Cubic and quartic polynomials (VCE Maths Methods Unit 1)

What this dot point is asking

VCAA wants you to sketch cubic and quartic polynomials from their factored or standard form, identify intercepts, end behaviour and turning/inflection points, and interpret root multiplicities geometrically.

End behaviour

The end behaviour of a polynomial is set by its leading term $ax^n$:

| $n$ even, $a > 0$ | Both ends $+\infty$ |
| $n$ even, $a < 0$ | Both ends $-\infty$ |
| $n$ odd, $a > 0$ | Left $-\infty$, right $+\infty$ |
| $n$ odd, $a < 0$ | Left $+\infty$, right $-\infty$ |

Root multiplicities

For a factor $(x - p)^k$:

• $k = 1$: graph crosses the $x$-axis at $x = p$.
• $k = 2$: graph touches the $x$-axis and bounces back (turning point on the axis).
• $k = 3$: graph crosses with a horizontal tangent (point of inflection on the axis).

Cubics

General cubic: $y = ax^3 + bx^2 + cx + d$.

Has up to two turning points (one local max and one local min) and at least one $x$-intercept (because every cubic has at least one real root).

Stationary point of inflection: a cubic of the form $y = a(x - h)^3 + k$ has one stationary point of inflection at $(h, k)$ and no other turning points.

Quartics

General quartic: $y = ax^4 + bx^3 + cx^2 + dx + e$.

Has up to three turning points. May have $0$, $1$, $2$, $3$ or $4$ real roots.

Special form: $y = a(x - h)^4 + k$. Single turning point at $(h, k)$, similar shape to a parabola but flatter at the vertex.

Building a polynomial from its roots

If a polynomial has roots $r_1, r_2, \ldots, r_n$ and leading coefficient $a$:

Multiplicities are written by repeating factors: $(x - p)^2$ for a double root at $p$.

Worked example

Sketch $y = -(x + 1)(x - 2)^2$.

Cubic. Leading coefficient $-1$ (expand: $-x^3 + \ldots$).

Roots: $x = -1$ (multiplicity $1$, crosses), $x = 2$ (multiplicity $2$, touches and turns).

$y$-intercept: $y(0) = -(1)(4) = -4$.

End behaviour: as $x \to +\infty$, $y \to -\infty$; as $x \to -\infty$, $y \to +\infty$ (odd degree, negative lead).

Local maximum exists at $x = 2$ (touch-and-turn at the double root from below).

Common traps

Confusing odd and even degree end behaviour

Even degree polynomials have both ends going the same direction; odd degree polynomials have opposite directions.

Treating a double root as just one root

A double root counts twice when counting roots with multiplicity and produces touch-and-turn behaviour rather than crossing.

Forgetting the sign of the leading coefficient

Negative leading coefficient flips the graph vertically.

Including more turning points than degree allows

A polynomial of degree $n$ has at most $n - 1$ turning points.

In one sentence

Cubic and quartic polynomials are sketched from their factored form by reading roots (with multiplicities determining crossing vs touching), the $y$-intercept by setting $x = 0$, and end behaviour from the leading term ($ax^n$ for $n$ even gives both ends in the same direction, for $n$ odd in opposite directions, with the sign of $a$ flipping vertically).

Examples in context

Example 1. Reading a factored cubic. Sketch $y = (x + 2)(x - 1)^2$. Roots: $x = -2$ (single, so the curve crosses) and $x = 1$ (double, so the curve touches and turns). The $y$-intercept is $y = (2)(1)^2 = 2$. Leading term: expanding gives $x^3$ as the highest power with positive coefficient, so as $x \to -\infty$, $y \to -\infty$ and as $x \to +\infty$, $y \to +\infty$.

Example 2. End behaviour of a quartic. For $y = -x^4 + 5x^2 - 4 = -(x^2 - 1)(x^2 - 4) = -(x - 1)(x + 1)(x - 2)(x + 2)$, the four roots are $x = \pm 1, \pm 2$ (all single, all crossings). The negative leading coefficient and even degree mean both ends point down ($y \to -\infty$). The $y$-intercept is $-4$.

Try this

Q1. State the roots of $y = (x - 3)(x + 1)^2$ and which is a turning point on the axis. [2 marks]

• Cue. Roots $x = 3$ (crosses) and $x = -1$ (double root, touches and turns).

Q2. Find the $y$-intercept and end behaviour of $y = 2x^3 - x$. [2 marks]

• Cue. $y$-intercept $0$; positive odd leading term, so $y \to -\infty$ as $x \to -\infty$ and $y \to +\infty$ as $x \to +\infty$.

Q3. A quartic has factored form $y = a(x + 2)(x - 1)^3$ and passes through $(0, 4)$. (a) Find $a$. (b) Describe the behaviour at each root. [2+2 marks]

• Cue. (a) $4 = a(2)(-1)^3 = -2a$, so $a = -2$. (b) At $x = -2$ single root (crosses); at $x = 1$ triple root (crosses with a flattening/stationary inflection).