# Graphing Square Root Functions

How to graph square root functions: their shapes, examples, and their solutions.

## Graphs

*y* = √*x* is the inverse function of *y* = *x*^{2} (*x* ≥ 0).

So the graphs of *y* = √*x* and *y* = *x*^{2} (*x* ≥ 0)

show the reflection in the line *y* = *x*.

Reflection in the line *y* = *x*

To prove this,

find the inverse of *y* = *x*^{2} (*x* ≥ 0).

Switch *x* and *y*:*x* = *y*^{2} (*y* ≥ 0)

Then *y* = √*x*.

± is removed because *y* ≥ 0.

So *y* = [right side] ≥ 0.

The graphs of square root functions

depend on the location of (-) signs.*y* = √*x*:*x* ≥ 0

(*x* is the radicand.)*y* ≥ 0

(*y* = [right side, +] ≥ 0)

*y* = √-*x*:*x* ≤ 0

(-*x* is the radicand, -*x* ≥ 0, *x* ≤ 0.)*y* ≥ 0

(*y* = [right side, +] ≥ 0)

It shows the reflection in the *y*-axis.

*y* = -√-*x*:*x* ≤ 0

(-*x* is the radicand, -*x* ≥ 0, *x* ≤ 0.)*y* ≤ 0

(*y* = [right side, -] ≤ 0)

It shows the reflection in the origin.

*y* = -√*x*:*x* ≥ 0

(*x* is the radicand.)*y* ≤ 0

(*y* = [right side, -] ≤ 0)

It shows the reflection in the *x*-axis.

## Example 1

Find the starting point of the graph.

Factor the radicand:

2*x* - 6 = 2(*x* - 3).

Then the starting point is (3, 1).

Starting from (3, 1),

draw *y* = √2*x*.*y* = √2(*x* - 3) + 1 is the image of *y* = √2*x*

under the translation (*x*, *y*) → (*x* + 3, *y* + 1).

Translation of a function

## Example 2

Find the starting point of the graph.

Factor the radicand:

-*x* + 3 = -(*x* - 3).

Then the starting point is (3, 0).

Starting from (3, 0),

draw *y* = √-*x*.*y* = √-(*x* - 3) is the image of *y* = √-*x*

under the translation (*x*, *y*) → (*x* + 3, *y* + 0).

Translation of a function