Square Root Function: Graph

How to graph the given square root function: graph, formula, 3 examples, and their solutions.

Graph

Graph

The graph of y = √x looks like this.

The starting point is the origin (0, 0).
It increase toward the quadrant I.

Coordinate Plane

The graph of y = √-x looks like this.

The coefficient of x is (-).

Then, to make the part inside the square root (+),
x should be minus.
(x ≤ 0)

So the graph increases
toward the quadrant II.

The graph of y = -√x looks like this.

The coefficient of the square root is (-).

Then the left side, y, is minus.
(y ≤ 0)

So the graph decreases
toward the quadrant IV.

The graph of y = -√-x looks like this.

Both x and y are (-).
(x ≤ 0, y ≤ 0)

So the graph decreases
toward the quadrant III.

As you can see,
the location of the minus sign
determines the direction of the graph.

Formula

Formula

For a square root function
y = √a(x - h) + k,

the starting point is
(h, k).

Example 1

Example

Solution

2x - 6 = 2(x - 3)

Common Monomial Factor

y = √2(x - 3) + 1

The starting point is (3, 1).

So draw the starting point (3, 1)
on the coordinate plane.

The coefficient of x is 2.
It's (+).

The coefficient of the square root is (+).

So the graph increases
toward the quadrant I.

So this is the graph of
y = √2(x - 3) + 1.

The graph covers x ≥ 3.
So the domain is {x|x ≥ 3}.

The graph covers y ≥ 1.
So the range is {y|y ≥ 1}.

Example 2

Example

Solution

-x + 4 = -(x - 4)

y = √-(x - 4)

The starting point is (4, 0).

So draw the starting point (4, 0)
on the coordinate plane.

The coefficient of x is -1.
It's (-).

The coefficient of the square root is (+).

So the graph increases
toward the quadrant II.

So this is the graph of
y = √-(x - 4).

The graph covers x ≤ 4.
So the domain is {x|x ≤ 4}.

The graph covers y ≥ 0.
So the range is {y|y ≥ 0}.

Example 3

Example

Solution

y = -√x + 3

The starting point is (0, 3).

So draw the starting point (0, 3)
on the coordinate plane.

The coefficient of x is 1.
It's (+).

The coefficient of the square root is (-).

So the graph decreases
toward the quadrant IV.

So this is the graph of
y = -√x + 3.

The graph covers x ≥ 0.
So the domain is {x|x ≥ 0}.

The graph covers y ≤ 3.
So the range is {y|y ≤ 3}.