Lesson 4 ~25 min Unit 2 · Linear +85 XP

Tables of Values

Tables are the bridge between rules and graphs. Learn to build them, read them, and use first differences to instantly test if a relationship is linear.

Table detective: Look at $x: 1, 2, 3, 4$ and $y: 3, 5, 7, 9$. As $x$ increases by 1, what happens to $y$? Is this linear? One simple test gives the answer instantly — no graph needed!

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Think First

warm-up

Before you read on — quickly: Look at these pairs of numbers carefully.

$x$	1	2	3	4
$y$	3	5	7	9

As $x$ increases by 1, what happens to $y$? Is this a linear relationship? What is the rule?

Record your answer in your workbook.

The Big Idea

+5 XP

A table of values shows how input ($x$) connects to output ($y$) through a rule. The secret to detecting linear relationships is first differences — the change in $y$ when $x$ increases by 1.

The linear test:

Calculate the change in $y$ when $x$ increases by 1
If all differences are constant, the relationship is linear (straight line)
If differences change, it’s non-linear (curve)

Think First answer: $y$ increases by 2 each time. First differences are all 2 — linear! Rule: $y = 2x + 1$.

$\Delta y$ constant ⇒ linear ⇒ straight line

Key insight

A table is a “preview” of the graph. If $y$ values change by a constant amount, you know the points form a straight line before drawing anything!

Real world

Taxi fares (distance vs cost) have constant first differences when the per-km rate is fixed. That makes them linear.

Shortcut

In $y = mx + c$, the first difference always equals $m$. When $x = 0$, $y = c$. Read the table to find the rule!

What You’ll Master

objectives

Know

A table shows $x$ and $y$ pairs systematically
Linear relationships have constant first differences
Non-linear relationships have changing first differences

Understand

How to use a table to predict whether points form a straight line
The connection between the rule, the table, and the graph
Why constant rate of change means linearity

Can Do

Create tables of values from rules
Calculate first differences between consecutive $y$ values
Determine if a relationship is linear from its table
Plot points from a table to visually confirm linearity

Words You Need

vocabulary

Table of valuesA table organising $(x, y)$ pairs with input in one column and output in another.

First differenceThe difference between consecutive $y$ values: $\Delta y = y_{n+1} - y_n$.

Linear relationshipA relationship with constant first differences. Points form a straight line when plotted.

Non-linear relationshipA relationship where first differences are NOT constant. Points form a curve.

Constant rateWhen $y$ changes by the same amount every time $x$ increases by 1. Equals the slope $m$.

The Linear TestCheck: are ALL first differences equal? Yes → linear. No → non-linear.

Spot the Trap

heads-up

Wrong: Calculating $y - x$ (e.g., $5 - 2 = 3$) instead of consecutive $y$ differences. First differences are ALWAYS $y_{n+1} - y_n$, not $y - x$.

Right: Calculate $\Delta y$ by subtracting each $y$ value from the one below it in the table. Only compare $y$ values to $y$ values.

Wrong: Checking only 2 differences and declaring it linear. Consider $y: 2, 4, 6, 9$ — the first two differences are 2, but the last is 3. It is not linear.

Right: Always check ALL first differences in the table. Use at least 4–5 rows for convincing evidence of linearity.

Creating Tables and Finding First Differences

+5 XP

Given a rule, create a table by substituting each $x$ value. Then check first differences to test linearity.

Worked example — Create a table for $y = 2x + 1$ with $x = 0$ to $5$

Set up the table with $x$ values from 0 to 5 and substitute into $y = 2x + 1$.
Calculate each $y$ value: $x=0 \to 1$; $x=1 \to 3$; $x=2 \to 5$; $x=3 \to 7$; $x=4 \to 9$; $x=5 \to 11$.
Check first differences: $3-1=2$; $5-3=2$; $7-5=2$; $9-7=2$; $11-9=2$. All equal 2 — the relationship is linear!

Step 1 of 3

$x$	$y = 2x+1$	First Difference
0	1	—
1	3	2
2	5	2
3	7	2
4	9	2
5	11	2

All first differences = 2. This is LINEAR. Rule: $y = 2x + 1$.

