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

These notes condense functions and models from the thinking with models framework of the IB Middle Years Programme Mathematics guide (first assessment 2022). The study guide has the explanations and fully worked examples. Test yourself with the practice set. Course page: MYP Mathematics. Progress tracker: checklist. To find weak spots first, try the all free 10-minute diagnostics.

Labels follow the framework: standard (MYP 4-5), extended (MYP 4-5) and enrichment (optional, not in eAssessments according to the guide).

Key definitions

Term	Meaning
Mapping	A pairing of inputs with outputs
Function	A mapping where each input has exactly one output
f(x)	The output when the rule f is applied to x
Model	A real situation described by expressions, equations or graphs
Asymptote	A line a graph approaches but does not reach
Domain, range (extended)	Allowed inputs; outputs that occur
Extrapolation	Using a model beyond the data that built it

Linear functions (standard)

Item	Result
Rule	f(x) = mx + c
Gradient	m = (y₂ - y₁)/(x₂ - x₁)
y-intercept	c
Lines that are parallel	same m
Perpendicular lines	m₁ × m₂ = -1

Method: find a line through two points.

1. Find the gradient using the coordinates of both points.
2. Substitute one point and rearrange to get c.
3. Test the second point in your equation.

Method: line perpendicular to a given line through a point.

1. Read m from the given line.
2. Flip it and change the sign: new gradient = -1/m.
3. Substitute the point to find c.

Reminder. Perpendicular to y = 4x - 1 through (8, 2): m = -1/4, so 2 = -2 + c, c = 4 and y = -x/4 + 4.

Simultaneous equations (standard)

The solution is the pair (x, y) that fits both equations, which is where the graphs cross. Parallel distinct lines give no solution.

Method	Use when
Substitution	one equation already says y = … or x = …
Elimination	x or y terms have equal or opposite coefficients, or can be made so
Graph	an estimate is enough, or you need to see the situation

Reminder. y = x + 4 and 2x + y = 13. Substitute: 2x + x + 4 = 13, so x = 3 and y = 7.

Quadratics (standard)

Form	Rule	Gives you
General	ax² + bx + c	y-intercept c; axis x = -b/(2a)
Factorised	a(x - p)(x - q)	crossing points p, q
Vertex	a(x - h)² + k	vertex (h, k)

• a > 0: opens up, minimum at the vertex. a < 0: opens down, maximum.
• Larger |a| gives a narrower parabola.
• The axis sits midway between x-intercepts.

Method: general form to vertex.

1. Find the axis x = -b/(2a) (or factorise and go halfway between roots).
2. Substitute that x to get k.
3. Write a(x - h)² + k and expand to check.

Reminder. f(x) = x² + 2x - 8 = (x + 4)(x - 2). Axis x = -1. f(-1) = -9. Vertex form (x + 1)² - 9.

Method: sketch a quadratic.

1. Decide whether it opens up or down from the sign of a.
2. Mark the y-intercept and any x-intercepts.
3. Mark the vertex on the axis halfway between the intercepts.
4. Join with a smooth curve and label the points.

Completing the square. x² + 6x + 2 = (x + 3)² - 9 + 2 = (x + 3)² - 7. For a leading coefficient other than 1, factor it out of the x terms first.

Exponential functions (standard)

Item	Result
Rule	f(x) = a × bˣ, b > 0, b ≠ 1
Start value	f(0) = a
Growth	b > 1
Decay	0 < b < 1
Horizontal asymptote	y = 0 (or y = k for a × bˣ + k)

Reminder. f(x) = 4 × 3ˣ - 5: start value without shift is 4, but f(0) = -1; asymptote y = -5; range f(x) > -5.

Method: model a situation with an exponential.

1. Identify the start value a.
2. Turn the percentage change into a factor b (add the rise to 1, or subtract the fall from 1).
3. Write y = a × bᵗ and test it on a known value.
4. State the limits of the model.

Model checks. Substitute known values to test the model. Ask whether the context allows negative or very large inputs. A population model that grows without limit fails once resources run out.

Domain and range (extended)

• Domain: take away inputs that break the rule (zero denominator, argument of a logarithm not positive) and any the context rules out.
• Range: use the vertex of a parabola or the asymptote of an exponential.

Function	Domain	Range
x² - 4x + 1 = (x - 2)² - 3	all reals	f(x) ≥ -3
2ˣ	all reals	f(x) > 0
1/x	x ≠ 0	f(x) ≠ 0

Rational functions (extended)

For f(x) = (ax + b)/(cx + d):

Feature	How to find it
Vertical asymptote	cx + d = 0, so x = -d/c
Horizontal asymptote	y = a/c
x-intercept	ax + b = 0, so x = -b/a
y-intercept	f(0) = b/d
Domain, range	x ≠ -d/c; f(x) ≠ a/c

Reminder. f(x) = (5x - 10)/(x + 2): vertical asymptote x = -2, horizontal y = 5, x-intercept 2, y-intercept -5.

