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

$f(x) = a x^2 + b x + c$ 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 $a(x - h)^2 + k$.

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

$f(x) = a x^4 + \dots$ 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 $x = q$.

For any polynomial sketch, mark all of:

$f(x) = x^n$ 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.

$f(x) = x^{-n} = \frac{1}{x^n}$ 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 $\mathbb{R} \setminus \{0\}$.

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

A double root means the graph touches the x-axis without crossing. A triple root means the graph crosses with a stationary point of inflection.

$\sqrt{x}$ is defined only for $x \geq 0$; $\sqrt[3]{x}$ is defined for all real $x$.

$|x|$ is never negative. $\sqrt{x^2} = |x|$, not $x$, when $x$ could be negative.

For an odd-degree polynomial with negative leading coefficient, the graph falls from top left to bottom right (opposite to the positive case).