The general cubic function
Theory
A cubic function has the general form \(y=ax^{3}+bx^{2}+cx+d\) with \(a\neq 0\). The sign of the leading coefficient \(a\) fixes the end behaviour, the factored form \(y=a(x-p)(x-q)(x-r)\) reads off the x-intercepts, and the constant \(d\) is the y-intercept. A cubic has up to three x-intercepts, and a repeated root makes the curve touch or flatten on the axis rather than cross it.
A cubic function is a polynomial of degree three, \(y=ax^{3}+bx^{2}+cx+d\), where \(a,b,c,d\) are constants and \(a\neq 0\). The \(x^{3}\) term dominates for large \(|x|\), so the ends of the graph run off in opposite directions. The leading coefficient \(a\) sets the end behaviour: if \(a\gt 0\) the curve falls from the bottom-left and rises to the top-right; if \(a\lt 0\) it rises from the top-left and falls to the bottom-right. Substituting \(x=0\) gives the y-intercept \(y=d\).
The factored form \(y=a(x-p)(x-q)(x-r)\) is the most useful for sketching, because setting \(y=0\) shows the x-intercepts at once: each factor gives one root, \(x=p\), \(x=q\) and \(x=r\). A cubic can have up to three x-intercepts; it always has at least one, since a cubic must cross the \(x\)-axis somewhere.
When a factor is repeated the curve meets the axis differently. A double factor \((x-p)^{2}\) makes the curve touch the \(x\)-axis at \(x=p\) and turn back, like the vertex of a parabola, rather than cross it. A triple factor \((x-p)^{3}\) makes the curve flatten as it passes through \(x=p\) — a stationary point of inflection.
The general (expanded) form of a cubic — the y-intercept is the constant term:
The factored form — the roots \(p,q,r\) are the x-intercepts:
Repeated-root forms — a double factor touches, a triple factor flattens through:
How to sketch a cubic from factored form
- Read the sign of \(a\). Decide the end behaviour: \(a\gt 0\) rises to the right, \(a\lt 0\) falls to the right.
- Find the x-intercepts. Set each factor to zero. Remember \((x-p)=0\) gives \(x=p\), and \((x+p)=0\) gives \(x=-p\).
- Check for repeated roots. A double factor \((x-p)^{2}\) touches the axis at \(x=p\); a triple factor \((x-p)^{3}\) flattens through it.
- Find the y-intercept. Put \(x=0\): in factored form this is \(y=a(-p)(-q)(-r)\); in expanded form it is the constant \(d\).
- Join smoothly. Mark the intercepts, then draw one continuous curve consistent with the end behaviour, crossing or touching as required.
| \(y\) | \(=\) | \(2(0)^{3}-5(0)^{2}+0-4\) |
| \(=\) | \(-4\) |
| \(a=2\) | \(\gt\) | \(0\) |
| \(x+1=0\) | \(\Rightarrow\) | \(x=-1\) |
| \(x-2=0\) | \(\Rightarrow\) | \(x=2\) |
| \(x-4=0\) | \(\Rightarrow\) | \(x=4\) |
| \(y\) | \(=\) | \((1)(-2)(-4)\) |
| \(=\) | \(8\) |
| \(x+3=0\) | \(\Rightarrow\) | \(x=-3\) (simple root, crosses) |
| \((x-1)^{2}=0\) | \(\Rightarrow\) | \(x=1\) (double root, touches) |
| \(y\) | \(=\) | \((3)(-1)^{2}\) |
| \(=\) | \(3\) |
| \(a=-1\) | \(\lt\) | \(0\) (falls to the right) |
| \(x+1=0,\;x-1=0,\;x-2=0\) | \(\Rightarrow\) | \(x=-1,1,2\) |
| \(y\) | \(=\) | \(-(1)(-1)(-2)\) |
| \(=\) | \(-2\) |
Common pitfalls
Frequently asked questions
What is the general form of a cubic function?
It is \(y=ax^{3}+bx^{2}+cx+d\) with \(a\neq 0\). The \(x^{3}\) term makes it a cubic; \(a\) controls the shape and the constant \(d\) is the y-intercept.
How do you find the x-intercepts of a cubic?
Write it as \(y=a(x-p)(x-q)(x-r)\) and set \(y=0\). Each factor gives one intercept, and \((x+3)\) gives \(x=-3\), not \(+3\).
How many x-intercepts can a cubic have?
One, two or three. Three distinct factors give three crossings; a double factor gives a touch (fewer crossings); and some cubics cross the axis only once.
What does a repeated root do to a cubic graph?
A double factor \((x-p)^{2}\) makes the curve touch the axis at \(x=p\) and turn back; a triple factor \((x-p)^{3}\) makes it flatten as it passes through — a stationary point of inflection.
How do you find the y-intercept of a cubic?
Substitute \(x=0\). In expanded form this is the constant \(d\); in factored form it is \(y=a(-p)(-q)(-r)\).
How does the sign of a affect a cubic graph?
If \(a\gt 0\) the curve falls from the bottom-left and rises to the top-right; if \(a\lt 0\) it is reversed. The size of \(a\) stretches the curve but does not move the intercepts.