D. Characteristics of Functions · Combining Functions and Modelling

Building and Using Function Models

D3.3 D3.1 D3.2About 75 minutes

Learning goals

I can follow the modelling cycle: understand, explore, choose a family, find parameters, check, use and interpret.
I can build a function model from data or from a description of a situation.
I can use a model to answer questions, solving equations and inequalities by algebra, graphs or numerical methods.
I can interpret results in context and state the limitations of a model.

Words to know

mathematical model
A function (or equation) that describes a real situation well enough to make predictions.

parameter
A number in a model, such as a, b and c in y = a·bᵗ + c, chosen to fit a particular situation.

regression
A technology method that finds the parameters of a chosen family that best fit data.

interpolate / extrapolate
To predict inside the range of the data (interpolate) or outside it (extrapolate). Extrapolation is riskier.

limitation
A condition under which a model stops being accurate or sensible, such as negative time or growth that cannot go on forever.

From data to decisions

A public-health inspector needs to know how fast food warms up on a counter. A harbour master needs to know when a boat can safely reach the dock. A small business wants to make the biggest box from a sheet of cardboard. Each question becomes a math question once you have a function model.

This final teaching lesson brings the whole unit (and course) together. You will choose a function family (lesson Comparing the Function Families), sometimes combine functions (lessons 1 to 5), compose them (lessons 6 to 10), and solve with algebra or with graphs and numerical methods (lesson When Algebra Is Not Enough).

The modelling cycle. It is a loop: if the model does not fit or gives unreasonable answers, go back and choose again.

Finding parameters by hand

Graphing technology can do regression, but you can find good parameters by hand from a few key facts:

Exponential y = a·bˣ + c: c is the level it approaches (asymptote); a = (starting value) − c; b comes from a ratio: if the gap above c shrinks from G₀ to G₁ in n steps, then bⁿ = G₁/G₀.
Sinusoidal y = a cos(k(x − d)) + c: a = (max − min)/2, c = (max + min)/2, k = 2π/period, and d = the x-value of a maximum (for cosine).
Polynomial: use known zeros for the factored form, then one more point for the leading coefficient; or use finite differences.
From a description: translate words into a formula, as in the box problem below.

t (min)	0	5	10	15	20	25	30
T (°C)	90.0	69.0	54.4	44.2	37.2	32.2	28.8
T − 21	69.0	48.0	33.4	23.2	16.2	11.2	7.8
ratio of gaps		0.696	0.696	0.695	0.698	0.691	0.696

Sample data: a mug of tea cooling in a 21°C room. The gaps above room temperature shrink by almost the same ratio every 5 minutes.

Worked example

Example 1: A cooling model from data

Use the table to build a model for the tea's temperature, check it, and find when the tea reaches 50°C.

Explore: the temperature drops quickly, then levels off near room temperature. The differences are not constant, but the gaps above 21°C have a nearly constant ratio, about 0.696 per 5 minutes. That points to an exponential plus a constant.
Choose and fit: T = a·bᵗ + c with c = 21 (room temperature) and a = 90 − 21 = 69. For b: b⁵ ≈ 0.696, so b ≈ 0.696^(1/5) ≈ 0.930. Model: T = 21 + 69(0.93)ᵗ.
Check: t = 20 gives 21 + 69(0.93)²⁰ ≈ 21 + 16.2 = 37.2 ✓ (matches the data).
Use: 21 + 69(0.93)ᵗ = 50 → 0.93ᵗ = 29/69 → t = log(29/69) / log 0.93 ≈ 11.9.
Interpret: about 12 minutes after it was poured, the tea is 50°C. The model is reasonable for t ≥ 0; it predicts the tea never quite reaches 21°C, which matches what happens in practice.

T = 21 + 69(0.93)ᵗ; the tea reaches 50°C after about 11.9 minutes.

The cooling data (dots) and the model T = 21 + 69(0.93)ᵗ (curve). The model approaches the room temperature (dashed line) and reaches 50°C at about t = 11.9 min.

Worked example

Example 2: Is the dock deep enough?

At a harbour dock (sample data), the water is deepest, 5.2 m, at 2:00 a.m., and shallowest, 1.0 m, about 6.2 hours later at 8:12 a.m. Tides like this repeat about every 12.4 hours. A sailboat needs at least 2 m of water. Build a model and find when the boat can use the dock between midnight and midnight.

