 sides need fencing). If $x$ is the width, $A(x) = x(40-2x)$ . Find the restricted domain, range, and the maximum area.

01
Restricted domain: $0 < x < 20$ (width must be positive, and the remaining length $40-2x$ must also stay positive).
02
Restricted range: $0 < A(x) \le 200$ (area is always positive, capped at the maximum).
03
Maximum area = 200 m², occurring at $x = 10$ m (making a 10 m × 20 m rectangle).
04
Note the assumption: 0 < $x < 20$ comes from us DECIDING that a garden with zero or negative width/length isn't meaningful — the bare equation $A(x)=x(40-2x)$ would happily accept any real $x$ .

Common slips

The domain and range "in context" should reflect the real-world limits of the scenario, not just "all real numbers" — profit can't sustain being negative, a flight doesn't climb forever.

When x-values in a data table are unevenly spaced, always convert to a rate (change ÷ time) before comparing — raw differences will mislead you.

Lock it in

Try the flashcards

8 cards · Choosing a model

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Recap card

6 lines to re-read the night before.

01
Finite differences determine polynomial degree — but convert to a rate first if x-values are unevenly spaced.
02
Geometry gives a shortcut: perimeter↔linear, area↔quadratic, volume↔cubic.
03
Ratios/division or a leveling-off shape suggest rational; a rule that changes at specific inputs suggests piecewise.
04
Always restrict domain and range to what's physically possible in the real-world context, not just what the algebra allows.
05
Assumption articulation: a restricted domain/range is a choice you make about the model, not a property of the equation itself — say so explicitly.
06
Piecewise functions in context often model distinct phases of a real process — describe each piece's behavior separately.

Practice 23 problems Next: Chapter 1.14 Function Model Construction and Application

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Function Model Selection and Assumption Articulation

Selecting Models from Real Contexts

Maya's Stickers

Theo's Followers

Priya's Views

Owen's Savings — Watch the Spacing!

Selecting a Model Type from Context Alone

Geometry Naturally Suggests a Degree

Restricted Domain and Range in Context

Piecewise Functions in Context

Practice Problems

Recap card

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