Module 5: Graphing Rational Functions



Section IV: Power, Polynomial, and Rational Functions

[pic]

Module 5: Graphing Rational Functions

In the previous module we studied the long-run behavior of rational functions. Now we will study the short-run behavior so that we can sketch the complete graphs of rational functions. Like polynomial functions, the short-term behavior of rational functions includes roots and y-intercepts. In addition, we must also pay attention to the values that are excluded from the domains of rational functions.

To get started, let’s consider some simple rational functions. In the next two examples we will sketch graphs of [pic] and [pic]. The reason to look at these examples is they are very similar (both have horizontal asymptote [pic] and domain [pic]). The only difference between these functions is that the factor of x in the denominator is squared in g but not in f. So these examples will help us determine how the degrees of the factors in the denominator affect the shape of the graphs of rational functions.

[pic]

[pic] example: Sketch a graph of the rational function [pic].

SOLUTION:

Let’s first establish the long-run behavior. Clearly, as [pic], [pic] gets closer and closer to zero, so [pic] is the horizontal asymptote. When [pic], [pic] is positive, so the graph is above the horizontal asymptote [pic], but when [pic], [pic] is negative, so the graph is below the horizontal asymptote [pic]. This information leads to the following graph of the long-run behavior:

|[pic] |

|Figure 1: The long-run behavior of [pic]. |

For the short-run behavior, we need to find the roots and the y-intercepts – but this function has NO roots and NO y-intercept! (There is no y-intercept since 0 is not in the domain of f, and there are no roots since [pic] for all x.) So the only other thing to look for are values that must be excluded from the domain of f. Obviously, 0 must be excluded from the domain of f. Let’s investigate what happens to the graph of f as x gets closer and closer to 0. In Table 1 we’ll see what happens as x gets close to zero from the right side of zero (i.e., from above 0) and in Table 2 we’ll see what happens as x gets close to zero from the left side of zero (i.e., from below 0).

|Table 1 | |Table 2 |

|x |[pic] | |x |[pic] |

|[pic] |10 | |[pic] |–10 |

|[pic] |100 | |[pic] |–100 |

|[pic] |1000 | |[pic] |–1000 |

|[pic] |10000 | |[pic] |–10000 |

|[pic] |100000 | |[pic] |–100000 |

We can see in Table 1 that as x gets close to 0 from the right side of 0, [pic] gets bigger and bigger (so f increases without bound). From Table 2 we can see that as x gets close to 0 from the left side of 0, [pic] gets smaller and smaller (so f decreases without bound). We can use this information to sketch the complete graph of f ; see Figure 2.

[pic]

Figure 2: The graph of [pic].

The line [pic] (i.e., the y-axis) is an important feature in the graph of f : the graph gets closer and closer to this line but never crosses it. This vertical line is an asymptote. Since it is a vertical line, it is called a vertical asymptote.

|[pic]DEFINITION: A vertical asymptote is a vertical line that the graph of a function never crosses. As the graph of a function approaches |

|this vertical line, the outputs either increase without bound or decrease without bound. |

[pic] example: Sketch a graph of the rational function [pic].

SOLUTION:

Just as with [pic] in the example above, [pic] gets closer and closer to 0 as [pic], so [pic] is the horizontal asymptote. But unlike f, as [pic], [pic] is always positive, so the graph is always above the line [pic]. The long-run behavior of g is graphed in Figure 3.

|[pic] |

|Figure 3: The long-run behavior of [pic]. |

Just like [pic], the function [pic] has no roots and no y-intercept. And since g isn’t defined at 0 it probably has vertical asymptote [pic] just like f. Let’s study at a couple of appropriate tables to make sure.

|Table 3 | |Table 4 |

|x |[pic] | |x |[pic] |

|[pic] |100 | |[pic] |100 |

|[pic] |10000 | |[pic] |10000 |

|[pic] |1000000 | |[pic] |1000000 |

|[pic] |100000000 | |[pic] |100000000 |

Tables 3 and 4 suggest that [pic] does in fact have a vertical asymptote at [pic]. But unlike [pic], g increases without bound on both the left and right side of 0. We can use this information to sketch a graph of g in Figure 4 below.