Choose: tides repeat, so use a sinusoid. A maximum at t = 2 suggests cosine: d(t) = a cos(k(t − 2)) + c, t in hours after midnight.
Parameters: a = (5.2 − 1.0)/2 = 2.1; c = (5.2 + 1.0)/2 = 3.1; k = 2π/12.4. So d(t) = 2.1 cos(2π(t − 2)/12.4) + 3.1.
Check: d(8.2) = 2.1 cos(π) + 3.1 = 1.0 ✓.
Use: d(t) ≥ 2 means cos(2π(t − 2)/12.4) ≥ −1.1/2.1 ≈ −0.5238. The angle θ with cos θ = −0.5238 is θ ≈ 2.122 rad, so −2.122 ≤ 2π(t − 2)/12.4 ≤ 2.122, which gives |t − 2| ≤ 2.122 × 12.4/(2π) ≈ 4.19 h.
So the depth is at least 2 m for 2 − 4.19 ≤ t ≤ 2 + 4.19, that is up to about t = 6.19 (6:11 a.m.). Adding the period 12.4 h gives the next window, 10.21 ≤ t ≤ 18.59, and the next one starts at 22.61.
Interpret: in the day, the dock is usable from midnight to about 6:11 a.m., from about 10:13 a.m. to 6:35 p.m., and from about 10:37 p.m. to midnight.

d(t) = 2.1 cos(2π(t − 2)/12.4) + 3.1. Safe windows: 12:00 a.m. to 6:11 a.m., 10:13 a.m. to 6:35 p.m., and 10:37 p.m. to 12:00 a.m.

The tide model d(t) (solid) and the 2 m safety line (dashed). In the shaded time intervals the water is at least 2 m deep.

Worked example

Example 3: A model from a description

A 30 cm by 20 cm sheet of cardboard is made into an open-top box by cutting a square of side x cm from each corner and folding up the sides. Find the volume function, its domain, and the cut that gives the largest volume.

Build: the base is (30 − 2x) by (20 − 2x) and the height is x. V(x) = x(30 − 2x)(20 − 2x), a cubic (a product of three linear functions).
Domain: every length must be positive: x > 0 and 20 − 2x > 0, so 0 < x < 10.
Use: a table gives V(3) = 3(24)(14) = 1008, V(4) = 4(22)(12) = 1056, V(5) = 5(20)(10) = 1000. The maximum is near x = 4.
Refine with a finer table (or graphing technology): the maximum is about 1056 cm³ at x ≈ 3.92 cm.
Interpret: cut squares about 3.9 cm on a side. (Calculus in MCV4U finds the exact value.) The model ignores the thickness of the cardboard and any overlap for gluing.

V(x) = x(30 − 2x)(20 − 2x), 0 < x < 10; largest volume about 1056 cm³ when x ≈ 3.92 cm.

The box net. Cutting a square of side x from each corner leaves a base (30 − 2x) by (20 − 2x), and the sides fold up to height x.

Every model has limits

Ask: For which inputs does the model make sense (domain in context)? Is it interpolating or extrapolating? Could the pattern change (an exponential can't grow forever; a tide model ignores storms)? Always state your answer in a sentence with units and, when it matters, a limitation.

Just for fun

The statistician George Box is often quoted as saying: "All models are wrong, but some are useful." Your cooling model will never be exactly right to the hundredth of a degree, but it will tell you when your tea is drinkable.

Hands-on activity

Bouncing ball model

You need: a ball that bounces well (a tennis ball or basketball), a tape measure, masking tape, a pencil, and a helper (a phone camera in slow motion helps).

Tape the tape measure to a wall, from the floor up to 1.5 m. Clear the area of anything breakable.
Hold the ball so its bottom is at 1.2 m and drop it (do not throw).
Record the height of the bottom of the ball at the top of each bounce for 4 or 5 bounces. Repeat 3 times and average.
Make a table of bounce number n and height h. Find the ratios of consecutive heights.
Build a model h(n) = 1.2rⁿ. Use it to predict when the bounce height will be under 10 cm.

What to look for: the ratios should be roughly constant (for a tennis ball often somewhere around 0.5 to 0.6; yours will differ). For example, heights 1.2, 0.66, 0.36, 0.20, 0.11 m give r ≈ 0.55 and h(n) = 1.2(0.55)ⁿ. Then 1.2(0.55)ⁿ < 0.1 gives n > log(1/12)/log 0.55 ≈ 4.2, so from the 5th bounce on. Limitations: the model counts whole bounces only, and a real ball eventually stops instead of bouncing forever.

Your turn

A podcast's subscribers in its first weeks (sample data): weeks 0 to 5: 120, 150, 188, 234, 293, 366. Which family fits? Write a model.

Use the podcast model to predict week 10 subscribers. Is this interpolation or extrapolation?

Give one reason the podcast model cannot hold for years.

A Ferris wheel has a maximum height of 33 m and a minimum of 3 m, and takes 80 s per turn. A rider is at the bottom at t = 0. Write a model h(t).

For V(x) = x(30 − 2x)(20 − 2x), why is x = 12 not allowed?

The tea model is T = 21 + 69(0.93)ᵗ. Find T(60) and explain why the model says T never equals 21.

More practice: warm-up

Data repeat every 24 hours. Which family?

Data level off at 15 and the gaps above 15 have a constant ratio. Which model form?

In y = a cos(k(x − d)) + c, how do you find c from the maximum and minimum?

Is predicting inside the range of your data called interpolation or extrapolation?

What is the period of y = cos(πt/40)?

More practice: on your own

Sample data for a cooling pie in a 20°C kitchen: t = 0, 10, 20 min gives T = 180, 116, 77.6°C. Build a model and find when it is 40°C.

A ball's height is modelled by h(t) = −4.9t² + 14t + 1.5 (m, t in s). Find h(1), h(2) and h(3), and explain what h(3) tells you about the domain.

Find a cubic model with zeros at x = 0, 4 and 9 (in thousands of units) for a company's profit P (in thousands of dollars), passing through (1, 48).

For the tide model, at what time after 2:00 a.m. is the water first 3.1 m deep, and what is special about this depth?

Two models for the podcast: exponential S = 120(1.25)ʷ and quadratic Q = 6w² + 19w + 120 (both fit weeks 0 to 5 fairly well). Compare their predictions at week 20.

Challenge

A model for drug concentration (mg/L) in the blood after one dose is C(t) = 8t(0.7)ᵗ, t in hours. Describe the combination, and use a table to estimate the peak and when C drops below 1 mg/L.

For the box, find the cut x that makes the volume exactly 900 cm³, using a numerical method (there are two answers). Explain the difference between the two boxes.

Hours of daylight at a northern Ontario location (sample model) are H(t) = 3.8 sin(2π(t − 80)/365) + 12.2, t = day of the year. On how many days is there more than 14 hours of daylight?

Check your understanding

Quiz

Data show gaps above a constant level shrinking by the same ratio each step. Which model form fits?

A tide is highest (6 m) at t = 1 h and lowest (2 m) 6.2 h later. Which model fits?

What is the domain of V(x) = x(40 − 2x)(25 − 2x) for an open box cut from a 40 cm by 25 cm sheet?

Predicting a value far outside the range of the data is called what?

The model T = 21 + 69(0.93)ᵗ predicts the temperature of tea. What does the 21 represent?

Two models fit the same five data points well but give very different predictions far in the future. What is the best next step?

For parents and teachers

Big idea: D3.3 asks students to solve problems, including real-world ones, by building a model (for example from data), using it to get mathematical results, and interpreting and communicating those results in context. This lesson draws on every unit of the course.

How to help: ask the three modelling questions out loud: "Why this family?", "Does it fit the data?" and "Where does the model stop making sense?" Have your teen state every final answer in a full sentence with units.

Common difficulties:

Choosing a family from the first few points only.
Forgetting to subtract the asymptote (room temperature) before taking ratios.
Mixing degrees and radians in sinusoidal models.
Reporting a number without context, units, or a limitation.

The tide, cooling and daylight data in this lesson are realistic samples made for practice, not measurements from a particular place or day. Safety: in the bouncing-ball activity, clear breakable objects and use a soft ball indoors.