MCF3M Functions & Applications · Unit 1 — Introduction to the Quadratic Function

Lesson 1.2

Comparing Rates of Change in Linear and Quadratic Functions

Some patterns grow steadily; others speed up. Use first and second differences to tell linear, quadratic and neither apart — and explain what that means in a real situation.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

calculate first and second differences
classify a relationship as linear, quadratic or neither
connect differences to degree
check that x-values are equally spaced before using differences
justify a classification and make predictions in context

Step 1 of 4

Warm Up

Notice how two patterns grow.

Two patterns

Look at these two patterns before using any formal words.

Pattern A: 5, 8, 11, 14 … Pattern B: 2, 3, 6, 11 …

Warm Up

Pattern A: 5, 8, 11, 14, … Pattern B: 2, 3, 6, 11, …

What do you notice about how each pattern grows?

aBoth grow by the same amount every step.bA grows by the same amount each step; B grows by more each step.cB grows by the same amount each step; A grows by more each step.dNeither pattern has any structure.

Step 2 of 4

Learn

First differences, second differences, and a data investigation.

Part 1 — Rate of change

Rate of change answers one question: how much does the output change when the input changes?

You've met this as slope: a line rises the same amount for each step to the right.

Part 2 — Linear relationships

A rain barrel is filling. Hours go up by 1 each row. How much does the water change each hour?

Rain barrel
hours	water (L)
0	20
1	35
2	50
3	65
4	80

First differences

35 − 20 = 15
50 − 35 = 15
65 − 50 = 15
80 − 65 = 15the same every time

Key term

First differences

The differences between consecutive y-values when the x-values go up by equal steps.

Constant first differences (15, 15, 15) → linear relationship (degree 1).

Try it

x: 0, 1, 2, 3 y: 7, 11, 15, 19

What is the (constant) first difference?

Your answer

Part 3 — Quadratic relationships

Now try a different table.

x	y
0	1
1	3
2	9
3	19
4	33

First differences: 2, 6, 10, 14 — not constant. Does that mean no pattern?

6 − 2 = 4difference of the differences
10 − 6 = 4
14 − 10 = 4constant!

Key term

Second differences

The differences between consecutive first differences.

Constant second differences → quadratic relationship (degree 2).

Constant first differences → linear (degree 1). Constant second differences → quadratic (degree 2).

Try it

x: 0, 1, 2, 3, 4 y: 1, 3, 9, 19, 33

What is the second difference?

Your answer

Part 4 — Neither

Some tables are neither. For y: 1, 2, 4, 8, 16 the first differences are 1, 2, 4, 8 and the second differences are 1, 2, 4. Neither is constant.

Check the x-values first

Differences only work when x goes up by equal steps. For x: 0, 1, 3, 4 and y: 1, 3, 7, 9 the differences 2, 4, 2 look messy — but y = 2x + 1 is linear. The jump from 1 to 3 skipped a value.

Try it

x: 0, 1, 2, 3, 4 y: 3, 4, 7, 15, 31

Classify the relationship.

aLinearbQuadraticcNeither

Real-world idea

Part 5 — Investigation: Difference Detective

A small company makes square rubber floor mats for photo booths. They charge $12 per square metre of mat plus an $8 cutting fee.

Their price list: side 1 m → $20, 2 m → $56, 3 m → $116, 4 m → $200, 5 m → $308.

Solve the case: what kind of relationship is this, and what will a 6 m mat cost?

Floor-mat price list
side (m)	cost ($)
1	20
2	56
3	116
4	200
5	308

Clue 1

Mat cost table — side s (m): 1, 2, 3, 4, 5 cost ($): 20, 56, 116, 200, 308

Difference Detective, clue 1: what is the FIRST first difference (from s = 1 to s = 2)?

Your answer ($)

Clue 2

First differences: 36, 60, 84, 108

Clue 2: what is the second difference?

Your answer ($)

Clue 3

Clue 3: classify the mat-cost relationship and justify it.

aLinear — the costs always increase.bQuadratic — side lengths increase by 1 and the second differences are constant (24).cQuadratic — because the first differences are constant.dNeither — the first differences change.

Clue 4

Clue 4: predict the cost of a 6 m mat by extending the differences.

Your answer ($)

Clue 5

Clue 5: what do the growing first differences mean for the business?

aEach extra metre of side length adds the same cost.bEach extra metre adds more cost than the one before, because the area grows faster than the side.cBigger mats are cheaper.dThe cost will eventually stop increasing.

Comparing Rates of Change in Linear and Quadratic Functions

Warm Up

Two patterns

Learn

Part 1 — Rate of change

Part 2 — Linear relationships

First differences

Part 3 — Quadratic relationships

Second differences

Part 4 — Neither

Part 5 — Investigation: Difference Detective

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.2 mastery check