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 $

GDC: graph $ y = 3x - 7 $ and $ y = 11, $ find intersection $ \to 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.

GDC: graph $ h(t) $; CALC maximum gives vertex $ (2, 21.5) $

Max height $ = 21.5 $ m at $ t = 2 $ s (check: $ h(2) = -5(4) + 40 + 1.5 = 21.5 $)

Hits ground: CALC zero of $ h(t) = 0 $ $ \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

Inverse Functions

The inverse $ f^{-1} $ reverses $ f $: if $ f(a) = b $ then $ f^{-1}(b) = a $. The graph of $ f^{-1} $ is the reflection of $ f $ in the line $ y = x, $ and the domain and range swap over.

Definition

$ f(a) = b \iff f^{-1}(b) = a $

Reflection of $ f $ in $ y = x $; domain & range swap.

Finding $ f^{-1}(k) $

solve $ f(x) = k $

The solution $ x $ is the value of $ f^{-1}(k). $

TI-84 Plus CE — Evaluating $ f^{-1}(k) $

Enter $ y = f(x) $ and $ y = k $ in [Y=]
[GRAPH], then [2nd][CALC] → 5: intersect → the $ x $-value is $ f^{-1}(k) $

TI-Nspire CX II — Evaluating $ f^{-1}(k) $

Graphs: enter $ f(x) $ and $ y = k $
[Menu] → Analyze Graph → Intersection → the $ x $-value is $ f^{-1}(k) $

Casio fx-CG50 — Evaluating $ f^{-1}(k) $

[MENU] → Graph: enter Y1 = $ f(x), $ Y2 = $ k $
[DRAW] (F6), [G-SOLV] (SHIFT+F5) → INTSECT → the $ x $-value is $ f^{-1}(k) $

Worked Example

Given $ f(x) = 3x - 7, $ find $ f^{-1}(0). $

Solve $ f(x) = 0: $ $ 3x - 7 = 0 \Rightarrow x = 7/3 \approx 2.33 $

Answer: $ f^{-1}(0) = 7/3 \approx 2.33 $

5

Solving Equations Graphically (Intersection)

In AI the standard way to solve any equation is to graph both sides and find where they meet. To solve $ g(x) = h(x), $ graph $ y = g(x) $ and $ y = h(x) $ and read off the $ x $-coordinates of the intersection points.

TI-84 Plus CE — Intersection

Enter both functions in [Y=], press [GRAPH]
[2nd][CALC] → 5: intersect → first curve, second curve, guess → $ (x, y) $

TI-Nspire CX II — Intersection

Graphs: enter both functions
[Menu] → Analyze Graph → Intersection → reads each $ (x, y) $

Casio fx-CG50 — Intersection

[MENU] → Graph: enter Y1, Y2, [DRAW] (F6)
[G-SOLV] (SHIFT+F5) → INTSECT (ISCT) → reads each $ (x, y) $

Worked Example

Solve $ 2^x = x + 3 $ using technology.

Graph $ y = 2^x $ and $ y = x + 3, $ find the intersections

Answer: $ x \approx -2.86 $ and $ x \approx 2.44 $ (from GDC)

6

Cubic Models

A cubic function can rise, dip, then rise again. Used for volumes/containers and data with two turning points.

Standard form

$ f(x) = ax^3 + bx^2 + cx + d $

Up to 3 real roots; up to 2 turning points (local max & local min).

GDC fit

CubicReg from a data table

Turning points via CALC max/min; roots via zero.

TI-84 Plus CE — Cubic regression

Enter data in [STAT] → Edit (L1, L2)
[STAT] → CALC → 6: CubicReg → gives $ y = ax^3 + bx^2 + cx + d $

TI-Nspire CX II — Cubic regression

Lists & Spreadsheet: enter data
[Menu] → Statistics → Stat Calculations → Cubic Regression

Casio fx-CG50 — Cubic regression

[MENU] → Statistics: enter List 1, List 2
[CALC] (F2) → [REG] (F3) → $ X^3 $ → gives the cubic coefficients

Worked Example

For $ f(x) = x^3 - 6x^2 + 9x, $ find the roots and turning points.

GDC zeros: $ x = 0 $ and $ x = 3 $

CALC maximum: $ (1, 4) $; CALC minimum: $ (3, 0) $

Answer: roots $ x = 0, 3 $; local max $ (1, 4) $, local min $ (3, 0) $

Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

7

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) = k a^x + c \quad (a > 1) $

or $ f(x) = k e^{rx} + c $ ($ r > 0 $)

Exponential decay

$ f(x) = k a^x + c \quad (0 < a < 1) $

or $ f(x) = k e^{rx} + c $ ($ r < 0 $)

The constant $ c $ sets the horizontal asymptote $ y = c $ (the limiting value). When $ c = 0 $ the asymptote is $ y = 0 $ and the initial value (at $ x = 0 $) is $ k $. Use the $ +c $ form for cooling and limiting-value contexts.

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] (F3) → EXP → gives $ y = ab^{x} $

8

Direct & Inverse Variation

Variation models use a power of $ x $. With inverse variation ($ n < 0 $) the $ y $-axis is a vertical asymptote.

Direct variation

$ y = a x^n \quad (n > 0) $

$ y $ grows as $ x $ grows; passes through the origin.

Inverse variation

$ y = a x^{-n} = a / x^{n} \quad (n > 0) $

Vertical asymptote $ x = 0 $; e.g. inverse-square law $ I = k / d^2. $

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Method: Substitute one known $ (x, y) $ pair to find the constant $ a, $ then use the model. On the GDC, graph the model to read off values or find intersections.

Worked Example

Intensity varies inversely with the square of distance: $ I = k / d^2. $ Given $ I = 4 $ when $ d = 1.5, $ find $ I $ when $ d = 3. $

Find $ k: $ $ k = 4 \times 1.5^2 = 4 \times 2.25 = 9, $ so $ I = 9 / d^2 $

At $ d = 3: $ $ I = 9 / 3^2 = 9 / 9 = 1 $

Answer: $ I = 1 $ W/m$^2$

Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

9

Sinusoidal Models

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

General sinusoidal model (SL)

$ f(x) = a \sin(bx) + d \quad$ or $\quad f(x) = a \cos(bx) + d $

$ a = $ amplitude period $ = 360^\circ / b $ $ d = $ vertical shift (principal axis)

At SL there is no phase-shift parameter $ c $: choose $\sin$ vs $\cos$ to match where the curve starts (use $\cos$ for a maximum at $ x = 0 $).

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 $

Depth is mid-level and rising at $ t = 0, $ so use $\sin$: $ D(t) = 4\sin(30t) + 6 $

Given a data table instead? Use SinReg on the GDC to fit $ a, b, d $ directly.

Answer: $ D(t) = 4\sin(30t) + 6 $ (degrees; high tide at $ t = 3 $)

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Common error: Using radians instead of degrees at SL. At AI SL, angles are always in degrees. Set degree mode before graphing or solving (TI-84: [MODE] → Degree; Casio: [SHIFT][MENU] SET UP → Angle: Deg).

10

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, cubic, exponential, sinusoidal, or variation) 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?

Functions & Modelling SL

IB Mathematics: Applications & Interpretation · Topic 2: Functions

JMaths

www.jmaths.xyz

10

Regression on the GDC

TI-84 Plus CE — Regression overview

[STAT] → CALC:
4: LinReg(ax+b) 5: QuadReg 6: CubicReg 0: ExpReg C: SinReg
[2nd][CATALOG] → DiagnosticOn shows $ r $/$ r^2 $ for linear/exp/ln/power; QuadReg & CubicReg give $ R^2 $ only; SinReg gives no $ r $/$ R^2 $ — judge sinusoidal fit by graphing the model over the data.

TI-Nspire CX II — Regression overview

Lists & Spreadsheet → enter data
[Menu] → Statistics → Stat Calculations → choose regression type
$ r $ and/or $ R^2 $ shown automatically — but Sinusoidal Regression reports no $ r $/$ R^2 $; judge the sinusoidal fit by graphing.

Casio fx-CG50 — Regression overview

[MENU] → Statistics → enter data
[CALC] (F2) → [REG] (F3) → choose: X (linear), $ X^2 $ (quadratic), $ X^3 $ (cubic), EXP, Sin
$ r $/$ r^2 $ displayed for linear/exp/power/poly; Sin regression gives no meaningful $ r $/$ r^2 $ — judge by graphing.

11

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, $ $ 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.