A cubic graph is the curved shape produced by any equation with x³ as its highest power. Unlike a straight line or parabola, cubics have a characteristic S-shape that rises through the origin, levels out briefly, then continues rising. At KS3 you plot them by calculating and joining coordinates.
What is a cubic function?
A cubic function has x³ as its highest power. The simplest cubic is:
$$y = x^3$$
You may also meet cubics like y = x³ + 2, y = 2x³, or y = x³ − 3x. All of these produce the same basic S-shaped curve, though shifted, stretched, or modified.
The name "cubic" comes from the word "cube." Just as a quadratic (x²) connects to area, a cubic (x³) connects to volume — cubing a length gives the volume of a cube with that side length.
What are the key features of y = x³?
| Feature | Detail |
|---|---|
| Passes through | (0, 0), (1, 1), (−1, −1), (2, 8), (−2, −8) |
| Shape | S-shaped (elongated, continuous) |
| Point of inflection | At the origin (0, 0): the curve changes from curving one way to the other |
| Rotational symmetry | 180° about the origin (the curve looks the same if rotated half a turn) |
| Crosses x-axis | Once — at x = 0 |
| As x → +∞ | y → +∞ |
| As x → −∞ | y → −∞ |
The most important distinction from a quadratic: y = x³ has no maximum or minimum — it just keeps rising to the right and falling to the left. Instead it has a point of inflection at the origin, where the gradient is zero but the curve does not turn back.
How do you plot y = x³?
Worked example: Complete a table of values and plot y = x³ for −3 ≤ x ≤ 3.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| x³ | −27 | −8 | −1 | 0 | 1 | 8 | 27 |
Notice:
- The values grow quickly in magnitude (cubing amplifies differences).
- Negative x gives negative y.
- The graph is symmetric: every point (x, y) has a corresponding point (−x, −y).
Plot all seven points and draw a smooth S-shaped curve through them. The curve is steeper at the extremes and almost flat near the origin.
How does y = x³ compare to y = x² and y = x?
Plotting all three on the same axes reveals clear differences:
| x | y = x | y = x² | y = x³ |
|---|---|---|---|
| −2 | −2 | 4 | −8 |
| −1 | −1 | 1 | −1 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 4 | 8 |
- y = x is a straight line, always rising.
- y = x² is a U-shaped parabola, symmetric about the y-axis, always non-negative.
- y = x³ is an S-shape, has the same sign as x, and passes through the same three points (−1, −1), (0, 0), (1, 1) as y = x.
For |x| > 1, the cubic grows fastest; for 0 < |x| < 1, the cubic gives the smallest y-value.
What does a modified cubic look like?
Once you know y = x³, you can sketch modified cubics by applying the rules you know about graph transformations:
- y = x³ + c: vertical translation. The S-shape shifts up by c.
- y = (x − a)³: horizontal translation. The point of inflection moves to (a, 0).
- y = −x³: reflection in the x-axis. The S-shape is flipped: it falls to the right and rises to the left.
Worked example: Describe the graph of y = x³ − 8.
This is y = x³ translated down by 8 units. The point of inflection moves from (0, 0) to (0, −8). The curve crosses the x-axis where x³ = 8, i.e. at x = 2. The overall S-shape is unchanged.
How do you identify where a cubic crosses the x-axis?
The x-intercepts (roots) of a cubic are where y = 0. For y = x³, this is only x = 0. For more complex cubics like y = x³ − 3x, you factorise:
$$x^3 - 3x = x(x^2 - 3) = x(x - \sqrt{3})(x + \sqrt{3})$$
Roots: x = 0, x = √3 ≈ 1.73, x = −√3 ≈ −1.73. (This level of factorising is more typical of GCSE or A-level.)
At KS3, you mainly need to plot the curve from a table and identify roots by looking at where y changes sign.
Frequently asked questions
Why does the cubic have an S-shape and not a U-shape?
The U-shape of a quadratic is caused by x² being always positive — the parabola is symmetric about the y-axis. Cubing preserves the sign of x (a negative cubed is negative), so the cubic passes through three quadrants and has rotational rather than reflective symmetry. The S-shape reflects this: the curve rises from bottom-left to top-right.
Is the point of inflection the same as a minimum or maximum?
No. At a minimum or maximum, the gradient equals zero and the curve turns back the other way. At a point of inflection, the gradient also equals zero but the curve continues in the same overall direction — it just flattens out momentarily then steepens again. On y = x³, the gradient at the origin is zero but y continues to increase for x > 0.
Does a cubic always pass through the origin?
Only if the constant term is zero. y = x³ passes through (0, 0). y = x³ + 5 passes through (0, 5). y = (x − 2)³ passes through (0, −8). The origin (0, 0) is only on the curve if substituting x = 0 gives y = 0.
How do cubic graphs appear in exams at KS3?
KS3 exam questions typically ask you to: complete a table of values, plot the points and draw the curve, read off a y-value for a given x, or read off x where y = 0. You might also be asked to match a graph to its equation (distinguishing linear, quadratic, cubic, and reciprocal). Accuracy in plotting and a smooth curve are the most important things.
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