Sketch polynomial functions using leading-term behaviour, intercepts and the multiplicity of each root

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 $m$.

• IMATH_23 (simple root): graph crosses the x-axis transversally.
• IMATH_24 (double root): graph touches the x-axis and bounces back (does not cross), like a parabola at its vertex.
• IMATH_25 (triple root): graph crosses with a flattening, like $y = x^3$ at the origin (horizontal point of inflection on the x-axis).
• IMATH_27 even: touch and bounce back, but flatter.
• IMATH_28 odd: cross with even flatter inflection at the axis.

The sign of the polynomial alternates across simple roots and does not change across roots of even multiplicity.

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.