Animated Solution for Mathematics - Functions: Draw the graph of $y=|x{|}^{1/2}$ for $-1\le x\le 1$.

Imagine you are standing on the Cartesian plane, ready to map out a function that seems simple but hides a beautiful geometric secret. We are tasked with graphing $y=|x{|}^{\frac{1}{2}}$ within the domain $x\in [-1,1]$.

This is not just a line; it is a journey into the heart of symmetry and piecewise behavior.

The Origin

Every great journey starts with a single step, and for this function, that step is the origin. When we test $x=0$, the math becomes incredibly elegant:

This confirms that our graph is anchored firmly at the point $(0,0)$. This point acts as the pivot for our entire construction.

Breaking the Barrier

The modulus function $|x|$ is the gatekeeper of this problem. It forces us to split our world into two distinct realities.

By definition, $|x|=x$ when $x\ge 0$, and $|x|=-x$ when $x<0$. We cannot treat this as one monolithic equation; we must respect the boundary at the y-axis.

The Parabolic Ascent

For the region where $x\ge 0$, our function simplifies beautifully to $y=x^{\frac{1}{2}}$, or $y=\sqrt{x}$. If you have spent time with conic sections, you will recognize this instantly.

It is the upper half of a parabola that opens to the right. As $x$ moves from $0$ to $1$, $y$ climbs from $0$ to $1$. It is a smooth, accelerating curve that defines the right side of our graph.

The Mirror Image

Now, we cross the y-axis into the territory where $x<0$. Here, the modulus demands that $|x|=-x$. Our function becomes $y=(-x{)}^{\frac{1}{2}}$, or $y=\sqrt{-x}$.

Do not be alarmed by the negative sign! Since $x$ is negative, $-x$ is positive, keeping our output firmly in the realm of real numbers. At $x=-1$, we find:

This creates a mirror image of our right-wing parabola, opening to the left.

When we step back and look at the two arcs meeting at the origin, we see a perfect balance. Mathematically, we have proven this is an Even Function because $f(-x)=f(x)$.

This symmetry is not just a visual treat; it is a fundamental property that makes the graph a mirror image across the y-axis. You have successfully navigated the piecewise definition, visualized the parabolic arcs, and understood the symmetry.

You have not just drawn a graph; you have understood the soul of the function.