Behaviour of polynomials at infinity

Understanding the behaviour of a polynomial function \(f(x)\) when \(x\) is large (that is, as \(x \to \pm \infty\)) helps us to sketch the graph of \(y=f(x)\).

We've seen from examples that, for polynomial functions up to degree 4, the graph of a polynomial \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\) behaves asymptotically like its leading term \(a_n x^n\).

Let's consider how this behaviour arises. Take a cubic polynomial that we saw earlier, \(f(x) = 2x^3 - 3x^2 + 5\).

One graph. f(x)= 2x cubed – 3x squared + 5, graph of cubic function, local maximum at (0,5) and local minimum at (1,4), x intercepts at (-1,0)
Detailed description of diagram

When \(x\) is large (positive or negative), \(x^2\) is much larger again, and \(x^3\) dwarfs them both. So the \(x^3\) term will 'dominate' the others. One way to see this algebraically is to write

\[ f(x) = 2x^3 - 3x^2 + 5 = 2x^3 \Big( 1 - \frac{3}{2x} + \frac{5}{2x^3} \Big). \]

When \(x\) is large, both \(\dfrac{3}{2x}\) and \(\dfrac{5}{2x^3}\) become very small, and so effectively the dominant term is \(2x^3\). This explains why, as seen on the graph,

\[ f(x) \to + \infty \ \text{as} \ x \to +\infty, \qquad \text{and} \qquad f(x) \to - \infty \ \text{as} \ x \to -\infty. \]

A similar argument applies to any polynomial \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\): for large \(x\), the \(a_n x^n\) term dominates the others.

Looking at the asymptotic behaviour, we can say that the graph \(y=x^n\) is roughly U-shaped when \(n\) is even, in the sense that \(x^n \to +\infty\) as \(x \to \pm \infty\). The actual graph may have many turning points, but if we imagine 'zooming out' and looking only at the large-scale picture when \(x\) is large, the graph is U-shaped. In a similar way, the graph of \(y=x^n\) is roughly /-shaped when \(n\) is odd. The behaviour of a polynomial graph \(y=f(x)\) when \(x\) is large will depend on whether the degree of \(f(x)\) is even or odd, and on the sign of the leading coefficient \(a_n\). We can summarise the outcomes, along with rough shapes of graphs when \(x\) is large, in a table.

Graph of a polynomial \(f(x)\) with leading term \(a_nx^n\)
If \(n\) is and \(a_n\) is then as \(x \to +\infty\) and as \(x \to -\infty\) so shape is roughly
even positive \(f(x) \to +\infty\) \(f(x) \to +\infty\) A curve like a letter U.
even negative \(f(x) \to -\infty\) \(f(x) \to -\infty\) A curve like an inverted letter U.
odd positive \(f(x) \to +\infty\) \(f(x) \to -\infty\) A line sloping upwards from left to right.
odd negative \(f(x) \to -\infty\) \(f(x) \to +\infty\) A line sloping downwards from left to right.

This confirms our Conjecture 5.


Sketch the graph of \(y= x(x-1)(x-2)(x-3)(x-4)\), marking all intercepts (but not turning points).


It's clear that the polynomial \(f(x) = x(x-1)(x-2)(x-3)(x-4)\) has roots at \(x=0,1,2,3,4\), and so these are the \(x\)-intercepts. The \(y\)-intercept is 0. Also, \(f(x)\) is a quintic polynomial with leading term \(x^5\). (This can be seen without fully expanding out the brackets.) Hence \(f(x)\) behaves like \(x^5\), for large~\(x\): as \(x \to + \infty\), \(f(x) \to +\infty\), and as \(x \to - \infty\), \(f(x) \to - \infty\). This is enough information to sketch the graph.

One graph. Graph of quintic function, local minimums in 4th quadrant, local maximums in 1st quadrant, x intercepts at (0,0), (1,0), (2,0), (3,0), (4,0).
Detailed description of diagram

Finally, we can use the asymptotic behaviour of a polynomial \(f(x)\) to obtain information about its \(x\)-intercepts. When we have an odd-degree polynomial, the graph of \(y=f(x)\) must go from \(-\infty\) to \(+\infty\), or vice versa. Therefore the graph of \(y=f(x)\) must cross the \(x\)-axis, giving at least one \(x\)-intercept. This proves that the graph of a polynomial of odd degree has at least one \(x\)-intercept, confirming Conjecture 2.

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