Rational Function Study
By: Amanda Sawyer
We will
study the formula:
To
study this equation, let us look at three separate situations. When we allow a = 3 and all other
valuables to equal one, we get the following graph.
When we allow all values to equal one, the equation becomes y
= 1 and we get the following graph.
When we allow value of c = 0 and all other values are one, the equation becomes a polynomial, and we can see this in the following graph.
From
the equation and its graph, we can clearly see that it can be
either a rational function, a polynomial, or the function y =
constant. To study this function,
we will look at its vertical asymptote, horizontal, asymptote, slant asymptote,
domain, range, areas of increasing and decreasing,
areas of concavity, xintercepts, and yintercepts. First letŐs look for the asymptotes.
1) Vertical Asymptote
To find
the vertical asymptote, we first need to simplify the equation and then set the
denominator equal to zero. This
value will become the vertical asymptote. If we assume that the numerator is
not a multiple of the denominator, then the vertical asymptote will be:
Otherwise,
we donŐt have a vertical asymptote, and our graph is a straight line or a
polynomial. We can observe
this behavior in the animation below where the value of c is varied from 5 to
5. In the graph below you will see the vertical asymptote in red from the
equation above and the general function in purple.
2) Horizontal Asymptote and
Slant Asymptotes
If a or c are not equal to 0, we will always have the same
degree on our numerator as well as our denominator. This allows us to easily calculate the Horizontal
Asymptote. We find this value by
taking the ratio of the leading coefficient of our numerator and
denominator. Our leading
coefficient of our numerator is a and the leading coefficient of our denominator is c, thus
the horizontal asymptote is:
If c = 0, we know that our graph is a polynomial, and it does not have any asymptotes. If a = 0, we know that our horizonal asymptote is y = 0.
Since
the degree of your numerator will always be the same as our denominator, we
will not have a slant asymptote.
In the
graph below you will see the horizontal asymptote in red from the equation
above and the general function in purple.
This helps to
determine our graphs Domain and Range.
3) Domain
If we
assume that the numerator is not a multiple of the denominator or c not equal
to 0, then graph has a vertical asymptote. From that information we can find the intervals for which
our domain exist by excluding those points. Therefore we can define the
interval to be:
Otherwise,
our graph will have a domain of all real numbers.
4) Range
We can
also determine the range of our graphs since we defined our horizontal
asymptote. When the numerator is a multiple of the denominator or c is not
equal to 0, its range will be equal to the single value of the multiple. Yet, if that situation does not occur,
we can find the range using its vertical asymptote. This equation gives us the
range of:
Next
letŐs study when our graph is increases and decreases:
5) Areas of Increasing and
Decreasing
We were
taught in calculus, that we could determine when a graph increases and
decreases by taking the derivative of our function. The derivative of our function is:
We can
tell from this derivative that our whole function will be increasing when our
numerator is positive, and our whole function will be decreasing when our
numerator is less than 0. When the
numerator is equal to zero, we have two different cases:
1) a =
c and d = b or a = b and c = d, then we will have a straight line.
2) If , the graph is increasing.
3) If , the graph is decreasing.
This
can be observed in the graphs below.
Equations 
Graph 


Now
letŐs study the concavity of our function:
6) Areas of Concavity
We can
determine the concavity of our function using the second derivative. When the values are positive in our
second derivative, the graph is concave up. When the values are negative in our second derivative, the
graph is concaved down. Our second
derivatives is:
This
gives us different cases where our graph is concave up and concave down.
If
a) for some value of x such that then our graph
is concave down.
b) for some value of x such that then our graph
is concave up.
If
a) for some value of x such that then our graph
is concave up.
b) for some value of x such that then our graph is
concave down.
Now,
letŐs investigate what happens to our xintercepts and yintercepts.
7) XIntercept
When
our function simplifies to a constant, then it does not have an
xintercept. Otherwise, the graph
has an xintercept when its numerator is equal to 0. We can find this value by setting the numerator equal to
zero and solving for its variable:
In the
graph below you will see the xintercept represented as the line in red from
the equation above and the general function in purple.
8) YIntercept
We can
find they yintercept for any of the functions by setting my x values equal to
0 and solving.
This
value gives us the yintercept.
We have now been
able to explore this equation and each of its parts using Graphing
Calculator. Through the use of its
utilities, we are able to show how each of our original graphs relates to its
different elements. This tool
makes it easier to check our solutions without having to manually create each
of our equations.