Graphing Square Root Functions

Graphing Square Root Functions

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

Graphs

The square root function is the inverse of the quadratic function (x >= 0).

y = √x is the inverse function of y = x2 (x ≥ 0).

So the graphs of y = √x and y = x2 (x ≥ 0)
show the reflection in the line y = x.

Reflection in the line y = x

To prove this,
find the inverse of y = x2 (x ≥ 0).

Switch x and y:
x = y2 (y ≥ 0)

Then y = √x.

± is removed because y ≥ 0.
So y = [right side] ≥ 0.

The graph of square root functions depend on the location of the (-) signs.

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

Graph the given function. y = square root (2x - 6) + 1

Find the starting point of the graph.

Factor the radicand:
2x - 6 = 2(x - 3).

Then the starting point is (3, 1).

Starting from (3, 1),
draw y = √2x.

y = √2(x - 3) + 1 is the image of y = √2x
under the translation (x, y) → (x + 3, y + 1).

Translation of a function

Example 2

Graph the given function. y = square root (-x + 3)

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