y = x^2

Graph of the Quadratic Function $y = x^2$

The graph of $y = x^2$ is an upward-opening parabola with its vertex at the origin¹. It is the simplest quadratic function: the general form $y = ax^2 + bx + c$ with $a = 1$ and $b = c = 0$ .

Domain and range

The domain is all real numbers
The range is $y \geq 0$
The minimum value $0$ is attained at $x = 0$
Even function

Squaring a real number never gives a negative result, which is what limits the range.

Symmetry

Since $(-x)^2 = x^2$ , the function is even and its graph is symmetric about the $y$ -axis: $x$ and $-x$ give the same value, and the axis of symmetry is the line $x = 0$ .

Monotonicity and vertex

The derivative is $y' = 2x$ : negative for $x < 0$ , where the function decreases, and positive for $x > 0$ , where it increases. It turns around at $x = 0$ and attains its minimum there, and that point is the vertex of the parabola. The second derivative $y'' = 2$ is positive everywhere, so the curve is convex throughout.

Rate of growth

$x$	$y = x^2$
$1$	$1$
$2$	$4$
$3$	$9$
$10$	$100$

Doubling $x$ multiplies $y$ by $4$ , and tripling $x$ multiplies it by $9$ : the value grows in proportion to the square of $x$ . The curve touches the $x$ -axis at its vertex, the only point it shares with that axis.

Relationships with other functions

Restricted to $x \geq 0$ the function is one-to-one, and its inverse is $y = \sqrt{x}$ ; the two graphs are reflections of each other in the line $y = x$ .

Range of $x$	Which is larger
$0 < \|x\| < 1$	$x^2 > x^4$
$\|x\| = 0$ or $1$	equal
$\|x\| > 1$	$x^2 < x^4$

As the archetype of the even powers it invites comparison with $y = x^4$ .

Applications

Parabolas appear throughout nature. Under gravity alone the distance fallen is $y = \dfrac{1}{2}gt^2$ , proportional to the square of the time, and a thrown object traces a parabolic path².

Projectile motion under gravity
Parabolic dishes and reflecting telescopes, which gather parallel rays into the focus
The sum of squared errors minimized in least-squares fitting

For $y = x^2$ the focus is $\left(0, \dfrac{1}{4}\right)$ and the directrix is the line $y = -\dfrac{1}{4}$ .

Quadratic function, Wikipedia
Projectile motion, Wikipedia

Graph of the Quadratic Function y=x2y = x^2y=x2