In this explainer, we will learn how to graph rational functions whose denominators are linear, determine the types of their asymptotes, and describe their end behaviors.
These are the functions , where , and are constants.
In order for to truly represent a rational function, we need to have . The easiest such example is when and , the reciprocal function. The figure shows the curve .
Geometrically, this is a hyperbola. Notice the two asymptotes:
- the horizontal asymptote: the line which the graph approaches as tends to and ;
- the vertical asymptote: the line which represents the value 0, missing from the domain of the function , and which is such that the curve approaches this line as tends to 0 either from above or from below.
Consider the curve
which is the same as
We will use both equations to examine what the graph is like:
- The denominator in both forms tells us that the function is not defined at . This locates the vertical asymptote.
- The form in (1) tells us that when is very large (positive or negative), the term is very small so that . This locates the horizontal asymptote as the line .
The following figure shows the curve together with its two asymptotes.
Notice that this curve is a hyperbola, which looks very much like . If we write (1) as rather than as , it is really much more like . Better still, . The negative multiple corresponds to the curve lying in the 2nd and 4th quadrants.
Perhaps the easiest way to decide is to ask, “As is going to infinity, is the curve above or below the horizontal asymptote?,” which we can answer with a little algebra, using the form in (2):
From this, we see the following.
- If is very big and positive (think ), then and is positive, while is also positive and close to 3. Therefore, is tending to 3 as is going to infinity, and the horizontal asymptote .
- How does the value
of at compare with 3? Is it greater than or
less than 3? Since , it follows that
The curve lies below the asymptote.
In summary, suppose we are to identify the curve .
Relating the Graph to the Expression 𝑎𝑥 + 𝑏/𝑐𝑥 + 𝑑 Where 𝑥 ≠ 0
- From , we see that the vertical asymptote is at .
- From , we see that the horizontal asymptote is .
- By considering the expression above for large positive values of , we decide whether the curve is above or below the asymptote as tends to .
Example 1: Identifying the Graphs of Simple Rational Functions
Which of the following graphs represents ?
From the denominator , we find the vertical asymptote must be at . Hence, graph (d) can not be the answer. The horizontal comes from dividing through by : which goes to when is large. Since these numbers are also positive when is large and positive, the curve must lie above the -axis. The graph (c) represents .
The only option is (c).
Suppose that, instead, we wanted to go in the other direction. We are given a graph, and we want to know what rational function it is the graph of. Assuming that it “comes from” the function , we have two ways to identify the function:
- by considering transformations of graphs: translations, dilations, and reflections in the axes,
- by considering the algebraic form and determining the coefficients.
As the second method is more applicable generally, let us have a look at it.
- Notice that we can always write in the form by dividing the numerator and the denominator by , which is not zero since the graph is a hyperbola, not a line. So we assume that .
- We know what is because the denominator is zero when , so that is where the vertical asymptote is found.
- We also know that, in the form above, the horizontal asymptote occurs at since, dividing throughout by , we have so the graph tells us what must be also.
- We are left with one additional piece of data which the graph must give us: the value of . One way to get this is by identifying a single point on the graph, be it the intercept with an axis or any other clearly identifiable point. This is because from we get the equation in which the only unknown is . Solve for and we are done.
Example 2: Determining the Equation of a Function from Its Graph
What function is represented in the figure below?
The vertical asymptote is , so the form of the function is with constant .
The horizontal asymptote is the line , which fixes , and the rational expression is now
We can see that the point is on the graph, so we solve for in the equation
The function graphed is