Functions & Relationships (Grade 9)

Input values, output values and rules shown as flow diagrams, tables, formulae and equations — how to move between these equivalent forms and how to find a missing input, output or rule.

A relationship links an input to an output by a rule. Grade 9 pushes further than Grade 8 in one important way: you must be able to switch between four ways of writing the same relationship — a flow diagram, a table, a formula and an equation — and see that they say the same thing.

1The four equivalent forms

A flow diagram, a table and a formula all describe the same relationship: here y = 3x + 1.

In words: "multiply by 3, then subtract 2"
Flow diagram: input → ×3 → −2 → output
Table: 1→1, 2→4, 3→7, 4→10
Formula / equation: y = 3x − 2

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The exam often gives you one form and asks for another. Getting comfortable translating between them is the whole skill.

2Finding the output (substitution)

Worked ExampleIf y = 3x − 2, find y when x = 6

1
Replace x with a bracket: y = 3( ) − 2.
Brackets stop sign errors.
2
y = 3(6) − 2 = 18 − 2.
Substitute the input.
3
y = 16.

3Finding the input (working backwards)

Worked ExampleIf y = 3x − 2, find x when y = 25

1
Write the equation: 3x − 2 = 25.
Substitute the known output.
2
Add 2: 3x = 27.
Undo the −2 first.
3
Divide by 3: x = 9.
Check: 3(9) − 2 = 25 ✓

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Going forwards you substitute; going backwards you solve an equation. Recognising which direction the question wants saves a lot of confusion.

4Finding the rule from a table

Worked ExampleFind the rule: 1→5, 2→9, 3→13, 4→17

1
The outputs go up by 4 each time.
A constant difference of 4 means the rule contains 4x.
2
Test 4x: input 1 gives 4, but the output is 5 — one more.
Find the adjustment.
3
Rule: y = 4x + 1.
Check input 4: 4(4) + 1 = 17 ✓

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A constant difference only gives a rule of the form y = ax + b. If the differences are not constant, the rule is not linear and this method will not work.

5Rules with squares

Worked ExampleIf y = x² + 1, find y when x = 5

1
Square first: 5² = 25.
Powers come before adding (BODMAS).
2
y = 25 + 1 = 26.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
If y = 2x + 7, find y when x = 5.
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10 + 7 = 17
2
If y = 5x − 3, find y when x = 4.
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20 − 3 = 17
3
If y = 5x − 3, find x when y = 32.
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5x = 35 → x = 7
4
If y = 4x + 1, find x when y = 21.
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4x = 20 → x = 5
5
Rule: ×3 then +4. Find the output for input 6.
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18 + 4 = 22
6
Find the rule: 1→4, 2→7, 3→10.
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Differences of 3 → y = 3x + 1
7
Find the rule: 1→2, 2→4, 3→6.
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y = 2x
8
If y = x² − 2, find y when x = 6.
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36 − 2 = 34
9
Write "multiply by 5 then subtract 1" as a formula.
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y = 5x − 1