# Graphing Square Root Functions

How to graph square root functions on a coordinate plane: their shapes, examples, and their solutions.

## Graph: *y* = √*x*

This is the graph of *y* = √*x*.

It starts from (0, 0).

And the graph

[increases] to the [right]

smoother and smoother.

## Graph: *y* = √*a*(*x* - *h*) + *k*

This is the graph of *y* = √*a*(*x* - *h*) + *k*.

It starts from (*h*, *k*).

To see why this is true,

move +*k* to the left side.

Then [*y* - *k* = √*a*(*x* - *h*)].

[*y* - *k* = √*a*(*x* - *h*)] is the image of [*y* = √*a**x*]

whose image is under the translation

(*x*, *y*) → (*x* + *h*, *y* + *k*).

Translation of a function

The starting point of [*y* = √*a**x*] is (0, 0).

By the translation (*x*, *y*) → (*x* + *h*, *y* + *k*),

the starting point becomes (*h*, *k*).

So the starting point of [*y* - *k* = √*a*(*x* - *h*)],

which is [*y* = √*a*(*x* - *h*) + *k*],

is (*h*, *k*).

## Graphs: *y* = √-*x*, *y* = -√*x*, *y* = -√-*x*

There are three other types of radical functions.

The shapes of the graphs are the same as [*y* = √*x*].

But, because of the (-) signs,

the directions of the graphs are different.

*y* = √-*x*

The coefficient of the square root is (+).

The coefficient of *x* is (-).

So the graph

[increases] to the [left]

smoother and smoother.

*y* = -√*x*

The coefficient of the square root is (-).

The coefficient of *x* is (+).

So the graph

[decreases] to the [right]

smoother and smoother.

*y* = -√-*x*

The coefficient of the square root is (-).

The coefficient of *x* is (-).

So the graph

[decreases] to the [left]

smoother and smoother.

## Example 1: Graph *y* = √2*x* - 6 + 1

To find the starting point,

factor 2*x* - 6.

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

So *y* = √2(*x* - 3) + 1.

So the starting point is (3, 1).

Draw [*y* = √2(*x* - 3) + 1]

on the coordinate plane.

Point the starting point (3, 1).

The coefficient of the square root is (+).

The coefficient of *x* is 2: (+).

So the graph

[increases] to the [right]

smoother and smoother.

## Example 2: Graph *y* = √-*x* + 4

To find the starting point,

factor -*x* + 4.

-*x* + 4 = -(*x* - 4)

So *y* = √-(*x* - 4).

So the starting point is (4, 0).

Draw [*y* = √-(*x* - 4)]

on the coordinate plane.

Point the starting point (4, 0).

The coefficient of the square root is (+).

The coefficient of *x* is -1: (-).

So the graph

[increases] to the [left]

smoother and smoother.