Just like linear equations have a basic form, $y=x$, radical functions have parent functions. These are the simplest versions from which all others are transformed.

The parent square root function is $f(x) = \sqrt{x}$. Its graph starts at the origin $(0,0)$ and gently curves upward as it moves to the right. Notice that the domain is limited: you can't take the square root of a negative number, so $x$ must be greater than or equal to zero.

The other key parent is the cube root function, $f(x) = \sqrt[3]{x}$. Unlike the square root, you can take the cube root of any real number, positive or negative. Its graph is an 'S' shape that passes through the origin and extends infinitely in both directions.

We can shift, stretch, compress, and reflect these parent functions using a single equation form. For the square root function, this blueprint is:

The point $(h, k)$ is the new starting point for the square root function, or the point of inflection for the cube root function. Let's break down what each parameter does.

Reflections are controlled by negative signs.

A negative sign on $a$ (outside the radical) reflects the graph vertically across the x-axis. A negative sign on $b$ (inside the radical) reflects the graph horizontally across the y-axis.

The absolute values of $a$ and $b$ determine how much the graph is stretched or compressed.

Vertical Stretches & Compressions ($a$):

• If $|a| > 1$, the graph is stretched vertically, making it appear steeper.
• If $0 < |a| < 1$, the graph is compressed vertically, making it appear flatter.

Horizontal Stretches & Compressions ($b$): This is also counter-intuitive, like the horizontal shift.

• If $|b| > 1$, the graph is compressed horizontally (squished inward).
• If $0 < |b| < 1$, the graph is stretched horizontally (pulled outward).

You can reverse the process to find a function's equation by looking at its graph. Here's the method:

1. Identify the vertex $(h, k)$. This is the starting point of the square root graph or the center point of the cube root graph. This gives you $h$ and $k$.

2. Determine reflections. Does the graph go up or down from the vertex? Left or right? This tells you the signs of $a$ and $b$.

3. Find the stretch/compression factor. Pick another clean point $(x, y)$ on the graph. Plug $x$, $y$, $h$, and $k$ into the equation $y = a\sqrt{x-h} + k$ (or the appropriate reflected version) and solve for $a$. For simplicity, we often assume $b=1$ unless there's a clear horizontal stretch.

Let's find the equation for the graph above.

1. The vertex is at $(-2, 1)$, so $h = -2$ and $k = 1$.

2. The graph opens down (so $a$ is negative) and to the right (so $b$ is positive). We'll assume $b=1$.

3. Our equation is $y = a\sqrt{x - (-2)} + 1$, or $y = a\sqrt{x+2} + 1$. We use the point $(2, -1)$ to find $a$.

So, the equation for the graph is $f(x) = -\sqrt{x+2} + 1$.

Understanding these transformations provides a powerful toolkit for analyzing and creating functions.