Free online graphing tool
Cubic Graph Calculator (y = ax³ + bx² + cx + d)
A cubic function is any polynomial whose highest power is three. Its graph always runs from bottom-left to top-right (or the reverse when a is negative) and can wiggle through one local maximum and one local minimum on the way.
Cubics can cross the x-axis once, twice or three times, which is why plotting one is the fastest way to see how many real roots a cubic equation has before you try to factorise it.
Key points of y = x^3 - 3x inside the visible window (-5 ≤ x ≤ 5). Pan or zoom to search a different range.
Roots (x-intercepts)
- x = -1.73205
- x = 0
- x = 1.73205
y-intercept
- y = 0
Turning points
- Max (-1, 2)
- Min (1, -2)
Intersections
- (-4.9933, -109.52)
- (-4.9867, -109.04)
- (-4.98, -108.57)
- (-4.9733, -108.09)
- (-4.9667, -107.62)
- (-4.96, -107.14)
- (-4.9533, -106.67)
- (-4.9467, -106.2)
- (-4.94, -105.73)
- (-4.9333, -105.27)
- (-4.9267, -104.8)
- (-4.92, -104.34)
Enter a value of x
What is a cubic graph calculator?
A cubic graph calculator plots y = ax³ + bx² + cx + d. Every cubic has at least one real root, at most three, and either two turning points or none at all. The curve always has a point of inflection where it changes from bending one way to bending the other.
Key facts at a glance
| General form | y = ax³ + bx² + cx + d, a ≠ 0 |
|---|---|
| Real roots | Always at least 1, never more than 3 |
| Turning points | Either 2 (a local max and a local min) or none |
| End behaviour, a > 0 | Falls to −∞ on the left, rises to +∞ on the right |
| End behaviour, a < 0 | Rises on the left, falls on the right |
| Point of inflection | At x = −b / 3a, where the curvature flips |
| y-intercept | (0, d) |
| Symmetry | y = x³ is an odd function — rotationally symmetric about the origin |
How to plot cubic graphs — step by step
- 1
Enter the cubic
Type y = x^3 - 3x. Powers use the ^ symbol, so x cubed is x^3.
- 2
Count the roots
The Key points tab lists every root inside the visible window. Zoom out if you suspect the curve crosses again further along.
- 3
Locate the turning points
The same tab reports the local maximum and minimum. A cubic with no turning points is strictly increasing or strictly decreasing.
- 4
Explore with sliders
The second equation, y = a x^3 + b x, has sliders. Set b positive and the wiggle disappears entirely — the two turning points merge and vanish.
Cubic examples you can plot right now
Copy any equation into the calculator above, or open it in the full calculator with one click.
The pure cubic
y = x^3
No turning points at all — just a point of inflection at the origin.
Plot this →Understanding the cubic graph
- Why at least one real root
- A cubic runs from −∞ to +∞ (or the reverse), so by the intermediate value theorem it must pass through zero somewhere. Even-degree polynomials like quadratics carry no such guarantee.
- Repeated roots on a graph
- Where a cubic touches the axis and turns back, that root is repeated. Where it cuts straight through with an S-bend, the root has multiplicity three.
- Point of inflection
- The place where the second derivative changes sign — visually, where the curve stops bending downwards and starts bending upwards. Every cubic has exactly one.
Cubic graph — frequently asked questions
How do I graph a cubic function?
Type it as y = x^3 - 3x in the calculator above. Use ^3 for the cube, and brackets for factorised forms such as y = (x - 1)(x + 2)(x - 3).
How many roots can a cubic have?
A cubic always has at least one real root and never more than three. Any roots that are not real come in a complex conjugate pair, so the possible counts are one or three (counting repeated roots).
Why does my cubic have no turning points?
When the derivative 3ax² + 2bx + c has no real roots, the curve never levels off — it rises or falls continuously. y = x^3 + x is the classic example.
How do I find the turning points of a cubic?
Read them from the Key points tab, which finds every local maximum and minimum inside the visible window numerically. Algebraically they are the roots of the derivative 3ax² + 2bx + c = 0.
Can I plot a quartic or higher polynomial?
Yes. Any power works — y = x^4 - 5x^2 + 4 or y = x^5 - x plot exactly the same way, and the Key points tab still finds the roots and turning points.
How do I plot a factorised cubic?
Type the brackets exactly as written: y = (x - 1)(x + 2)(x - 3). The multiplication signs are optional, so adjacent brackets are multiplied automatically.
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