Chapter 6.9: Rational Functions
Learning Objectives
In this section, you will:
- Use arrow notation.
- Solve applied problems involving rational functions.
- Find the domains of rational functions.
- Identify vertical asymptotes.
- Identify horizontal asymptotes.
- Graph rational functions.
Suppose we know that the cost of making a product is dependent on the number of items, produced. This is given by the equation If we want to know the average cost for producing items, we would divide the cost function by the number of items,
The average cost function, which yields the average cost per item for items produced, is
Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
Using Arrow Notation
We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in (Figure), and notice some of their features.
Several things are apparent if we examine the graph of
- On the left branch of the graph, the curve approaches the x-axis
- As the graph approaches from the left, the curve drops, but as we approach zero from the right, the curve rises.
- Finally, on the right branch of the graph, the curves approaches the x-axis
To summarize, we use arrow notation to show that or is approaching a particular value. See (Figure).
Symbol | Meaning |
---|---|
approaches from the left ( but close to ) | |
approaches from the right ( but close to ) | |
approaches infinity ( increases without bound) | |
approaches negative infinity ( decreases without bound) | |
the output approaches infinity (the output increases without bound) | |
the output approaches negative infinity (the output decreases without bound) | |
the output approaches |
Local Behavior of
Let’s begin by looking at the reciprocal function, We cannot divide by zero, which means the function is undefined at so zero is not in the domain. As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in (Figure).
–0.1 | –0.01 | –0.001 | –0.0001 | |
–10 | –100 | –1000 | –10,000 |
We write in arrow notation
As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in (Figure).
0.1 | 0.01 | 0.001 | 0.0001 | |
10 | 100 | 1000 | 10,000 |
We write in arrow notation
See (Figure).
This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line as the input becomes close to zero. See (Figure).
Vertical Asymptote
A vertical asymptote of a graph is a vertical line where the graph tends toward positive or negative infinity as the inputs approach We write
End Behavior of
As the values of approach infinity, the function values approach 0. As the values of approach negative infinity, the function values approach 0. See (Figure). Symbolically, using arrow notation
Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line See (Figure).
Horizontal Asymptote
A horizontal asymptote of a graph is a horizontal line where the graph approaches the line as the inputs increase or decrease without bound. We write
Using Arrow Notation
Use arrow notation to describe the end behavior and local behavior of the function graphed in (Figure).
Show Solution
Notice that the graph is showing a vertical asymptote at which tells us that the function is undefined at
And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at As the inputs increase without bound, the graph levels off at 4.
Try It
Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.
Show Solution
End behavior: as Local behavior: as (there are no x– or y-intercepts)
Using Transformations to Graph a Rational Function
Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.
Show Solution
Shifting the graph left 2 and up 3 would result in the function
or equivalently, by giving the terms a common denominator,
The graph of the shifted function is displayed in (Figure).
Notice that this function is undefined at and the graph also is showing a vertical asymptote at
As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at
Analysis
Notice that horizontal and vertical asymptotes are shifted left 2 and up 3 along with the function.
Try It
Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.
Show Solution
The function and the asymptotes are shifted 3 units right and 4 units down. As and as
The function is