Transforming quadratics (extended)

Rule	Effect on the graph of f
f(x) + k	up k
f(x - h)	right h
-f(x)	reflect in the x-axis
f(-x)	reflect in the y-axis
a f(x)	vertical dilation, factor a
f(bx)	horizontal dilation, factor 1/b

Reminder. y = 3(x + 2)² - 7 is y = x² moved 2 left, dilated vertically by 3, then moved 7 down. Vertex (-2, -7).

Shapes and asymptotes (extended)

Function	Features to quote
Cubic	up to three x-intercepts; ends go in opposite directions
Rational 1/x	asymptotes x = 0 and y = 0
sin x	wave between -1 and 1, period 360°
tan x	vertical asymptotes at 90° + multiples of 180°
log₂ x	passes (1, 0); vertical asymptote x = 0

Enrichment summary (optional)

Idea	Key fact
Transforming cubics and trig	same rules; 3 sin x - 2 has values from -5 to 1
e and ln	e ≈ 2.718; ln x undoes eˣ
Composite g∘f	do f first, then g; order matters
Inverse f⁻¹	swap x and y, solve for y; mirror image in y = x
Logarithm	logarithm base b of a asks which power of b gives a

Must-know distinctions

• Function and relation. A circle is a relation; a line, parabola or exponential curve is a function.
• Zero, intercept, root. For a function these all point to the same place on the graph: an x-intercept.
• Vertex and y-intercept. The y-intercept is f(0). The vertex is on the axis of symmetry, not at x = 0 unless the form is symmetric.
• Growth factor and growth rate. A 12% rise each year has factor 1.12. A 12% fall has factor 0.88.
• Vertical and horizontal asymptotes. Vertical ones come from zero denominators. Horizontal ones describe behaviour for very large or very negative inputs.

Quick self-test

1. If f(x) = 7 - 2x, find f(-3) and solve f(x) = 1.
2. Find the equation of the line through (0, 6) and (3, -3).
3. Give the gradient of a line perpendicular to y = 0.25x + 6.
4. Solve x + y = 10 and x - y = 4.
5. Write x² - 8x + 12 in factorised form and vertex form.
6. State the vertex and whether it is a maximum or minimum for -(x - 1)² + 6.
7. The value of a machine is 5000 × 0.8ᵗ dollars. Find the value after 2 years and the horizontal asymptote.
8. (Extended) State the domain and range of f(x) = 2ˣ + 3.
9. (Extended) Find both asymptotes of f(x) = (4x - 1)/(2x + 6).
10. (Enrichment) With f(x) = x + 5 and g(x) = 2x, find g∘f(1) and f⁻¹(x).

Answers

1. f(-3) = 7 + 6 = 13. For 7 - 2x = 1, x = 3.
2. m = (-3 - 6)/(3 - 0) = -3; c = 6; y = -3x + 6.
3. Negative reciprocal of 0.25 is -4.
4. Adding: 2x = 14, x = 7, y = 3.
5. (x - 2)(x - 6). Axis x = 4, f(4) = -4, so (x - 4)² - 4.
6. Vertex (1, 6). a is negative, so it is a maximum.
7. 5000 × 0.64 = 3200 dollars. Asymptote y = 0.
8. Domain all reals. Range f(x) > 3.
9. Vertical x = -3; horizontal y = 4/2 = 2.
10. g∘f(1) = g(6) = 12. f⁻¹(x) = x - 5.

Where marks are usually lost

• Writing a point as (x, f(x)) the wrong way round, or reading the vertex of (x - 3)² - 4 as having x-coordinate -3.
• Forgetting that a line’s perpendicular gradient needs the minus sign as well as the flip.
• Solving only for x in simultaneous equations and leaving out y.
• Giving the y-intercept of a quadratic as the vertex’s y-value.
• Naming the asymptote of a shifted exponential as y = 0 when a constant has been added.
• Stating the horizontal asymptote as x = a/c; it is y = a/c.
• Giving a range that includes values the function never reaches (for example including -3 for 2ˣ - 3).
• Using a model outside its context without comment, such as negative time or a population that never stops growing.
• Using f(a + 1) = f(a) + 1 as if the rule were always additive.

For criteria in practice, see criteria in practice: explain your reasoning and state what a result means in context.

Official syllabus

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