Everything covered in this lesson, in one place - useful for revision or printing.
\( \text{input} \rightarrow \text{rule} \rightarrow \text{output} \)
A relationship connects two sets of numbers: every input is turned into an output by a rule.
In this lesson you will learn to:
\( 3 \rightarrow \boxed{\times 2} \rightarrow 6 \)
Think of a rule as a machine. You feed in a number (the input), the machine applies its rule, and a new number comes out (the output).
\( x \rightarrow \text{rule} \rightarrow y \)
We usually call the input \(x\) and the output \(y\).
\( y = x + 5 \)
If the rule is “add 5”, then for input \(x=3\):
\( \text{input} \xrightarrow{\ \text{rule}\ } \text{output} \)
Key idea: a relationship links every input to exactly one output through a rule. Next we will draw this idea as a flow diagram.
\( x \rightarrow \boxed{\times 3} \rightarrow \boxed{+1} \rightarrow y \)
A flow diagram shows the rule as a chain of operations, applied left to right.
\( x \rightarrow \boxed{\times 6} \rightarrow \boxed{-4} \rightarrow y \)
Use the diagram for \(x = 1, 2, 3\):
\( x \rightarrow \boxed{\times 6} \rightarrow \boxed{-4} \rightarrow y \quad\Longleftrightarrow\quad y = 6x - 4 \)
Every flow diagram can be written as a formula. Read the operations in order:
\( y = 6x - 4 \)
Key idea: a flow diagram and a formula say the same thing. The diagram shows the steps; the formula compresses them into algebra.
\(x\) 1 2 3 4 \(y\) 5 8 11 14
A table lists inputs in the top row and outputs below. This table shows the rule \( y = 3x + 2 \):
\( y = 2x + 4 \)
Complete the outputs for \( x = 1, 2, 3, 10, 50 \):
\( x:\ 1,\ 2,\ 3,\ 4 \qquad y:\ 6,\ 11,\ 16,\ 21 \)
The outputs increase by 5 each time, so the rule starts with \(5x\). Test \(x=1\): \(5(1)=5\), but the output is 6 — so add 1.
\( \text{difference } d \Rightarrow y = dx + c \)
Key idea: when outputs change by a constant difference \(d\), the rule is \( y = dx + c \). Find \(c\) by testing the first input.
\( y = 3x - 2 \)
To use a formula, substitute the input and calculate. For \( x = -4 \):
\( y = x^2 + 1 \)
Rules can include powers. For \( x = 3 \):
\( C = 6k + 18 \)
A taxi charges R18 plus R6 per kilometre. The formula gives the cost \(C\) for \(k\) kilometres.
\( \text{substitute} \rightarrow \text{calculate} \rightarrow \text{check} \)
Key idea: substitution turns a formula into a specific answer. Use brackets, follow the order of operations, and sense-check the result.
\( y = 2x + 5, \quad y = 25 \)
Sometimes you know the output and must find the input. Set up an equation and solve:
\( x \rightarrow \boxed{\times 3} \rightarrow \boxed{+4} \rightarrow 100 \)
On a flow diagram, work backwards using inverse operations, right to left:
\( 3(32) + 4 = 100 \)
Always check by running your answer forwards through the rule:
\( \text{forwards: substitute} \qquad \text{backwards: solve} \)
Key idea: finding an output is substitution; finding an input is solving an equation (or reversing the flow diagram with inverse operations).
\( \text{words} \leftrightarrow \text{flow diagram} \leftrightarrow \text{table} \leftrightarrow \text{formula} \)
The same relationship can be shown four ways:
\( y = 4n - 5 \)
From the formula \( y = 4n - 5 \):
\( \text{Which form is best?} \)
Each form has a job:
\( y = 2x + 4 \)
Key idea: being able to move between words, flow diagrams, tables and formulae is the heart of Functions & Relationships.
\[ \text{input} \xrightarrow{\ \text{rule}\ } \text{output} \qquad y = dx + c \]
You now know how to: