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Sample Test: Modeling Data with Functions

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides a sample test related to modeling data with functions.

Exercise 1

Sketch a vertical parabola through the three points in each graph.

Figure 1
(a) (b) (c) (d)
Three scattered points on a Cartesian coordinate system for plotting a parabola.Three scattered points on a Cartesian coordinate system for plotting a parabola.Three scattered points on a Cartesian coordinate system for plotting a parabola.Three scattered points on a Cartesian coordinate system for plotting a parabola.

Exercise 2

Three cars and an airplane are traveling to New York City. But they all go at different speeds, so they all take different amounts of time to make the 500-mile trip. Fill in the following chart.

Table 1
Speed (ss)—miles per hour Time (tt)—hours
50  
75  
100  
500  
  • a. Is this an example of direct variation, inverse variation, or neither of the above?
  • b. Write the function s(t)s(t).
  • c. If this is one of our two types, what is the constant of variation?

Exercise 3

There are a bunch of squares on the board, of different sizes.

Table 2
ss—length of the side of a square AA—area of the square
1  
2  
3  
4  
  • a. Is this an example of direct variation, inverse variation, or neither of the above?
  • b. Write the function A(s)A(s).
  • c. If this is one of our two types, what is the constant of variation?

Exercise 4

Anna is planning a party. Of course, as at any good party, there will be a lot of ppp on hand! 50 Coke cans fit into one recycling bin. So, based on the amount of Coke she buys, Anna needs to make sure there are enough recycling bins.

Table 3
cc—Coke cans Anna buys bb—recycling bins she will need
50  
100  
200  
400  
  • a. Is this an example of direct variation, inverse variation, or neither of the above?
  • b. Write the function b(c)b(c).
  • c. If this is one of our two types, what is the constant of variation?

Exercise 5

Gasoline has gotten so expensive that I’m experimenting with using alcoholic beverages in my car instead. I’m testing different beverages to see how fast they make my car go, based on the “proof” of the drink.1 pp is the proof of the drink, ss is the maximum speed of my car.

A typical beer is 10-proof. With beer in the tank, my car goes up to 30 mph.

The maximum proof of wine (excluding “dessert wines” such as port) is 28-proof. With wine in the tank, my car goes up to 75 mph.

  • a. Based on those two data points, create a model—that is a function s(p)s(p) that will predict, based on the proof of the drink, how fast I can get my car to go.
  • b. Test your model to make sure it correctly predicts that a 28-proof drink will get me up to 75 mph.
  • c. Use your model to predict how fast I can get with a 151-proof drink.

Exercise 6

Oops! I gave it a try. I poured Bacardi 151 into the car (so-called because it is indeed 151-proof). And instead of speeding up, the car only went up to 70 mph. Your job is to create a new quadratic model s(p) = ap2 + bp + c s(p)=ap2+bp+c based on this new data.

  • a. Write simultaneous equations that you can solve to find the coefficients aa, bb, and cc.
  • b. Solve, and write the new s(p)s(p) function.
  • c. Test your model to make sure it correctly predicts that a 151-proof drink will get me up to 70 mph.

Exercise 7

Make up a word problem involving inverse variation, on the topic of skateboarding.

  • a. Write the scenario.
  • b. Label and identify the independent and dependent variables.
  • c. Show the function that relates the dependent to the independent variable. This function should (of course) be an inverse relationship, and it should be obvious from your scenario!

Exercise 8

I found a Web site (this is true, really) that contains the following sentence:

[This process] introduces an additional truncation error [directly] proportional to the original error and inversely proportional to the gain (gg) and the truncation parameter (qq).

I don’t know what most of that stuff means any more than you do. But if we use TT for the “additional truncation error” and EE for the “original error,” write an equation that expresses this relationship.

Exercise 9

Which of the following correctly expresses, in words, the relationship of the area of a circle to the radius?

  • A. The area is directly proportional to the radius
  • B. The area is directly proportional to the square of the radius
  • C. The area is inversely proportional to the radius
  • D. The area is inversely proportional to the square of the radius

Exercise 10

Now, suppose we were to write the inverse of that function: that is, express the radius as a function of the area. Then we would write:

The radius of a circle is ___________ proportional to _____________________ the area.

Exercise 11

Death by Cholera

In 1852, William Farr reported a strong association between low elevation and deaths from cholera. Some of his data are reported below.

Table 4
EE: Elevation (ft) 10 30 50 70 90 100 350
CC: Cholera mortality (per 10,000) 102 65 34 27 22 17 8
  • a. Use your calculator to create the following models, and write the appropriate functions C(E)C(E) in the blanks.
  • Linear: C=C=
  • Quadratic: C=C=
  • Logarithmic: C=C=
  • Exponential: C=C=
  • b. Which model do you think is the best? Why?
  • c. Based on his very strong correlation, Farr concluded that bad air had settled into low-lying areas, causing outbreaks of cholera. We now know that air quality has nothing to do with causing cholera: the water-borne bacterial Vibrio cholera causes the disease. What might explain Farr’s results without justifying his conclusion?

Footnotes

  1. “Proof” is alcohol percentage, times 2: so if a drink is 5% alcohol, that’s 10 proof. But you don’t need that fact for this problem.

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