|[pic] |

|Figure 4: The graph of [pic]. |

[pic]

Before we move on, let’s observe what we can learn from these last two examples. Both [pic] and [pic] have vertical asymptotes at [pic] (this is because x is a factor in the denominator). In function f, x s a linear factor and the behavior at the vertical asymptote [pic] is different on the left and right sides of 0, while in function g, x is a quadratic factor (i.e., it is squared) and the behavior is the same on both sides of 0. It turns out that this is always the case: if the factor is squared then the behavior is the same on both sides of the asymptote but if the factor is linear, the behavior is different on both sides of the asymptote. (You’ve probably already realized that if the factor has any even power, the behaviors will be the same on both sides of the asymptote, and if the factor has an odd power, the behaviors will be different on both sides of the asymptote.)

[pic] example: Sketch the graph of the function [pic].

SOLUTION:

In the previous module we determined that the long-run behavior for h; see Figure 5 below.

|[pic] |

|Figure 5: The long-term behavior of [pic]. and its horizontal |

|asymptote [pic]. |

To sketch the short-run behavior of h, we need to find the roots, y-intercept, and the vertical asymptotes. To find the roots, we need to find all x such that [pic]. Of course, the only way that [pic] is if the numerator of h is 0. So we need to solve [pic]:

[pic]

Thus, 2 and –2 are roots.

To find the y-intercept, we need to evaluate [pic]:

[pic]

So the y-intercept is [pic].

To find the vertical asymptotes, we need to determine what values make the denominator 0, so we need to solve [pic]. It’s easy to see that the only solution is 1. Therefore, the only vertical asymptote is the line [pic]. Since the factor in the denominator that gives

us this vertical asymptote is a quadratic factor (i.e., it is squared), we know that the graph has the same behavior on both sides of the vertical asymptote. When we plot all of the information we’ve determined thus far (see Figure 6), we can see that in order to avoid contradicting what we’ve concluded about the roots and the horizontal asymptote, the only possibility is that the graph decreases without bound on both sides of the vertical asymptote.

|[pic] |

|Figure 6: The long- and short-run behavior of [pic]. |

Guided by the roots, y-intercept, vertical and horizontal asymptotes, we can sketch the complete graph of h; see Figure 7.

|[pic] |

|Figure 7: The graph of [pic], along with asymptotes [pic] and |

|[pic]. |

Be sure to convince yourself that based on what we discovered about the long-term behavior and the roots and vertical asymptote, the graph of [pic] MUST look like the one pictured in Figure 7. You should also graph this function yourself on your graphing calculator for practice.

Notice that we were forced to cross the horizontal asymptote at [pic] in order to have the appropriate long-term behavior as well as the root [pic].

|[pic] Key Point: It is perfectly reasonable for the graph of a rational function to cross its horizontal asymptote when the x-values |

|aren’t very large. |

[pic] example: Sketch the graph of the function [pic].

SOLUTION:

In the previous module we determined that the long-run behavior for k; see Figure 8.

|[pic] |

|Figure 8: The long-run behavior of [pic]. |

To sketch the short-run behavior of k, we need to find the roots, y-intercept, and the vertical asymptotes. To find the roots, we need to find all x such that [pic]. Of course, the only way that [pic] is if the numerator of k is 0, which obviously occurs when [pic], so 0 is the only root.

To find the y-intercept, we need to evaluate [pic]. Since 0 is a root, we know that [pic], so the y-intercept is [pic]. (So the x-intercept and the y-intercept are the same point.)

