Graphing
Radical Functions: More Examples (page
3 of 3)

Graph

First
I'll find the domain: Again, the contents of the square-root function
are a quadratic, so it may be simplest to find the domain of the
radical function by looking at the graph of the quadratic inside
the square root.

In this case, 16
– x^{2}
is positive between –4
and 4,
so this will be the domain of the radical.

Next, I'll find
some additional plot points:

Note: I used a
calculator to approximate the y-values.

Finally, I'll do
the graph:

This graph is just what
it looks like: the top half of a circle. As a matter of fact, it's the
top half of the circle centered at the origin and having radius
r = 4. The negative
of this square-root function would give you the bottom half of the same
circle.

First, I'll find
the domain: it may be simplest to determine the domain of the
radical function by looking at the graph of the quadratic argument
of the function:

This quadratic
is positive (higher than the x-axis)
before x
= 0 and after
x
= 4. In other
words, the domain of the radical is split into TWO pieces. This
means that the graph of the radical function will also be in two
pieces: one part on the left, stopping at x
= 0, and another
part on the right, starting at x
= 4. There
will be nothing but blank space between these two pieces.

Keeping this domain
restriction in mind, I'll carefully find some plot-points:

Finally, I'll do the graph:

Here's an example of a
cube-root function:

Graph

There are no domain constraints
with a cube root, because you can
graph the cube root of a negative number. So you don't have to find the
domain; the domain is "all x".
(Note: Since you can take the fifth root, seventh root, ninth root, etc.,
of negative numbers, there are no domain considerations for any odd-index
radical function. You only have to find the domain whenever you are dealing
with even-index
radical functions: a
square root, a fourth root, a sixth root, etc.)

There are no domain
constraints, so I'll go straight to finding some plot points:

Note that you can
find the x-values
that give "neat" y-values
by setting the argument of the cube root equal to a perfect cube,
such as 1,
8, or 27.

Warning: Radicals graph
as curved lines. Don't succumb to the temptation of trying to put a straight
line through these points. Instead, use enough plot-points to clearly
show the shape of the graph, and then draw the graph complete with its
curves:

For radical graphs, it
is well worth taking the time to find lots of plot-points for your T-chart.
Then draw very neat axes and scales, and draw your line carefully. Don't
just slap these graphs together, because if you do, you'll probably get
many of them wrong.