Listed below are three examples of rational functions used to model particular situations. While the applications are not related, you will discover that the functions all exhibit a similar behavior.
In many situations, a model is approximate. In the first two examples below, the functions were derived empirically, meaning they were determined by considering the actual, real-world results of a similar activity in the past. The model approximates what would happen in a future, similar situation.
Other models, like the third example below, may be exact relations determined by known, fixed material and labor costs. That is, for physically meaningful values of n – you cannot make a negative number of DVDs or an infinite number of DVDs – the output C(n) is exact.
Also, a model is often limited in its applicability. You will see this in all three of these examples, and the implications of these limitations will be a part of the resulting discussion.
In the following examples, each function has a denominator, making them all rational functions, and so the denominator becoming zero has to be avoided because dividing by zero is undefined, or not allowed.
- The cost C(x), in thousands of dollars, to remove x percent of a particular toxic pollutant from a certain river is modeled by:
- The cost C(x), in thousands of dollars, to inoculate xpercent of the population of a large city against a strain of the flu is modeled by the rational function:
- The cost C(n), in dollars per dvd, for a particular company to manufacture n copies of a certain DVD is given by:
Note: It is standard to use n as the independent variable name when the variable must be an integer.
Let’s begin our discussion:
Before your first post, please watch the video below:
Evaluating a Rational Expression in which m = 4 Compared with the Same Expression in which m = –4 (click the image to open the video [Video, 1:15 mins]):
For this activity, you will need to create and post a graph. The easiest way to plot points and graph a function and then generate an image of the work is to use Desmos, a free, online graphing calculator (Links to an external site.). The following information will help you with this work:
- Written directions on using the online calculator [PDF, 461 KB]
- Video directions on using the online calculator [Video, 3:32 mins]
For your initial post, please do the following:
- Choose one of the rational functions in the examples given above.
- Select eight values of the independent variable for your chosen function, and then compute the corresponding output of the function. Show your calculations.
- For examples 1 and 2, use this random number generator (Links to an external site.) to select your x-values. This will assure you’ve chosen a variety of values so your calculations will give you an idea of the range of the function. Make sure to specify Generate 8 random integers with a value between 1 and 100.
- For example 3, the random number generator is less useful. Just be sure to include some large numbers.
- For your chosen rational function, plot the eight data points you computed in (2) and then plot the entire function so that the graph passes through the points you plotted. This will require that you specify the window of your graph.
- For examples 1 and 2, the specified window of your graph should include all the possible values of the independent variable x. You will not be able to include all y values (you will see why), but specify the range of the y-axis so that you can see what happens when x is close to 100.
- For example 3, you cannot include all possible values of the independent variable x. But extend the x-axis out far enough to see what happens as xgets increasingly large.
- In your post:
- Show your calculations from part (2).
- Describe what the graph tells you about your corresponding example problem.
- Finally, insert the image of your graph (see instructions above).