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GRAPHS · LESSON 10

Cubic and Inverse Graphs

Recognise advanced graph shapes and use them to solve equations.

⏱️ 18–22 min 📘 Challenge ✏️ Worked examples included

Learning Objectives

Introduction

This lesson develops Cubic and Inverse 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.

Why Learn Different Graph Shapes?

Not every relationship is linear or quadratic. Recognising a familiar shape helps us predict how a function behaves.

A graph's shape tells us how its output changes, including where it rises, falls, crosses an axis or approaches a value.
🇿🇼 Zimbabwe example: The same graph skills can model mobile-data use, ZESA electricity units, rainfall in Mutare, kombi journeys and school attendance.

Cubic Graphs

y = x³

A basic cubic graph passes through the origin and has an S-like shape. Negative x-values produce negative y-values, while positive x-values produce positive y-values.

x−2−1012
x³−8−1018
Cubic graphs may have one or three roots, depending on the equation.

Inverse Graphs

y = 1/x

The graph has two separate branches. It never touches the x-axis or y-axis because x cannot equal zero and 1/x can never equal zero.

x−4−2−1124
1/x−0.25−0.5−110.50.25
Do not substitute x = 0 into an inverse function such as 1/x. Division by zero is undefined.

Drawing the Curves

  1. Calculate enough values to reveal the shape.
  2. Choose scales that show both large and small values.
  3. Plot accurately.
  4. Draw one smooth curve through the pattern.
  5. For inverse graphs, draw the two branches separately.

Solving Equations Graphically

The roots of y = x³ − 4x are the x-values where the curve crosses the x-axis.

To solve x³ = 2x + 3, draw y = x³ and y = 2x + 3 on the same axes. The x-coordinates of their intersections are the solutions.

Graphical solutions may not be exact. State an appropriate decimal accuracy.

✅ Quick Check

Describe the basic shape of y = x³.
Why is x = 0 excluded from y = 1/x?
Find y when x = −2 for y = x³ + 1.

🧠 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 = x³ for −3 ≤ x ≤ 3.
  2. Complete a table for y = 2/x using x = −4, −2, −1, 1, 2, 4.
  3. State which axes y = 1/x never touches.
Developing
  1. Draw y = x³ − 4x and estimate its roots.
  2. Draw y = 4/x and use it to estimate x when y = 1.5.
Challenge
  1. Explain what happens to 1/x as x becomes a very large positive number.

✅ Check Your Work

Attempt the questions before opening the solutions.

Show answers and working

Foundation

  1. For x=−3,−2,−1,0,1,2,3, y=−27,−8,−1,0,1,8,27.
  2. For x=−4,−2,−1,1,2,4, y=−0.5,−1,−2,2,1,0.5.
  3. Neither the x-axis nor the y-axis.

Developing

  1. y=x³−4x=x(x−2)(x+2), so roots are −2,0,2.
  2. 4/x=1.5, so x≈2.67.

Challenge

  1. It approaches 0 from the positive side but never becomes 0.

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