Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

1

Function Notation, Domain & Range

A function assigns each input exactly one output. We write $ f(x) $ to mean "the output of function $ f $ when the input is $ x $".

Domain

The set of all allowed inputs ($ x- $values).

e.g. for $ f(x) = \sqrt{}x, $ domain is $ x \geq 0 $

Range

The set of all possible outputs ($ y- $values).

e.g. for $ f(x) = x^2, $ range is $ y \geq 0 $

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GDC tip: Graph the function and use the window/trace to identify the domain and range visually. Look for where the graph exists horizontally (domain) and vertically (range).

Worked Example

Given $ f(x) = 3x - 7, $ find $ f(4) $ and solve $ f(x) = 11 $.

$ f(4) = 3(4) - 7 = 12 - 7 = 5 $

$ f(x) = 11 \Rightarrow 3x - 7 = 11 \Rightarrow 3x = 18 \Rightarrow x = 6 $

Answer: $ f(4) = 5, x = 6 $

2

Linear Models

A linear function produces a straight-line graph. Used when there is a constant rate of change.

Gradient–intercept form

$ f(x) = mx + c $

$ m = $ gradient (rate of change), $ c = y- $intercept

Gradient formula

$ m = (y_2 - y_1) / (x_2 - x_1) $

Worked Example

A plumber charges a $60 call-out fee plus $45 per hour. Write a model for the cost $ C $ after $ t $ hours, and find the cost for 3.5 hours.

$ C(t) = 45t + 60 $ (gradient = 45, $ y- $intercept = 60)

$ C(3.5) = 45(3.5) + 60 = 157.5 + 60 = 217.50 $

Answer: $ C(t) = 45t + 60 $; cost = $217.50

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Interpretation: Always state what $ m $ and $ c $ mean in context. E.g. "The gradient 45 means the cost increases by $45 for each additional hour."

3

Quadratic Models

A quadratic function produces a parabola. Used for projectile motion, area problems, and any quantity that increases then decreases (or vice versa).

Standard form

$ f(x) = ax^2 + bx + c $

$ a $ > 0: opens upward $ a $ < 0: opens downward

Axis of symmetry

$ x = -b / (2a) $

Vertex = ($ x, f(x)) $ at this value

TI-84 Plus CE — Finding vertex & roots

Enter equation in [Y=]
Press [GRAPH] to view the parabola
[2nd][CALC] → 3: minimum (or 4: maximum) → set bounds → vertex
[2nd][CALC] → 2: zero → set bounds → each root

TI-Nspire CX II — Finding vertex & roots

Open Graphs, type equation and press [Enter]
[Menu] → Analyze Graph → Minimum (or Maximum)
[Menu] → Analyze Graph → Zero → set bounds for each root

Casio fx-CG50 — Finding vertex & roots

[MENU] → Graph → enter Y1 = equation
[DRAW] (F6), then [G-SOLV] (SHIFT+F5)
Select MIN/MAX for vertex, ROOT for $ x- $intercepts

Worked Example

A ball is thrown upward. Its height is modelled by $ h(t) = -5t^2 + 20t + 1.5 $. Find the maximum height and when it hits the ground.

Axis of symmetry: $ t = -20 / (2 \times -5) = 2 $ seconds

Max height: $ h(2) = -5(4) + 20(2) + 1.5 = 21.5 $ m

Hits ground: solve $ h(t) = 0 $ using GDC $ \to t = 4.07 $ s (positive root)

Answer: Maximum height = 21.5 m at $ t = 2 $ s; hits ground at $ t = 4.07 $ s

Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

4

Exponential Models

Used when a quantity grows or decays by a constant percentage over equal time intervals. The graph has a horizontal asymptote.

Exponential growth

$ f(x) = A \times b^x (b > 1) $

or $ f(x) = A \times $ $ e^{kx} $ ($ k $ > 0)

Exponential decay

$ f(x) = A \times b^x (0 < b < 1) $

or $ f(x) = A \times $ $ e^{kx} $ ($ k $ < 0)

$ A = $ initial value (when $ x = 0) $. The horizontal asymptote is $ y = 0 $ (unless the function is shifted vertically).

Worked Example

A population of bacteria doubles every 3 hours. Initially there are 500 bacteria. Write a model and find the population after 12 hours.

Doubling every 3 hours: $ P(t) = 500 \times 2 $^{$ t/3 $}

$ P(12) = 500 \times 2^{12/3} = 500 \times 2^4 = 500 \times 16 = 8000 $

Answer: $ P(t) = 500 \times 2 $^{$ t/3 $}; population = 8000 after 12 hours

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Half-life: For decay problems, "half-life of $ h $ hours" means $ f(t) = A \times (0.5) $^{$ t/h $}. The base is 0.5 and the exponent divides by the half-life period.

