Quadratic patterns: constant second differences

Warm-up

Three quick questions from earlier lessons.

Discover

Squares of tiles: 1, 4, 9, 16 tiles. The differences are 3, 5, 7. What comes next in the pattern?

Key idea

Quadratic functions have constant second differences.

A quadratic function has the form

f(x) = ax^2 + bx + c

with

a \neq 0

. Over equal steps in

x

, its first differences change, but its second differences are constant (equal to

2a

for steps of 1). Linear: constant first differences; exponential: constant ratios.

Worked example

Values 2, 5, 10, 17, 26.

3, 5, 7, 9

First differences.

2, 2, 2: quadratic

Second differences.

a = 1

f(x) = x^2 + 1

for

x = 1, 2, …

2a = 2.

Your turn, guided

Your turn: values 0, 3, 12, 27, 48. The second difference.

6

Differences 3, 9, 15, 21.

Practice ★

A tile pattern has 6, 9, 14 and 21 tiles in steps 1 to 4 (a square of tiles plus 5 extra). How many tiles are in step 10?
A tile pattern has 10, 13, 18 and 25 tiles in steps 1 to 4 (a square of tiles plus 9 extra). How many tiles are in step 10?

For whole numbers $x$ from 1 to 14, which quantity is a quadratic function of $x$ ?
1. The area of a square with side $x$ cm
2. The perimeter of a square with side $x$ cm
3. The number of cells after $x$ doublings

Watch out!

Practice ★★

For $x = 0, 1, 2, 3, 4$ a function takes the values 5, 8, 13, 20, 29. What is the (constant) second difference?
For $x = 0, 1, 2, 3, 4$ a function takes the values 0, 9, 28, 57, 96. What is the (constant) second difference?

Exit ticket

Three questions to check you have got it.

A tile pattern has 6, 9, 14 and 21 tiles in steps 1 to 4 (a square of tiles plus 5 extra). How many tiles are in step 10?
A tile pattern has 10, 13, 18 and 25 tiles in steps 1 to 4 (a square of tiles plus 9 extra). How many tiles are in step 10?
A tile pattern has 5, 8, 13 and 20 tiles in steps 1 to 4 (a square of tiles plus 4 extra). How many tiles are in step 10?

Summary card

Quadratic patterns

Constant second differences mean quadratic.

Watch out: growing differences are not linear.

End of the lesson

Next lesson: Adding, subtracting and multiplying polynomials Back to the chapter

About this lesson