Functions & Relationships
We investigate input-output relationships using flow diagrams, tables of values, and formulae. We determine, interpret, and justify equivalence between different representations of the same relationship.
3.1 Input-Output, Flow Diagrams & Tables
- Determine output values from given input values using rules
- Represent relationships using flow diagrams and tables of values
- Determine the rule (formula) from a table of values
Real-World Connection
A function is like a vending machine: you put in money (input), the machine applies a rule (multiply cost, dispense item), and you get something out (output). Every time you put in the same input, you get the same output — that's the definition of a function. Spreadsheet formulas, GPS routing, and app algorithms all work this way.
Definition
Function / Relationship
A relationship between inputs and outputs where each input gives exactly one output. The rule describes how to calculate the output from the input.
To find the rule from a table: (1) Look at the differences between consecutive outputs. If constant → linear rule (y = ax + b). (2) Find 'a' from the constant difference. (3) Use one input-output pair to find 'b'.
💡 Tip
For a linear rule y = ax + b: the value of 'a' equals the constant difference in y when x increases by 1. The value of 'b' is the y-value when x = 0 (the y-intercept).
Worked Example
Finding the rule from a table
Problem
Worked Example
Flow diagrams with inverse operations
Problem
Worked Example
Identifying a non-linear rule
Problem
CAPS Cognitive Level Distribution
3.2 Formulae and Equivalent Representations
- Describe and justify relationships using formulae, equations, tables, and words
- Determine equivalence between different representations
- Use relationships to solve problems in context
Real-World Connection
The same relationship can be shown as a table, a graph, a formula, or in words. A weather forecast might give you a table of temperatures, a line graph, and a formula — all representing the same relationship. Scientists, engineers, and economists choose the representation that makes the pattern most obvious for their audience.
Four Representations of a Relationship
- Words: 'The cost is R50 per hour plus a R100 fixed fee.'
- Formula: C = 50h + 100
- Table of values: h = 0,1,2,3 → C = 100,150,200,250
- Graph: straight line with gradient 50 and y-intercept 100
ℹ️ Note
All four representations are equivalent — they describe the same relationship. Each has advantages: formulas allow quick calculation, tables show specific values, graphs show trends and patterns, words provide context.
Worked Example
Converting between representations
Problem
Worked Example
Showing equivalence between different representations
Problem
Worked Example
Determining the rule from words and checking equivalence
Problem
CAPS Cognitive Level Distribution
3.3 Numeric and Geometric Patterns
- Investigate and extend numeric and geometric patterns looking for relationships between terms
- Describe and justify the general rule for patterns with a constant difference (arithmetic) and constant ratio (geometric)
- Use the general term Tₙ to find any term in a sequence without listing all terms
- Describe patterns in words, diagrams, and general term formulas
Real-World Connection
Patterns are the language of nature and finance. The arrangement of petals on a sunflower follows a Fibonacci pattern. Bacteria double every 20 minutes — a geometric pattern (×2 each step). A savings account grows by a fixed amount monthly — arithmetic. A loan with compound interest grows geometrically. Recognising which type of pattern you have determines which formula to use.
Definition
Sequence / Number Pattern
An ordered list of numbers where each number is called a term. Terms are labelled T₁ (first term), T₂ (second term), Tₙ (nth term). The position number n tells you where a term sits in the sequence.
Definition
Arithmetic Sequence (Constant Difference)
A sequence where you ADD the same number each time to get the next term. The fixed number added is called the common difference (d). These patterns produce linear (straight-line) graphs.
General Term of an Arithmetic Sequence
Tₙ = the nth term; a = T₁ = first term; d = common difference; n = position number. This formula lets you find ANY term without listing all the terms before it.
Why does Tₙ = a + (n-1)d work? You start with the first term (a) and add d exactly (n−1) times to reach the nth position. For the 1st term: a + 0d = a. For the 2nd: a + d. For the 3rd: a + 2d. Pattern: always (n−1) lots of d.
Definition
Geometric Sequence (Constant Ratio)
A sequence where you MULTIPLY by the same number each time. The fixed multiplier is called the common ratio (r). These patterns produce exponential (curved) graphs.
💡 Tip
Quick test: Calculate the differences between consecutive terms. If differences are CONSTANT → arithmetic (linear). If differences are NOT constant but RATIOS are constant → geometric (exponential). If neither → look for another pattern (quadratic, etc.).
Arithmetic vs Geometric Patterns
Arithmetic
Geometric
Operation
Add constant d each step
Multiply by constant r each step
Example
3 ; 7 ; 11 ; 15 ; … (d=+4)
3 ; 6 ; 12 ; 24 ; … (r=×2)
General term
Tₙ = a + (n−1)d
Tₙ = a × r^(n−1)
Graph type
Straight line (linear)
Curved (exponential)
🚨 Common Mistake
When using Tₙ = a + (n−1)d, the position n must be a positive integer (1, 2, 3, …). Never substitute n=0 unless specifically asked about the 'zeroth term'. Also note: d can be negative (decreasing sequence) or fractional.
Worked Example
Finding the general term of an arithmetic sequence
Problem
Worked Example
Identifying and using a geometric sequence
Problem
Worked Example
Pattern from a geometric figure
Problem
CAPS Cognitive Level Distribution