TI-84 Plus CE — Exponential regression

Enter data in [STAT] → Edit (L1, L2)
[STAT] → CALC → 0: ExpReg
ExpReg L1, L2 → gives $ y = $ $ ab^{x} $ with values of $ a $ and $ b $

TI-Nspire CX II — Exponential regression

Enter data in Lists & Spreadsheet
[Menu] → Statistics → Stat Calculations → Exponential Regression
Select X List and Y List → gives $ y = $ $ ab^{x} $

Casio fx-CG50 — Exponential regression

[MENU] → Statistics → enter data in List 1, List 2
[CALC] (F2) → REG → EXP → gives $ y = $ $ ab^{x} $

5

Sinusoidal Models

Used for periodic (repeating) phenomena: tides, temperatures, Ferris wheels, daylight hours.

General sinusoidal model

$ f(x) = a \sin(b(x - c)) + d $

$ a = $ amplitude period = 360° / $ b $ $ c = $ horizontal shift $ d = $ vertical shift (principal axis)

Amplitude

$ a = (max - min) / 2 $

Principal axis

$ d = (max + min) / 2 $

Worked Example

The depth of water in a harbour varies between 2 m and 10 m with a period of 12 hours. Write a model for depth $ D(t) $.

Amplitude: $ a = (10 - 2) / 2 = 4 $

Principal axis: $ d = (10 + 2) / 2 = 6 $

Period = 12, so $ b = 360 / 12 = 30 $

$ D(t) = 4 $ sin(30$ t) + 6 $ (assuming high tide at $ t = 3) $

Answer: $ D(t) = 4 $ sin(30$ t) + 6 $

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Common error: Using radians instead of degrees at SL. At AI SL, angles are always in degrees. Check your GDC is in degree mode before graphing or solving.

Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

6

The Modelling Process

The IB expects you to understand the four-stage modelling cycle: develop a model, fit parameters, test against data, then use the model to make predictions.

Stage 1 — Develop

Choose an appropriate model type (linear, quadratic, exponential, sinusoidal) based on the shape of the data or the real-world context.

Stage 2 — Fit

Use regression on the GDC or known information to find the parameters ($ a, b, c, $ etc.).

Stage 3 — Test

Check the model against the original data. Does it give reasonable values? Is $ r $ or $ R^2 $ close to 1? Do residuals look random?

Stage 4 — Use

Use the model for interpolation (predicting within the data range) or extrapolation (predicting outside — less reliable).

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IB Exam tip: When asked to "comment on the reliability" of a prediction, consider: Is it interpolation or extrapolation? Is $ r $ close to $ \pm 1 $? Does the model make physical sense at extreme values?

TI-84 Plus CE — Regression overview

[STAT] → CALC:
4: LinReg(ax+b) 5: QuadReg 0: ExpReg C: SinReg
Turn on Diagnostic: [2nd][CATALOG] → DiagnosticOn → shows $ r $ and $ r^2 $

TI-Nspire CX II — Regression overview

Lists & Spreadsheet → enter data
[Menu] → Statistics → Stat Calculations → choose regression type
$ r $ and $ R^2 $ shown automatically in results

Casio fx-CG50 — Regression overview

[MENU] → Statistics → enter data
[CALC] (F2) → REG → choose: X (linear), X2 (quadratic), EXP, SIN
$ r $ and $ R^2 $ displayed with regression output

7

Common Errors & Exam Traps

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Domain restrictions in context. If $ t $ represents time, then $ t \geq 0. $ If $ x $ is a number of people, then $ x $ must be a positive integer. Always state practical domain limits.

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Extrapolation dangers. A model fitted to data for 0 $ \leq t \leq 10 $ may give nonsensical predictions at $ t = 100 $. Exponential models are especially unreliable for long-term extrapolation.

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Confusing the model type. Constant rate of change = linear. Constant percentage change = exponential. Rises then falls (or repeats) = quadratic or sinusoidal. Look at the $ context, $ not just the numbers.

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Forgetting units. The IB deducts marks if your answer lacks units. "The height is 21.5" loses marks; "$ h = 21.5 $ m" does not.

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Not reading the graph window. When using GDC, adjust Xmin, Xmax, Ymin, Ymax to see the full shape. A poor window can hide roots or the vertex entirely.

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Formula booklet: The equation of a straight line, quadratic formula, and exponential/logarithmic relationships are given. The sinusoidal parameters ($ a, b, c, d) $ are NOT given — you must know how to find them from max, min, and period.

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3 sf rule: Unless the question specifies otherwise, give answers correct to 3 significant figures. When using regression, keep full GDC precision for intermediate calculations.