IB MYP Mathematics: Functions and models: linear, quadratic, exponential and rational (MYP Mathematics)

This guide teaches functions and models, one topic area inside the thinking with models part of the IB Middle Years Programme Mathematics guide (first assessment 2022). The framework columns used are the standard and extended levels for years 4 and 5, plus the optional enrichment list. The framework lists nothing for MYP 1-3 here. Linear programming and networks sit in other units and are not covered. Use it with the revision notes and the practice set. The course page is MYP Mathematics, the checklist tracks progress, and all free 10-minute diagnostics show where to start.

The guide treats a model as a depiction of a real event using expressions, equations or graphs. A function is a relation between variables. You should be able to show one idea as a table, a graph and a rule, and say what each tells you. The guide connects this branch to related concepts such as change, equivalence, patterns, quantity, representation and systems.

What this guide covers

Topic	Framework column
Mapping, function notation	Standard
Linear functions f(x) = mx + c, parallel and perpendicular lines	Standard
Simultaneous equations, algebraically and graphically	Standard
Quadratics in three forms and their parameters	Standard
Exponential functions and horizontal asymptotes	Standard
Domain and range; rational functions (ax + b)/(cx + d)	Extended
Transforming quadratics: translation, reflection, dilation	Extended
Shape of cubic, rational, trigonometric and logarithmic functions, with asymptotes	Extended
Transforming cubic and trigonometric functions; ln and e; composite, inverse and logarithmic functions	Enrichment (optional, and the guide says not in eAssessments)

Mapping and function notation

A mapping pairs each input with an output. It is a function when every input has exactly one output. Two inputs may share an output (many-to-one). One input may not give two outputs. On a graph, a vertical line crossing the curve twice shows the relation is not a function.

Function notation names the rule. In f(x) = 6 - 4x, the letter f is the name, x is the input and f(x) is the output. The guide says y = is also accepted where the question does not insist on f.

Worked example. For f(x) = 6 - 4x:

• f(3) = 6 - 4(3) = -6
• f(a + 1) = 6 - 4(a + 1) = 2 - 4a
• f(x) = -10 means 6 - 4x = -10, so -4x = -16 and x = 4

The relation x² + y² = 25 is not a function: x = 3 gives y = 4 or y = -4.

Linear functions

A linear function is f(x) = mx + c. The gradient m is the change in output for each 1 added to the input. The constant c is where the line meets the y-axis. Between points (x₁, y₁) and (x₂, y₂), m = (y₂ - y₁)/(x₂ - x₁).

Worked example. Find the line through (2, 7) and (6, 19).

1. m = (19 - 7)/(6 - 2) = 12/4 = 3
2. Substitute (2, 7): 7 = 3(2) + c, so c = 1
3. The line is y = 3x + 1

Parallel and perpendicular lines. Two lines with the same gradient are parallel. For perpendicular lines the gradients multiply to -1, so the second gradient is the negative reciprocal of the first.

• Parallel to y = 3x + 1 through (1, -4): m = 3, so -4 = 3 + c, c = -7, giving y = 3x - 7.
• Perpendicular to y = -2x + 5 through (4, 3): m = 1/2, so 3 = 2 + c, c = 1, giving y = x/2 + 1.

As a model. A taxi fare is a fixed fee c plus a rate m per kilometre, valid only for distances of zero or more.

Simultaneous equations

The solution of two equations is the pair of values that satisfies both. Graphically it is the intersection of the two lines. Parallel lines never meet, so there is no solution.

Substitution. Solve y = 2x + 1 and 3x + 2y = 16.

Replace y: 3x + 2(2x + 1) = 16, so 7x + 2 = 16 and x = 2. Then y = 2(2) + 1 = 5.

Elimination. Solve 2x + 3y = 12 and 5x - 3y = 9. Adding removes y: 7x = 21, so x = 3. Then 6 + 3y = 12 gives y = 2. Check in the other equation: 15 - 6 = 9. Correct.

