Grade 9 Patterns and Functions

Key concepts

• Constant difference patterns and their general rule Tₙ = bn + c
• Patterns in shapes, such as matchsticks or tiles
• Input–output relationships in flow diagrams and tables
• Writing a rule as an equation, e.g. y = 2x − 3
• Finding a missing input or output

Worked examples

Find the general term of 4; 7; 10; 13; …

The constant difference is 3, so Tₙ = 3n + c. When n = 1 the term is 4, so 3 + c = 4 and c = 1. Tₙ = 3n + 1.

For the rule y = 2x − 3, what input x gives the output y = 11?

Solve 2x − 3 = 11: add 3 to get 2x = 14, then divide by 2: x = 7.

Pattern 1 uses 4 matches (one square), pattern 2 uses 7 and pattern 3 uses 10. How many matches does pattern 20 use?

Each new square adds 3 matches, so Tₙ = 3n + 1. T₂₀ = 3(20) + 1 = 61 matches.

Complete the table for y = x² + 1 when x = −2, 0 and 3.

x = −2 gives y = 4 + 1 = 5; x = 0 gives y = 1; x = 3 gives y = 9 + 1 = 10.

Common mistakes

• Using the first term as the constant c (for 4; 7; 10, the rule is 3n + 1, not 3n + 4).
• Mixing up the position n and the value of the term.
• Forgetting that (−2)² = 4, not −4.
• Not checking the rule with the first two terms.

Exam tips

• Write the terms in a table with their positions above them.
• Test your rule with n = 1 and n = 2 before using it.
• For shape patterns, think about what is added each time.

Frequently asked questions

What is a general term?

A rule that gives any term of a pattern from its position n, such as Tₙ = 3n + 1. It saves you from writing out every term.

How are patterns linked to graphs?

If you plot the position n against the term Tₙ for a constant-difference pattern, the points lie on a straight line. That links patterns to linear graphs.

Do patterns continue in high school?

Yes. Grade 10 studies linear patterns, Grade 11 quadratic patterns, and Grade 12 sequences and series.