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GRAPHS · LESSON 05

Tables of Values and Linear Graphs

Move from a formula to coordinates, and from coordinates to a straight-line graph.

⏱️ 18–20 min 📘 Developing ✏️ Worked examples included

Learning Objectives

Introduction

This lesson develops Tables of Values and Linear Graphs from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

🤔 Think About This...

Consider the rule: “double the number and add one”.

If the input is 0, 1, 2 and 3, what outputs will the rule produce?

The rule can be written as y = 2x + 1. A table helps us organise the inputs and outputs before drawing the graph.

📈

Build the Line y = 2x + 1

Change x and see the matching coordinate appear on the graph.

From an Equation to a Table

y = 2x + 1
x−2−1012
y−3−1135
Key idea: Every column becomes one coordinate: (−2, −3), (−1, −1), (0, 1), (1, 3), (2, 5).

Worked Example: Completing the Table

Find y when x = −2 for y = 2x + 1.

1

Substitute x = −2.

2

y = 2(−2) + 1.

3

y = −4 + 1 = −3.

4

The coordinate is (−2, −3).

Plotting the Straight Line

  1. Choose scales that include all x- and y-values.
  2. Plot every coordinate carefully.
  3. Check that the points follow a straight pattern.
  4. Use a ruler to draw one straight line through the points.
If one point lies far away from the pattern, check the substitution before bending the line to include it.

Why Does the Graph Form a Line?

In y = 2x + 1, every increase of 1 in x creates an increase of 2 in y. The rate of change is constant, so the plotted points follow a straight line.

constant change in y ÷ change in x → straight-line relationship

Reading Values from the Graph

Once the graph has been drawn, it can estimate values not included in the original table.

  1. To find y for a chosen x, move vertically to the line, then horizontally to the y-axis.
  2. To find x for a chosen y, move horizontally to the line, then vertically to the x-axis.
Values read from a graph may be estimates. Write a sensible level of accuracy.

✅ Quick Check

For y = 3x − 2, find y when x = 4.
What coordinate is produced when x = 0 in y = 2x + 5?
Why should a ruler be used for a linear graph?

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

Practice Questions

Foundation
  1. Complete a table for y = x + 3 using x = −2, −1, 0, 1, 2.
  2. Write the resulting coordinate pairs.
  3. Plot the points and draw the line.
Developing
  1. Complete and draw y = 3x − 1 for −2 ≤ x ≤ 3.
  2. Use your graph to estimate y when x = 1.5.
  3. Use your graph to estimate x when y = 5.
Challenge
  1. Without plotting, explain why y = 4x − 7 will form a straight line.

✅ Check Your Work

Attempt the questions before opening the solutions.

Show answers and working

Foundation

  1. For x=−2,−1,0,1,2, the y-values are 1,2,3,4,5.
  2. (−2,1), (−1,2), (0,3), (1,4), (2,5).
  3. The plotted points lie on one straight line.

Developing

  1. For x=−2,−1,0,1,2,3, y=−7,−4,−1,2,5,8.
  2. When x=1.5, y=3.5.
  3. When y=5, x=2.

Challenge

  1. It has the form y=mx+c with a constant gradient m=4, so equal changes in x give equal changes in y.

Common Mistakes

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary