EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Cubic and reciprocal graphs
Graphs · Lesson 7 of 8
Last Lesson and Before
Answer each one, then check.
1. Work out (−2)³.
−8
2. Last lesson: what shape is the graph of y = x²?
A U-shaped parabola
3. Work out 12 ÷ 0.5.
24
4. What is the reciprocal of 3?
⅓
Learning Objectives
1. Complete a table of values for a cubic and plot its graph.
2. Complete a table of values for a reciprocal and plot its graph.
3. Recognise and sketch linear, quadratic, cubic and reciprocal graphs.
4. Use a graph to solve equations.
Cubic Graphs
A cubic has an x³ term and no higher power.
▸ y = x³. Passes through the origin: (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8).
▸ S-shape. Positive x³: rises from bottom left to top right. Negative x³: falls from top left to bottom right.
▸ Up to three roots. y = x³ − 4x = x(x − 2)(x + 2) crosses the x-axis at −2, 0 and 2.
▸ Cube negatives carefully. (−3)³ = −27, so x³ − 4x at x = −3 is −27 + 12 = −15.
A Table of Values for y = x³ − 4x
|
x |
x³ |
−4x |
y |
|---|---|---|---|
|
−3 |
−27 |
12 |
−15 |
|
−2 |
−8 |
8 |
0 |
|
−1 |
−1 |
4 |
3 |
|
0 |
0 |
0 |
0 |
|
1 |
1 |
−4 |
−3 |
|
2 |
8 |
−8 |
0 |
|
3 |
27 |
−12 |
15 |
Reciprocal Graphs
A reciprocal graph has the form y = k/x.
▸ No value at x = 0. You cannot divide by 0, so there is no point on the y-axis.
▸ Two separate branches. For positive k, one branch is in the top right and one in the bottom left.
▸ Asymptotes. The curve gets closer and closer to both axes but never touches them.
▸ Negative k. The branches are in the top left and bottom right instead.
Plotting y = 12/x
|
Complete a table of values for y = 12/x for x = 1, 2, 3, 4, 6, 12, and describe the graph. |
1. Divide 12 by each value
12, 6, 4, 3, 2, 1
2. Plot and join with a smooth curve
It falls steeply, then levels off
3. Negative values
x = −1 gives −12, x = −2 gives −6: the same shape, upside down, in the bottom left
Answer: y = 12, 6, 4, 3, 2, 1. The curve never touches either axis.
Four Graph Shapes
|
Know these four shapes by sight. The highest power of x tells you which: power 1 is a straight line, power 2 a parabola, power 3 an S-shaped cubic, and x on the bottom of a fraction a reciprocal with two branches. |
Linear, quadratic, cubic and reciprocal. |
Match the Equation to the Graph
|
Equation |
Shape of graph |
|---|---|
|
y = 3 − 2x Straight line sloping down |
|
|
y = x² − 1 U-shaped parabola |
|
|
y = −x² + 4 ∩-shaped parabola |
|
|
y = x³ + 1 S-shaped curve rising to the right |
|
|
y = 5/x Two branches, top right and bottom left |
|
|
y = −5/x Two branches, top left and bottom right |
Key Terms
|
Cubic An expression whose highest power of x is x³. |
Reciprocal graph A graph of the form y = k/x. |
|
Asymptote A line that a curve gets closer and closer to but never touches. |
Root An x-value where a graph crosses the x-axis. |
Your Task: Graph Sketch Race
10 minutes
|
Your teacher calls out an equation. Sketch its graph on a mini-whiteboard in 20 seconds: y = 4, y = x³, y = −x², y = 2/x, y = 2x − 3, y = −x³, x = −1, y = x² + 2, y = −3/x. 1. Decide the type from the highest power. 2. Decide the direction from the sign. 3. Mark any intercepts. |
A good answer shows: Horizontal line; S-shape rising; ∩-shape through the origin; branches top-right and bottom-left; straight line through (0, −3), gradient 2; S-shape falling; vertical line; U-shape with minimum (0, 2); branches top-left and bottom-right.
Can I...?
☐ Complete a table of values for a cubic.
☐ Plot a cubic graph.
☐ Complete a table of values for a reciprocal graph.
☐ Plot a reciprocal graph.
☐ Recognise linear, quadratic, cubic and reciprocal graphs.
☐ Explain why y = k/x has no point at x = 0.
Summary
✓ Cubic: S-shaped, up to three roots.
✓ Reciprocal y = k/x: two branches, never touching the axes.
✓ The highest power of x tells you the type of graph.
|
EXAM FOCUS Match each equation to one of the graphs A, B, C and D: y = 3 − 2x, y = x² − 1, y = x³, y = 5/x. (2 marks) In a matching question, start with the equations you are surest of - the straight line and the reciprocal are usually easiest - and eliminate. |