On a graph, draw both lines on one set of axes and read where they cross. A graph gives an approximate answer, so use algebra when exact values are needed.

Quadratic functions and their parameters

A quadratic has an x² term and graphs as a parabola. The guide lists three forms:

Form	Rule	Shows directly
General	f(x) = ax² + bx + c	y-intercept c; axis of symmetry x = -b/(2a)
Factorised	f(x) = a(x - p)(x - q)	where it crosses the x-axis: p and q
Vertex	f(x) = a(x - h)² + k	vertex (h, k)

Parameters. If a > 0 the parabola opens upward and has a minimum. If a < 0 it opens downward and has a maximum. A larger |a| makes it narrower. In vertex form, h shifts the graph sideways and k shifts it up or down. The axis of symmetry lies halfway between the two x-intercepts.

Worked example 1. f(x) = x² - 6x + 5.

• Factorise: (x - 1)(x - 5), so the x-intercepts are 1 and 5.
• Axis: halfway, x = 3. Then f(3) = 9 - 18 + 5 = -4, so the vertex is (3, -4).
• Vertex form: f(x) = (x - 3)² - 4. Expanding returns x² - 6x + 5.
• y-intercept: 5.

Worked example 2. f(x) = -2(x + 1)(x - 3).

• Intercepts at x = -1 and x = 3, and a = -2 so it opens downward.
• Axis x = 1. f(1) = -2(2)(-2) = 8. The maximum is 8 at (1, 8).
• Expanding: -2(x² - 2x - 3) = -2x² + 4x + 6.

Completing the square. To write 3x² - 6x + 1 in vertex form, take out the 3 from the x terms: 3(x² - 2x) + 1. Complete the bracket: 3[(x - 1)² - 1] + 1 = 3(x - 1)² - 3 + 1 = 3(x - 1)² - 2. The vertex is (1, -2).

As a model. A ball is thrown so its height in metres after t seconds is h = -5t² + 20t + 1.

• Axis: t = -20/(2 × -5) = 2. Then h(2) = -20 + 40 + 1 = 21, so the greatest height is 21 m.
• Lands when h = 0: t = (20 + √420)/10 ≈ 4.05 s (the negative root is rejected).
• Limits: the model ignores air resistance and only applies from throwing until landing, so t ≥ 0 and h ≥ 0.

Exponential functions and horizontal asymptotes

An exponential function has the variable in the power: f(x) = a × bˣ with b > 0 and b ≠ 1. If b > 1 it shows growth. If 0 < b < 1 it shows decay. When x = 0, f(0) = a, so a is the starting value.

The graph of 3 × 2ˣ rises ever more steeply to the right and flattens towards the x-axis on the left without touching it. The line y = 0 is a horizontal asymptote: the graph approaches it without reaching it. Adding a constant moves the asymptote. For f(x) = 5 × (0.5)ˣ + 2 the graph falls towards y = 2 and never reaches it.

Worked example. A laptop is worth V = 900 × 0.85ᵗ dollars after t years.

• Start: V(0) = 900.
• After 3 years: 900 × 0.85³ = 552.71, so about $553.
• Value below $300: V(6) = 339.43 and V(7) = 288.52, so during the seventh year.
• Limit: the model says the value never reaches zero and gives a smooth curve, but real second-hand prices drop in steps and can fall suddenly if the model is discontinued.

Using models well. Name the assumptions, check the answer is sensible for the context, and be wary of extrapolation, which means using the model beyond the data that built it. Exponential growth cannot continue forever, because algae run out of space.

Extended: domain and range

The domain lists every input the rule accepts and the context allows. The range lists every output the rule produces.

• f(x) = x² - 6x + 5 has domain all real numbers. Its minimum is -4, so the range is f(x) ≥ -4.
• g(x) = 2ˣ has domain all reals and range g(x) > 0, because of the asymptote y = 0.
• A real context restricts the domain: for the ball above the domain is 0 ≤ t ≤ 4.05.

Extended: rational functions (ax + b)/(cx + d)

A rational function has a linear expression over a linear expression. The denominator cannot be zero, so there is a vertical asymptote where cx + d = 0.

Worked example. f(x) = (4x + 3)/(x - 1).

• Vertical asymptote: x - 1 = 0, so x = 1. The domain is x ≠ 1.
• Rewrite: (4x + 3)/(x - 1) = 4 + 7/(x - 1). For very large x the fraction shrinks towards zero, so the horizontal asymptote is y = 4. In general it is y = a/c. The range is f(x) ≠ 4.
• x-intercept: 4x + 3 = 0, so x = -3/4. y-intercept: f(0) = 3/(-1) = -3.
• The graph has two separate branches, one each side of x = 1.

Extended: transforming quadratics

Starting from y = x², these changes act on the graph:

Change to the rule	Effect
f(x) + k	translation k units up (down if k is negative)
f(x - h)	translation h units right (left if h is negative)
-f(x)	reflection in the x-axis
f(-x)	reflection in the y-axis
a f(x)	vertical dilation, scale factor a
f(bx)	horizontal dilation, scale factor 1/b

Worked example. Describe y = -2(x - 3)² + 1 as a transformation of y = x².

1. Translate 3 units right.
2. Dilate vertically by factor 2.
3. Reflect in the x-axis (the negative sign).
4. Translate 1 unit up.

The vertex is (3, 1) and the maximum value is 1, so the range is y ≤ 1. Vertex form is the transformation written as a rule. Reflecting y = x² in the y-axis changes nothing, because the parabola is symmetric about that axis.

Extended: shapes of other functions

Function	Shape and key features
Cubic, for example (x + 1)(x - 2)(x - 4)	up to three x-intercepts; rises on one end and falls on the other; y-intercept is the product at x = 0, here 8
Rational, 1/x	two branches; asymptotes x = 0 and y = 0
Trigonometric, sin x	repeating wave between -1 and 1; period 360°
Trigonometric, tan x	repeating curve; vertical asymptotes at 90°, 270° and every 180° after that
Logarithmic, log₂ x	passes (1, 0) and (2, 1); vertical asymptote x = 0; only positive inputs

An asymptote is read from the rule: look for inputs that make a denominator zero, a tangent undefined, or a logarithm’s argument zero. For a rational function that is the vertical asymptote. For exponentials it is the horizontal asymptote.

Enrichment (optional)

• Transforming cubic and trigonometric functions. The same rules apply. y = 2 sin x + 1 stretches the wave to amplitude 2 and lifts it by 1, so its values lie between -1 and 3.
• ln and e. The number e is about 2.718. The function ln x is the inverse of eˣ, so eˣ = 7 gives x = ln 7 ≈ 1.95.
• Composite functions. g∘f(x) means g(f(x)): apply f first. With f(x) = 2x + 3 and g(x) = x², g∘f(3) = g(9) = 81, but f∘g(3) = f(9) = 21. Order matters.
• Inverse functions. The inverse f⁻¹ reverses f. For f(x) = 2x + 3, write y = 2x + 3, swap to x = 2y + 3, and solve: f⁻¹(x) = (x - 3)/2. Mirror the graph of f in the line y = x to get the graph of f⁻¹.
• Logarithmic functions. The logarithm of a to base b is the power that b needs to reach a. So log₂ 32 = 5, and 2ˣ = 20 gives x = log₂ 20 ≈ 4.32.

Common errors

• Calling a relation a function when one input has two outputs.
• Writing f(a + 1) as f(a) + 1.
• Taking the reciprocal but forgetting the minus sign for a perpendicular gradient.
• Reading the vertex of (x - 3)² - 4 as (-3, -4). The x-coordinate is +3.
• Giving the horizontal asymptote of a rational function as the vertical one.
• Trusting an exponential model far outside the data it was built from.

Official syllabus

Source: Mathematics guide of the Middle Years Programme, International Baccalaureate Organization; first assessment 2022. Section used: the framework for thinking with models.