Radical Functions: More Examples (page
3 of 3)
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
is positive between –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
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
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:
is positive (higher than the x-axis)
= 0 and after
= 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
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
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
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.