EDEXCEL GCSE MATHS · HIGHER
Cubic equations
Equations and graphs · Lesson 5 of 6
Warm-up
Answer each one, then check.
1. Work out (−2)³.
−8
2. Factorise x² − 4.
(x − 2)(x + 2)
3. Work out 2³ − 4 × 2.
0
4. What is a root of a graph?
Where y = 0
5. What shape is y = x²?
A parabola
Learning Objectives
1. Recognise the shape of a cubic graph.
2. Complete a table and draw a cubic graph.
3. Find roots by factorising or from the graph.
4. Solve f(x) = k by drawing a horizontal line.
Cubic Graph
A cubic graph has an x³ term and a smooth S-shaped curve. It can cross the x-axis up to three times.
With a positive x³ term the curve rises from bottom left to top right.
A Cubic Graph
|
Three roots, one maximum and one minimum. |
The graph of y equals x cubed minus 4x showing three roots and two turning points.
Table of Values
y = x³ − 4x.
|
x |
−3 |
−2 |
−1 |
1 |
|---|---|---|---|---|
|
y |
−15 |
0 |
3 |
0 | −3 | 0 | 15 |
Roots of a Cubic
|
Find the roots of y = x³ − 4x. |
1. Set y = 0
x³ − 4x = 0
2. Factorise
x(x² − 4) = x(x − 2)(x + 2)
3. Solve
x = 0, 2, −2
Answer: The roots are x = −2, 0 and 2.
Solving a Cubic from the Graph
|
Use the graph of y = x³ − 4x to solve x³ − 4x = 1. |
1. Draw
The line y = 1
2. Read the three intersections
x ≈ −1.9, −0.3 and 2.1
3. Check
2.1³ − 4 × 2.1 = 0.861, close to 1
Answer: x ≈ −1.9, x ≈ −0.3 and x ≈ 2.1
How Many Solutions?
|
How many solutions does x³ − 4x = 5 have? |
1. Local maximum
About 3.1, below 5
2. The line y = 5
Meets the curve once, on the right
Answer: One solution.
Common Mistakes
Draw carefully.
▸ Sharp corners. The curve is smooth: no straight segments or points.
▸ Negative values. Cube negatives carefully: (−2)³ = −8.
▸ Missing solutions. A cubic can have up to three solutions.
▸ Turning point values. Read them from the graph or a fine table.
Key Terms
|
Cubic An expression or equation with a highest power of x³. |
Root A solution of f(x) = 0. |
|
Local maximum A turning point at the top of a hill. |
Local minimum A turning point at the bottom of a valley. |
|
Intersection Where two graphs meet. |
Estimate An approximate value from a graph. |
Your Task: Cubic Table
12 minutes
|
Complete a table for y = x³ − 3x for x = −2 to 2. Sketch the graph. How many roots does it have? 1. Cube each x carefully. 2. Join with a smooth curve. |
A good answer shows: y = −2, 2, 0, −2, 2. The curve crosses the x-axis at x = −√3, 0, √3: three roots.
Can I...?
☐ Recognise a cubic graph.
☐ Complete a table.
☐ Draw a smooth curve.
☐ Find roots.
☐ Identify turning points.
☐ Draw a line to solve an equation.
☐ Count the solutions.
☐ Avoid common mistakes.
Summary
✓ A cubic has an x³ term.
✓ It can have up to three roots and two turning points.
✓ Solve f(x) = k with the line y = k.
✓ Factorise to find exact roots.
|
EXAM FOCUS Use the graph of y = x³ − 4x to find estimates for the solutions of x³ − 4x = 1. (3 marks) Draw the horizontal line y = 1 and read all three intersections. |