y = x^2(x-1)

Graph of the Cubic Function $y = x^2(x-1)$

$y = x^2(x-1)$ expands to $x^3 - x^2$ . The factored form shows a double root at $x = 0$ and a simple root at $x = 1$ , and the graph touches the $x$ -axis at the former while crossing it at the latter¹. It is about as clean an illustration as one could want of how the multiplicity of a root shows up in a picture.

Intercepts and sign

Since $x^2 \geq 0$ , the sign of the value is decided by $x - 1$ alone.

Range of $x$	Sign of $y$
$x < 0$	negative
$x = 0$	$0$ , touches the axis
$0 < x < 1$	negative
$x = 1$	$0$ , cuts through
$x > 1$	positive

Around $x = 0$ the sign does not change, so the curve merely brushes the axis and drops back below it.

Why a double root means tangency

When $f(x)$ has $(x-a)^2$ as a factor, not only $f(a) = 0$ but also $f'(a) = 0$ . Indeed $y' = 3x^2 - 2x = x(3x-2)$ , which vanishes at $x = 0$ . A point where both the value and the slope are zero is exactly a point where the $x$ -axis is the tangent line. A repeated root is therefore not just two coincident solutions but a geometric statement about tangency.

Monotonicity and extrema

$x$	$\cdots$	$0$	$\cdots$	$\dfrac{2}{3}$	$\cdots$
$y'$	$+$	$0$	$-$	$0$	$+$
$y$	increasing	maximum $0$	decreasing	minimum $-\dfrac{4}{27}$	increasing

Note that the double root is itself the location of the maximum: touching the axis and turning around are the same event.

Concavity and centre of symmetry

The second derivative is $y'' = 6x - 2$ , changing sign at $x = \dfrac{1}{3}$ , so the inflection point is $\left(\dfrac{1}{3}, -\dfrac{2}{27}\right)$ . It lies exactly midway between the two extrema: its $x$ -coordinate is the average of $0$ and $\dfrac{2}{3}$ , and its $y$ -coordinate the average of $0$ and $-\dfrac{4}{27}$ . Since the inflection point of any cubic is its centre of symmetry, rotating the graph by $180^\circ$ about that point maps it onto itself.

Classification by multiplicity

Roots	Behaviour at the $x$ -axis	Example
three distinct	crosses at three places	$x^3 - 3x$
double and simple	touches at one, crosses at another	$x^2(x-1)$
triple	touches and crosses at one point	$x^3$

This function is the representative of the middle case.

Counting solutions

The number of real solutions of $x^2(x-1) = k$ can be read off as intersections with a horizontal line.

Range of $k$	Real solutions
$-\dfrac{4}{27} < k < 0$	$3$
$k = -\dfrac{4}{27}$ or $k = 0$	$2$
otherwise	$1$

Being able to read repeated roots off the extrema, rather than computing a discriminant, is basic to how cubics are handled.

Zero of a function, Wikipedia

Graph of the Cubic Function y=x2(x−1)y = x^2(x-1)y=x2(x−1)