Here is the mathematical notation for a rational function:
R(x) = where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0
Graphing rational functions can be tricky. The denominator can never be equal to zero, because division by zero is not allowed. Each graph has a vertical asymptote at the point(s) that could make the denominator = 0, as defined by Q(x) ≠ 0 (assuming the function has already been reduced.) Horizontal asymptotes may also exist within a rational function. It is when the graph reaches a line horizontally that it will not cross, but comes closer and closer to. Here’s a picture:
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. The ironic thing is that neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.
Vertical Asymptote
Example:
It is established that the acceleration due to gravity, g (in meters/sec2), at a height h meters above sea level is given by:
g(h) = where 6.374 × 106 is the radius of Earth in meters.
Where is the function undefined? (Locate its vertical asymptote.)
Solution:
Set the denominator = 0 and solve for h, = 0
= 0
h = − meters
Problems:
1. The Refrigerator: How much does it really cost?
On the web site http://www.eren.doe.gov/consumerinfo/energy_savers/appliancesbody.html
there is a list of yearly cost for electricity for common household appliances.
Appliance Average Cost/year in
electricity
Home Computer $9
Television $13
Microwave $13
Dishwasher $51
Clothes Dryer $75
Washing Machine $79
refrigerator that lasts for 15 years. Assume the only costs associated with the refrigerator are it purchase cost and electricity.
b. Develop a general function that gives the annual cost of a refrigerator as a function of the number of years you own the refrigerator.
c. Sketch a graph of that function. What is an appropriate window? d. Find the horizontal and vertical asymptotes of this function,
e. Explain the meaning of the horizontal asymptote in terms of the refrigerator. f. If a company offers a refrigerator that costs $1200, but says that it will last at least
twenty years, is the refrigerator worth the difference in cost?
2. The function C(t) = describes the concentration of a drug in the blood stream
over time. In this case, the medication was taken orally. is measured in micrograms per milliliter and is measured in minutes.
a. Sketch a graph of the function over the first five minutes after the dose is given. Label axes.
b. Determine when the maximum amount of the drug is in the body and the amount at that time.
c. Explain within the context of the problem the shape of the graph between taking the medication orally ( ) and the maximum point. What does the shape of the graph communicate between the maximum point and afterwards?
d. What are the asymptotes of the rational function C(t) = What is the
meaning of the horizontal and vertical asymptotes within the context of the problem? Explore with different values of t to see how the body reacts to the drug over time.
Answers:
1a. C(15) = = $128.67
1b. C(x) = where x = number of years owned and C(x) represents the annual cost.
But since, x = number of years owned, x > 0, an adjustment to the graph would be:
1d. The vertical asymptotes of the rational function is x = 0 and y = 92. The n-value asymptote can be seen from the domain; you cannot substitute in the function. The asymptote associated with C is more of a challenge
1e. The yearly expense of electricity continues no matter how many years the refrigerator works. The cost will never go below the $92, but the cost approaches $92.
1f. No. Graphing the two functions C(x) = and C(x2) = together or
reviewing a table of values will show the more expensive refrigerator remains more expensive annually although both approach $92 as x approaches infinity.
Note: the graph appears to be linear when t is small; which implies, it the body is absorbing the drug at a constant rate (at first.)
2b. Look at the graph from 0 < t < 40 and zoom in to find the maximum point:
The maximum concentration appears to be around 16 - 20 seconds. Zooming in (I used www.wolfram.alpha zooming in and narrowing it between 18 < t< 19) the maximum is 18.2 minutes, and substituting this into the equation, the body is 13.76 micrograms.
2c. The graph shows that the concentration is 0 at time . Within the first 15 minutes, the concentration rises sharply and then after reaching the maximum, it slows down and the body appears to be removing the drug, cleansing the body of the drug.
There is no vertical asymptote since the denominator will never be = 0.
Sources:
1. Precalculus, Sullivan, Michael, 8th Edition, Pearson Prentice Hall, New Jersey, p. 194
