Graphing Square Root Functions

Graphing Square Root Functions

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

Graph: y = √x

Graph: y = Square root [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

y = Square root [a(x - h)] + k starts from (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 = √ax]
whose image is under the translation
(x, y) → (x + h, y + k).

Translation of a function

The starting point of [y = √ax] 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

Graphs: y = Square root [-x], y = -Square root [x], y = -Square root [-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 = √2x - 6 + 1

Graph the given function. y = Square root [2x - 6] + 1

To find the starting point,
factor 2x - 6.

2x - 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

Graph the given function. y = Square root [-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.