To find the vertical asymptotes, we need to determine what values make the denominator 0, so we need to solve [pic]:

[pic]

Therefore, the lines [pic] and [pic] are the vertical asymptotes of k. Let’s plot all of the graphical behavior we have determined thus far:

|[pic] |

|Figure 9: The long-run behavior of [pic], along with its |

|the vertical asymptotes [pic] and [pic], and the point |

|[pic]. |

Since the factors in the denominator that gives us the vertical asymptote are linear factors (i.e., both [pic] and [pic] have an (invisible) exponent of 1), we know that the graph has the different behavior on both sides of the vertical asymptote. (So if the graph increases without bound on one side of [pic] then it will decrease without bound on the other side of [pic], and the same is true about [pic].) Using what we’ve plotted in Figure 9, we can finish the graph of k by making sure that all of the behavior we’ve determined is satisfied and avoiding any contradictions; see Figure 10.

|[pic] |

|Figure 10: The graph of [pic], along with its the vertical |

|asymptotes [pic]and [pic]. |

Be sure to convince yourself that based on what we discovered about the long-run behavior and the roots and vertical asymptote, the graph of [pic] MUST look like the one pictured in Figure 10. You should also graph this function yourself on your graphing calculator for practice.

[pic]

[pic] example: Sketch a graph of the function [pic].

SOLUTION:

In the previous module we determined that the long-run behavior for m; see Figure 11.

|[pic] |

|Figure 11: The long-run behavior of [pic] along with its oblique|

|asymptote [pic]. |

To sketch the short-run behavior of m, we need to find the roots, y-intercept, and the vertical asymptotes. To find the roots, we need to find all x such that [pic]. So we need to solve [pic]:

[pic]

Thus, 0, 4 and – 4 are roots.

To find the y-intercept, we need to evaluate [pic]. Since 0 is a root, we know that [pic], so the y-intercept is [pic].

To find the vertical asymptotes, we need to determine what values make the denominator 0, so we need to solve [pic]:

[pic]

Therefore, the lines [pic] and [pic] are the vertical asymptotes of m. Let’s plot all of the graphical behavior we have determined thus far:

|[pic] |

|Figure 12: The long-run behavior of [pic], along with its oblique asymptote [pic],|

|its vertical asymptotes [pic] and [pic], and the points [pic], [pic], and [pic]. |

Using what we’ve plotted in Figure. 12, we can finish the graph of m by making sure that all of the behavior we’ve determined is satisfied and avoiding any contradictions; see Figure 13.

|[pic] |

|Figure 13: The graph of [pic], along with its oblique asymptote [pic]|

|and its vertical asymptotes [pic] and [pic]. |

Be sure to convince yourself that based on what we discovered about the long-run behavior and the roots and vertical asymptote, the graph of [pic] MUST look like the one pictured in Figure 13. You should also graph this function yourself on your graphing calculator for practice.

[pic]

[pic] example: Find an algebraic rule for the function g graphed in Figure 14.

[pic]

Figure 14: The graph of [pic].

SOLUTION:

Since the graph has vertical asymptotes, we know that g is a rational function. Since the vertical asymptotes are [pic] and [pic], we know that the factors [pic] and [pic] must appear in the denominator. Since g has roots –3 and 5, its rule must have factors [pic] and [pic] in the numerator. This tells us that the rule for g looks like

[pic].

Now we need to find k. Let’s use the horizontal asymptote [pic] to find k. Since [pic] is the horizontal asymptote, we know that for extreme x values (i.e., as [pic]), [pic]. So…

[pic].

Therefore, [pic].

[pic] Try this one yourself and check your answer.

Find an algebraic rule for the function h graphed in Figure 15.

[pic]

Figure 15: The graph of [pic].

SOLUTION:

Since the graph has vertical asymptotes, we know that h is a rational function. Like the graph of h above, since the vertical asymptotes are [pic] and [pic], we know that the factors [pic] and [pic] must appear in the denominator. But in this case, since the behavior is the same on both sides of the asymptote [pic], we know that the factor [pic] must be squared (or raised to any even power.) Since h has roots –2 and 8, its rule must have factors [pic] and [pic] in the numerator. This tells us that the rule for h looks like

[pic].

Now we need to find k. In this case we cannot use the horizontal asymptote [pic] to find k. (We need to use a graph feature that is NOT on the x-axis. You should try to find k using the horizontal asymptote [pic] to convince yourself that it isn’t possible.) Instead, let’s use the y-intercept [pic]. Since [pic] is the y-intercept we know that [pic]. So…

[pic]

Therefore, [pic].

[pic]

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