Graphing polynomials: leading-term behaviour, intercepts and root multiplicity

What this dot point is asking

NESA wants you to sketch the graph of a polynomial in factored form by combining three pieces of information: the leading-term behaviour at $\pm \infty$, the x-intercepts (with multiplicities), and the y-intercept.

The answer

End behaviour from the leading term

For large $|x|$, a polynomial behaves like its leading term $a_n x^n$.

• If $n$ is even and $a_n > 0$: both ends go to $+\infty$.
• If $n$ is even and $a_n < 0$: both ends go to $-\infty$.
• If $n$ is odd and $a_n > 0$: left end goes to $-\infty$, right end goes to $+\infty$.
• If $n$ is odd and $a_n < 0$: left end goes to $+\infty$, right end goes to $-\infty$.

Sketching the two end arms first locks in the shape.

Root multiplicity

If $(x - a)^m$ is a factor of $P(x)$, then $a$ is a root of multiplicity $m$, and the multiplicity decides exactly how the curve meets the x-axis there.

• $m = 1$ (simple root): graph crosses the x-axis transversally, like a straight line through the axis.
• $m = 2$ (double root): graph touches the x-axis and bounces back (does not cross), like a parabola at its vertex.
• $m = 3$ (triple root): graph crosses with a flattening, like $y = x^3$ at the origin (horizontal point of inflection on the x-axis).
• $m \ge 4$ even: touch and bounce back, but flatter.
• $m \ge 5$ odd: cross with an even flatter inflection at the axis.

The pattern reduces to two questions: is the multiplicity odd or even, and is it $1$ or greater than $1$? Odd multiplicity crosses, even multiplicity touches; multiplicity greater than $1$ flattens the curve against the axis (a horizontal tangent at the root). The reason is local: near $x = a$ the curve behaves like $c(x - a)^m$ for some constant $c$, so it inherits the shape of $y = x^m$ near the origin. A simple root looks like $y = x$ (slices through), a double root like $y = x^2$ (touches and turns), a triple root like $y = x^3$ (flattens then crosses).

The sign of the polynomial alternates across simple roots and does not change across roots of even multiplicity. That single fact is the bridge to the sign-diagram method used for polynomial inequalities.

y-intercept

Set $x = 0$ to find the y-intercept, which is the constant term of the polynomial.

Turning points and shape

A polynomial of degree $n$ has at most $n - 1$ turning points (where the derivative changes sign) and at most $n - 2$ points of inflection (where $P''$ changes sign). For Extension 1 sketches, you do not need to compute these precisely. You combine end behaviour with the root analysis to draw a continuous curve that hits the intercepts in the right way.

A practical recipe

1. Sketch the end behaviour as two arrows on the axes.
2. Mark every x-intercept and label its multiplicity.
3. Mark the y-intercept.
4. Draw a smooth curve from the left end, through the intercepts (crossing at simple and odd roots, touching at even roots), to the right end, alternating sign appropriately.

Sketch a polynomial, stage by stage

The reliable way to sketch is to build the picture up in the order of the recipe, rather than trying to draw the whole curve at once. Below, the cubic $P(x) = (x + 2)(x - 1)^2$ is sketched one stage at a time. It is degree $3$ with a positive leading coefficient, a simple root at $x = -2$ (so the curve crosses), and a double root at $x = 1$ (so the curve touches and turns).

Stage 1, draw the end behaviour first. The degree is $3$ (odd) and the leading coefficient is positive (multiplying the leading $x$ from each factor gives $+x^3$), so the left arm falls away to $-\infty$ and the right arm rises to $+\infty$. Sketch those two arrows before plotting anything else; they fix which way the curve enters and leaves.

Stage 2, mark the roots with their multiplicities and the y-intercept. From the factors, $x = -2$ is simple (multiplicity $1$, the curve will cross) and $x = 1$ is double (multiplicity $2$, the curve will touch and turn). The y-intercept is $P(0) = (0 + 2)(0 - 1)^2 = 2$. Label each root with what the curve must do there before drawing.

Stage 3, join the dots respecting cross versus touch. Come up the left arm, cross the axis transversally at $x = -2$, rise to a local maximum, fall through the y-intercept $(0, 2)$, and arrive at the double root $x = 1$ where the curve must touch and turn, not cross. Because $x = 1$ is even multiplicity, the curve stays on the same side of the axis just before and just after, so it bounces back up and heads off to $+\infty$.

Stage 4, label the finished sketch. Add the equation and the key coordinates: x-intercepts at $-2$ (cross) and $1$ (touch), y-intercept $(0, 2)$, both confirmed against the end behaviour. A cubic has at most $n - 1 = 2$ turning points, and this sketch uses exactly two (one local maximum, one local minimum at the touch), which is a useful sanity check.

How exam questions ask about graphing

The same skill is tested through a handful of recurring wordings. Match the phrasing to the move:

• "Sketch the graph of $y = P(x)$, showing all intercepts and their behaviour." Run the four-stage recipe: end behaviour, roots with multiplicities, y-intercept, then a smooth curve. Markers award marks for correct end arms, the cross-versus-touch decision at each root, and a labelled y-intercept, even if the curviness between roots is rough.
• "Describe the behaviour of the graph at $x = a$." This is a multiplicity question. Read the power of $(x - a)$: odd crosses, even touches, and any power $> 1$ flattens (horizontal tangent). A factor $(x - a)^3$ gives a horizontal point of inflection on the axis.
• "For what values of $k$ does $P(x) = k$ have [one / two / three] solutions?" This is a horizontal-line question: $P(x) = k$ has as many solutions as the line $y = k$ has intersections with the curve. Sketch $y = P(x)$, then count how the count changes as $k$ passes the local maximum and local minimum values.
• "The graph of $y = P(x)$ is shown. Find a possible equation." Read the x-intercepts and their behaviour off the picture (touch means an even power, cross an odd power), write $P(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}\dots$, then pin down the leading constant $a$ using one other known point such as the y-intercept.
• "Show that $P(x)$ has exactly one real zero." Combine the shape argument with sign: if a cubic is increasing throughout (no turning points, or a single horizontal inflection) it crosses the axis once. Often paired with a quick derivative or a sign check at two points.