Worked example — Test if $x: 0,1,2,3,4$ and $y: 4,7,10,13,16$ is linear

Verify $x$ increases by 1 each step: $1-0=1$, $2-1=1$, $3-2=1$, $4-3=1$. ✓
Calculate first differences: $7-4=3$; $10-7=3$; $13-10=3$; $16-13=3$.
All differences = 3 (constant). The relationship is linear. Rule: $y = 3x + 4$ (since $m = 3$ and $c = 4$ when $x = 0$).

Step 1 of 3

The Linear Test and Non-Linear Relationships

+5 XP

Worked example — Linear Test: compare Table A and Table B

Table A: $x: 1,2,3,4$ and $y: 5,8,11,14$. Differences: $8-5=3$, $11-8=3$, $14-11=3$. All equal 3 → LINEAR. Rule: $y = 3x + 2$.
Table B: $x: 1,2,3,4$ and $y: 2,4,8,16$. Differences: $4-2=2$, $8-4=4$, $16-8=8$. NOT equal → NON-LINEAR. Rule: $y = 2^x$.
Why? A linear rule $y = mx + c$ gives $\Delta y = m$ always. For non-linear rules (like $y = x^2$ or $y = 2^x$), the change in $y$ depends on where you are in the table.

Step 1 of 3

Non-linear examples:

$x$	$y = x^2$	$\Delta y$
0	0	—
1	1	1
2	4	3
3	9	5
4	16	7

Differences: 1, 3, 5, 7 — NOT constant → NON-LINEAR

Common Pitfalls

Mistakes to Avoid

heads-up

Calculating $y - x$ instead of $\Delta y$: First differences are ALWAYS between consecutive $y$ values. Calculate $y_{n+1} - y_n$, never $y - x$. For $x=2, y=5$: the first difference is $5 - \text{previous y}$, not $5 - 2 = 3$.
Checking only 2 differences: One mismatch breaks the pattern. Always check every first difference. For $y: 2, 4, 6, 9$: the first two differences are 2, but $9 - 6 = 3$ — not linear!
Using too few rows: Use at least 4–5 rows in your table. This gives 3–4 first differences to compare, which is convincing evidence.

Copy Into Books

Summary Notes — Tables of Values

Creating a Table from a Rule

Write $x$ values in order (increasing by 1)
Substitute each $x$ into the rule to find $y$
Fill in the table systematically

The Linear Test

Check $x$ increases by 1 each step
Calculate ALL first differences ($y_{n+1} - y_n$)
All equal? → LINEAR
Not equal? → NON-LINEAR

Reading the Rule from a Table

First difference = $m$ (slope)
$y$ when $x = 0$ = $c$ (constant)
Rule: $y = mx + c$
Check: verify with another row

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Tables of Values

4 problems

Four drill problems to sharpen your table skills. Work each, then reveal the answer.

1 First differences: $y = 5, 9, 13, 17$. What is $\Delta y$?

$9-5=4$, $13-9=4$, $17-13=4$. All equal.$\Delta y = 4$ (linear)
2 Is this linear? $y: 2, 5, 9, 14$ (find all differences first).

Differences: $5-2=3$; $9-5=4$; $14-9=5$. Not equal.No — non-linear (diffs: 3, 4, 5)
3 Table: $x=1,2,3$ and $y=6,10,14$. Find the rule.

Diff $= 4$, so $m = 4$. For $n=1$: $6 = 4(1) + c \Rightarrow c = 2$.$y = 4x + 2$
4 Linear table: constant diff $= 3$, at $x=1$, $y=2$. What is $y$ when $x=5$?

$x=1\to2$; $x=2\to5$; $x=3\to8$; $x=4\to11$; $x=5\to14$. Adding 3 each time.14

Complete in your workbook.

Quick Check · 5 questions

Is this relationship linear? $x: 1,2,3,4$ and $y: 2,5,8,11$

+10 XP

What is the first difference in this table? $x: 0,1,2,3,4$ and $y: 5,9,13,17,21$

+10 XP

Which rule will produce a linear table of values?

