Polynomial, power and modulus functions: VCE Math Methods Unit 3

What this dot point is asking

VCAA wants the standard graph shapes for polynomial functions up to quartic, the rational-power family $f(x) = x^n$ for $n \in Q$ (including the square root and cube root), and the modulus function. You need to identify key features (intercepts, turning points, end behaviour, asymptotes if any) and sketch transformed versions cleanly.

Polynomial functions

A polynomial of degree $n$ has the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ with $a_n \neq 0$. The end behaviour is dictated by the leading term $a_n x^n$.

Quadratics

$f(x) = a x^2 + b x + c$ with one turning point at $x = -\frac{b}{2a}$. The discriminant $\Delta = b^2 - 4ac$ tells you the number of real roots: two if $\Delta > 0$, one (repeated) if $\Delta = 0$, none if $\Delta < 0$. Standard forms: general $a x^2 + b x + c$, factored $a(x - p)(x - q)$, turning-point $a(x - h)^2 + k$.

Cubics

$f(x) = a x^3 + b x^2 + c x + d$ has up to two stationary points and at least one real root. Standard factored forms:

• Three distinct real roots: $f(x) = a(x - p)(x - q)(x - r)$.
• A repeated root: $f(x) = a(x - p)^2 (x - q)$. The graph touches the x-axis at $x = p$ (without crossing) and crosses at $x = q$.
• One real root: $f(x) = a(x - p)(x^2 + b x + c)$ with the quadratic factor irreducible.

For a positive leading coefficient, the cubic rises from bottom left to top right. For a negative leading coefficient, it falls from top left to bottom right.

Quartics

$f(x) = a x^4 + \dots$ has up to three stationary points and 0, 2 or 4 real roots. Common factored form $f(x) = a(x - p)^2 (x - q)^2$ has a positive-leading "W" shape if $a > 0$, touching the x-axis at $x = p$ and $x = q$.

Key features to identify

For any polynomial sketch, mark all of:

• x-intercepts (solve $f(x) = 0$ via the factor theorem)
• y-intercept ($f(0)$)
• Stationary points ($f'(x) = 0$, classify with $f''$ or sign analysis)
• End behaviour from the leading term

Power functions

A power function has the form $f(x) = x^n$ for some rational $n$. The shape depends on the sign and form of $n$.

Positive integer powers

$f(x) = x^n$ for $n \in \{1, 2, 3, \dots\}$. Even powers ($x^2$, $x^4$) are symmetric in the y-axis with a minimum at the origin. Odd powers ($x^3$, $x^5$) are symmetric about the origin with a stationary point of inflection at the origin.

Negative integer powers

$f(x) = x^{-n} = \frac{1}{x^n}$ for $n \in \{1, 2, 3, \dots\}$. These have a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. Maximal domain is $\mathbb{R} \setminus \{0\}$.

For $n = 1$: $f(x) = \frac{1}{x}$, hyperbola in quadrants 1 and 3.

For $n = 2$: $f(x) = \frac{1}{x^2}$, both branches in the upper half-plane.

Fractional powers (roots)

$f(x) = x^{1/n}$ for $n \in \{2, 3, 4, \dots\}$.

Square root $f(x) = \sqrt{x} = x^{1/2}$. Maximal domain $[0, \infty)$, range $[0, \infty)$. The graph starts at the origin and curves up and right.

Cube root $f(x) = \sqrt[3]{x} = x^{1/3}$. Maximal domain $\mathbb{R}$, range $\mathbb{R}$. Passes through the origin with a vertical tangent. Odd-symmetric (rotation symmetric about the origin).

General fractional powers $f(x) = x^{p/q}$ with $\gcd(p, q) = 1$ are defined on $[0, \infty)$ when $q$ is even and on $\mathbb{R}$ when $q$ is odd.

The modulus function

$f(x) = |x|$ is defined piecewise:

The graph is a V-shape with vertex at the origin, range $[0, \infty)$.

Standard transformations follow the usual form $y = a|b(x - h)| + k$. For example, $y = |x - 2| - 1$ is the V-shape translated 2 right and 1 down, with vertex at $(2, -1)$.

A useful identity: $|x|^2 = x^2$, and $\sqrt{x^2} = |x|$ (not $x$).

Examples in context

Example 1. Reading a factored cubic model. A profit model is $P(x) = -(x - 1)^2(x - 5)$ (thousand dollars) for production level $x$. The roots are $x = 1$ (double, where the curve touches the axis) and $x = 5$ (single, where it crosses). The $y$-intercept is $P(0) = -(1)(-5) = 5$. The negative leading term means the cubic falls from top left to bottom right, so profit is positive for small $x$ and turns negative beyond $x = 5$.

Example 2. Domain of a square-root model. The time for an object to fall a distance $d$ is $t = \sqrt{\frac{d}{5}}$ seconds (a power function with $n = \tfrac12$). Its domain is $d \ge 0$ and range $t \ge 0$. Doubling the drop height does not double the time: from $d = 20$ ($t = 2$ s) to $d = 80$ ($t = 4$ s) requires four times the height, reflecting the square-root shape.

Try this

Q1. State the $x$-intercepts and their type for $y = (x + 3)(x - 2)^2$. [2 marks]

• Cue. $x = -3$ (single, crosses); $x = 2$ (double, touches and turns).

Q2. State the maximal domain and range of $f(x) = \sqrt{x + 4}$. [2 marks]

• Cue. Domain $[-4, \infty)$; range $[0, \infty)$.

Q3. The graph of $y = |x|$ is transformed to $y = |x - 1| + 2$. (a) State the vertex. (b) Find $y$ when $x = 4$. [2+1 marks]

• Cue. (a) Vertex $(1, 2)$. (b) $|4 - 1| + 2 = 5$.