Graphs of Rational Functions Name:
Algebra 2/Trigonometry Period:
Mr. Dien’s Guide to Graphing Rational Functions
A rational function is a quotient of two polynomial functions. It has the form where . Before we can sketch/graph the rational function we need to do some ground work drawing up boundaries so see how the function behaves. These include, zeros (crosses the x axis) of the rational function point discontinuities (holes), vertical and horizontal asymptotes.
Zeros (where the function crosses the x-axis)
Zero’s occur when there are x (domain) values that cause the numerator to equal zero (as well as the whole function at that point).
Example. 1 Find the zeros then graph the function on your calculator to see the rest of the function.
Sketch the function.
Practice 1 :Find the zeros of ,
then using your graphing calculator sketch the graph.
(Hint: Factor the numerator)
Point Discontinuities (Holes)
Point Discontinuities occur when the rational function has variable factor in the numerator that reduces with a factor in the denominator such as for any x other than , the function’s factors will reduce to . To graph the function we use the reduced form, but when the graph gets to we will run into the case of which is undefined. So at that point we use an open dot. You can have multiple holes for a function.
Example. 2 Graph function
What will the reduced function equation look like?
(The reduced form is the one you graph)
At what x will the hole lie?
Then state the coordinates of the point discontinuity.
Practice 2: Graph the function
What will the reduced function equation look like?
(The reduced form is the one you graph)
At what x’s will the hole lie?
Then state the coordinates of the point discontinuity.
Vertical Asymptotes
These appear when there are values for domain x that will make the denominator zero. Usually this is looks like a dotted vertical line. Now the graph can get close to this line and shoots up to either positive or negative infinity depending from which direction you are approaching the x that causes the asymptote. The function will never cross a vertical asymptote.
Example. 3
Graph the vertical asymptote for and show how the function behaves around that asymptote.
1. There is a vertical asymptote at
2. As from the left side side (such as )
then a small negative number, therefore
3. As from the right side (such as )
then a small positive number, therefore
Practice 3
Graph the asymptotes and sketch the functions behavior near the
vertical asymptotes.
Sometimes the numerator can play a factor in the sign of the
function as it approaches the asymptote. As it approaches the
asymptote one factor at the bottom gets really small but what
are the signs of the other factors?
Horizontal Asymptotes
A Horizontal Asymptote to a function is like what the Pirate’s Code is in the “Pirates of the Caribbean”, its more of a guideline than a rigid rule, in the sense that the a horizontal
asymptote is a line or a curve that the function approaches for large values of x. For small x, the horizontal asymptote has nothing much to do with the curve.
In general the way to find horizontal asymptotes can be broken down to three simple rules. To figure them out you just need to think about what happens to the functions as x approaches infinity and negative infinity (source: Doctor Pat, The Math Forum)
a) the highest power is in the numerator ---> the curve diverges to
plus/minus infinity and there is no horizontal asymptote. (see slant asymptote section for cases when the greatest power of the numerator is exactly one bigger than greatest power for the denominator.)
Ex.
b) the highest power is in the denominator ---> the curve converges
to zero as a horizontal asymptote
Ex.
c) both the numerator and the denominator have the same power --->
the curve converges and has an asymptote at a value L where L is
the simplified ratio of the coefficients of the highest power
terms.
Ex.
Example 4: Find the horizontal asymptotes for the following functions
a. b. c.
Slant Asymptote
The slant asymptote is a special kind of asymptote that occurs when the greatest power of the numerator is exactly one bigger than greatest power for the denominator. To figure out what the equation of the asymptote you have to long divide and then see what happens when
Example 5: Find the equation of the slant asymptote
It has a asymptote by looking at the highest degree term in the numerator and denominator. Now we do long division…
Which gives us and as
it will become, so the equation of the slant
asymptote is .
(Note there is still a vertical asymptote at .)
Practice 5:
a. Find the slant asymptote for
b. Find the slant asymptote for