+10 XP

Complete the table for $y = 3x - 2$ with $x = 1, 2, 3, 4$.

+10 XP

A table has constant first differences of 4. If $y = 7$ when $x = 1$, what is $y$ when $x = 4$?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Create a table of values for $y = 2x + 3$ with $x$ from 0 to 5. Calculate the first differences and determine whether the relationship is linear.

Answer in your workbook.

Reason Medium 3 MARKS

Q7. A table has $x: 1,2,3,4,5$ and $y: 1,4,9,16,25$. Explain why this is NOT linear using first differences.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. A linear relationship passes through $(1, 3)$ and $(2, 5)$. Complete the table for $x = 3, 4, 5$ and state the rule.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Linear. Differences $5-2=3$, $8-5=3$, $11-8=3$ are all equal.

2. C — 4. Every consecutive $y$ difference is 4.

3. A — $y = x + 3$. Only rule in the form $y = mx + c$.

4. C — $1, 4, 7, 10$. Substitute $x = 1,2,3,4$ into $3x - 2$.

5. B — 19. Start at $y=7$, add 4 three times: $7 \to 11 \to 15 \to 19$.

Show Your Working Model Answers

Q6 (3 marks): Table: $x=0\to3$; $x=1\to5$; $x=2\to7$; $x=3\to9$; $x=4\to11$; $x=5\to13$ [1]. First differences: $5-3=2$, $7-5=2$, $9-7=2$, $11-9=2$, $13-11=2$ [1]. All differences equal 2 — the relationship is linear [1].

Q7 (3 marks): Calculate first differences: $4-1=3$; $9-4=5$; $16-9=7$; $25-16=9$ [1]. The differences are $3, 5, 7, 9$ — NOT constant (they increase by 2 each time) [1]. For a linear relationship, ALL first differences must be equal. Since $3 \neq 5 \neq 7 \neq 9$, this is non-linear (the rule is $y = x^2$) [1].

Q8 (3 marks): First difference $= 5 - 3 = 2$ [1]. Continue adding 2: $x=3\to7$; $x=4\to9$; $x=5\to11$ [1]. Rule: slope $m = 2$. Using $(1,3)$: $3 = 2(1) + c \Rightarrow c = 1$. Rule: $y = 2x + 1$ [1].

Stretch Challenge · +25 XP, +10 coins

Finding Rules from Tables

Challenge 1: This table is linear. Find the rule $y = mx + c$.

$x$	0	1	2	3
$y$	5	8	11	14

Challenge 2: Complete the table and find the rule for this linear relationship.

$x$	1	2	3	4	5
$y$	6	?	?	?	22

Challenge 3: A table starts with $x = 0$, $y = -1$ and has constant first differences of $-3$ (decreasing by 3).
(a) Write the first 5 rows. (b) Find the rule. (c) What is $y$ when $x = 10$?

Reveal solutions

C1: First differences $= 3$, so $m = 3$. When $x=0$, $y=5$, so $c=5$. Rule: $y = 3x + 5$.

C2: Total change $= 22 - 6 = 16$ over 4 steps. First difference $= 16 \div 4 = 4$. Values: $6, 10, 14, 18, 22$. Find $c$: $6 = 4(1) + c \Rightarrow c = 2$. Rule: $y = 4x + 2$.

C3: (a) $y$ values: $-1, -4, -7, -10, -13$. (b) $m = -3$, $c = -1$. Rule: $y = -3x - 1$. (c) $y = -3(10) - 1 = -31$.

Quick Review

Table of values

Organised $(x, y)$ pairs from a rule, with $x$ increasing by 1

First difference

$\Delta y = y_{n+1} - y_n$ (consecutive $y$ values only)

Linear test

All $\Delta y$ equal? → Linear. Any different? → Non-linear

Reading the rule

First diff $= m$; $y$ at $x=0$ gives $c$; rule: $y = mx + c$

Check ALL diffs

Don’t stop after 2 matches — verify every difference

Plotting confirms

Linear table → straight line. Non-linear table → curve

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