For graphing higher-power
polynomials, you will of course need plenty of points. The best points
are the intercepts;
to find these, use all the factoring tools
that you have. Then also pick some x's
between the x-intercepts,
and plot. If you keep in mind the end-behavior
of polynomials, then these graphs can actually be not too hard to do.
For example, let y
= x^{4} –
13x^{2} + 36. This is
a positive even power ("to the fourth"), so the graph will be
up on both ends (like the quadratic above). Factoring the polynomial,
we get y = (x
+ 3)(x –
3)(x + 2)(x –
2), so the zeroes (x-intercepts)
are –3, –2,
2, and 3. If you plot
a few other points on your T-chart, it will be no trouble to graph this:

The following is often
what happens, if the student is careless:

Okay
T-chart

Incorrect
graph

By not paying attention
to the domain and by not plotted negative x-values,
the student fooled himself into thinking that this graphed as a straight
line. Instead, you want to do the graph like this:

Better
T-chart

Correct
graph

Note that radical-function
graphs are generally curvey like this; they are not straight lines. (For
further information, please study the lesson on "Graphing
Radical Functions".)

Before you even draw
a T-chart for a rational function, you first have to find the asymptotes
and intercepts.
Once you have successfully done that, you can then choose x's
between the x-intercepts
and vertical asymptotes, to give you the additional information necessary
to graph the function.

Actually, as bad as these
functions look, they are quite easy to graph. For instance, suppose you
have:

From what you've learned
about rational functions, you know that the vertical
asymptotes are
at x
= 2 and x = –2,
the horizontal
asymptote is at
y
= 2, the x-intercepts
are at x
= –3 and x
= 3, and the y-intercept
is at y
= 4.5. Now plot a few
additional points between these other points:

T-chart

Graph

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(Remember that horizontal
asymptotes
are just "suggestions" off to the sides; they mean next to nothing
in the "middle", and you're quite welcome to cross them.)

How did I know which way
to go at the vertical asymptotes? Go back and look at the x-intercepts
we had. We can only cross the x-axis
at an intercept; therefore, if there is no intercept, then there is no
crossing of the axis. So, on the left, we knew the graph traced along
the horizontal asymptote, came down to cross at x
= –3, and then stayed
down, because there was no place to cross to get back up. In the middle,
there were no x-intercepts,
but there were points above the x-axis,
so the graph was always above. On the right, the graph works the same
as it did on the left. (Occasionally the graph just touches the x-axis
at an intercept, instead of going though.That's why we checked points
between the x-intercepts
and the vertical asymptotes.) (For further information, please study the
lesson on "Graphing
Rational Functions".)

Since piecewise
functions are defined
in pieces, then you have to graph them in pieces, too. For instance, suppose
you have:

Since this has two pieces,
you may find it helpful to do two T-charts; if it had more pieces, you
could do more T-charts. The break between the two "halves" of
the function (the point at which the function changes rules) is at x
= 1, so that is where
your T-charts will break. The procedure looks like this:

T-chart
1

T-chart
2

Graph

Why did I list that last
point in T-chart 1 in parentheses? Technically speaking, x
= 1 does not belong
on that chart, but it is often helpful to know where the function "almost"
is at the end. That's why, on the graph, I drew that point as an open
circle, meaning that the graph is everything up to, but not including,
that point. This can be especially important when, as in this case, the
pieces of the function don't join up at the ends.