Mathematics • Year 8 • Unit 2 • Lesson 4

Tables of Values

Build a table of values from a rule, use first differences to test for linearity, and read coordinate pairs straight from the table. One worked example, one guided example with blanks, then eight independent problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

The Linear Test from the lesson: are ALL first differences equal? Yes → linear. No → non-linear.

Problem. Build a table of values for y = 2x + 1 using x = −1, 0, 1, 2, 3. Apply the Linear Test, then write the five (x, y) coordinate pairs.

Step 1 — Substitute each x into the rule.

x = −1: y = 2(−1) + 1 = −1. x = 0: y = 2(0) + 1 = 1.

x = 1: y = 3. x = 2: y = 5. x = 3: y = 7.

Step 2 — Write the table.

x: −1 0 1 2 3

y: −1 1 3 5 7

Step 3 — Compute first differences (Δy).

1 − (−1) = 2, 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2.

Reason: first difference is "the change in y for a 1-unit change in x".

Step 4 — Apply the Linear Test.

All first differences equal 2 → LINEAR. Constant rate = 2.

Step 5 — Coordinate pairs (read straight from the table).

(−1, −1), (0, 1), (1, 3), (2, 5), (3, 7).

Answer: linear, rate +2 per unit x.

Stuck? Revisit lesson § Key Terms — "first difference" and "the Linear Test".

2. We do — fill in the missing steps

Same shape as Section 1. Fill every blank. 4 marks

Problem. Build a table for y = 3x − 2 using x = 0, 1, 2, 3, 4. Apply the Linear Test.

Step 1 — Substitute:

x = 0: y = 3(0) − 2 = ____. x = 1: y = ____. x = 2: y = ____.

x = 3: y = ____. x = 4: y = ____.

Step 2 — Table:

x: 0 1 2 3 4

y: ___ ___ ___ ___ ___

Step 3 — First differences:

Δy = ____, ____, ____, ____.

Step 4 — Linear Test: All differences equal ____. So the relationship is __________ (linear / non-linear).

Step 5 — Coordinate pairs:

( __ , __ ), ( __ , __ ), ( __ , __ ), ( __ , __ ), ( __ , __ ).

Stuck? Substitute carefully and watch the − 2 at the end.

3. You do — independent practice

Show your reasoning. 3.1–3.4 are foundation (build a small table from a rule). 3.5–3.6 are standard (apply the Linear Test). 3.7–3.8 are extension (reason from the table back to the rule).

Foundation — build the table

3.1 Build a table for y = x + 4 using x = 0, 1, 2, 3. List the four (x, y) pairs. 1 mark

3.2 Build a table for y = 2x using x = −2, −1, 0, 1, 2. List the five (x, y) pairs. 1 mark

3.3 Build a table for y = 10 − x using x = 0, 2, 4, 6, 8. List the five (x, y) pairs. 1 mark

3.4 Build a table for y = −x + 5 using x = 0, 1, 2, 3, 4. List the five (x, y) pairs. 1 mark

Standard — apply the Linear Test

3.5 Test whether the following table represents a linear relationship. Show first differences. 2 marks

x: 1 2 3 4 5

y: 3 7 11 15 19

3.6 Test whether the following table represents a linear relationship. Show first differences. 2 marks

x: 1 2 3 4 5

y: 1 4 9 16 25

Extension — reason from the table back to the rule

3.7 A table shows: x = 1 → y = 8, x = 2 → y = 13, x = 3 → y = 18, x = 4 → y = 23. (a) Show it's linear. (b) Write the rule connecting y to x. 2 marks

3.8 A linear rule produces these values: x = 0 → y = 7, x = 1 → y = 5, x = 2 → y = 3. (a) State the first difference. (b) Predict y when x = 5. (c) Write the rule. 2 marks

Stuck on 3.8? First difference is NEGATIVE here (y goes DOWN as x goes UP). The constant rate is the coefficient of x in the rule.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do y = 3x − 2

Step 1: y values −2, 1, 4, 7, 10.
Step 3: Δy = 3, 3, 3, 3.
Step 4: All differences equal 3. So linear.
Step 5: (0, −2), (1, 1), (2, 4), (3, 7), (4, 10).

3.1 — y = x + 4

(0, 4), (1, 5), (2, 6), (3, 7).

3.2 — y = 2x

(−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4).

3.3 — y = 10 − x

(0, 10), (2, 8), (4, 6), (6, 4), (8, 2).

3.4 — y = −x + 5

(0, 5), (1, 4), (2, 3), (3, 2), (4, 1).

3.5 — Linear Test (y: 3, 7, 11, 15, 19)

First differences: 7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4. All equal 4 → linear. Constant rate +4.

3.6 — Linear Test (y: 1, 4, 9, 16, 25)

First differences: 3, 5, 7, 9 — NOT equal → non-linear. (These are the square numbers, y = x².)

3.7 — Rule for y: 8, 13, 18, 23

(a) Differences: 5, 5, 5 → all equal → linear.
(b) Constant rate is 5, so try y = 5x + c. At x = 1, y = 8 → 5 + c = 8 → c = 3. Rule: y = 5x + 3. Check x = 4: 20 + 3 = 23 ✓.

3.8 — Rule for y: 7, 5, 3 (at x = 0, 1, 2)

(a) First difference: 5 − 7 = −2 (constant).
(b) Continue the pattern: x = 3 → 1, x = 4 → −1, x = 5 → −3.
(c) Constant rate −2, y-intercept (x = 0) is 7. Rule: y = −2x + 7. Check x = 5: −10 + 7 = −3 ✓.