Everyday Math Demystified, 2nd edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 7

1. A true mathematical function can never have more than one value of the dependent variable for any single value of the independent variable. For example, if time is the independent variable on the graph of a function while temperature is the dependent variable, you'll never see more than one temperature value plotted for a single point in time. The question asks, in effect, which of the four characteristics (A, B, C, or D) represents an impossible situation. That's choice B. You might observe a certain temperature more than once during the course of the day (for example, at 10:00 a.m. and 7:00 p.m.), so choice A represents a plausible scenario. You might encounter an unusual day when the temperature trends generally downward for the entire 24-hour period, so scenario C is possible. You might experience a "freak day" when the temperature rises continuously over the whole 24 hours (that's what "strictly nondecreasing" means), so scenario D can occur. Once again, only choice B represents a situation where you don't have a function.

2. On a continuous-curve graph, the independent variable usually runs along the horizontal axis, increasing from left to right. The dependent variable usually runs along the vertical axis, increasing from bottom to top. Therefore, both choices A and B must be wrong, because the question asks for something that never occurs. A continuous-curve graph will generally show more precision than a point-to-point graph of the same function, so choice C is wrong. Now we have only choice D remaining. We can expect that a continuous-curve graph will reveal more detail than than a vertical bar graph will reveal. But choice D says the opposite thing! Because the question asks for something that can never happen, D is the right choice here.

3. All four options involve graphs based on data sampled at regular time intervals. For any type of graph, the precision improves as the time interval decreases. That fact should make us suspect that choice A is best because it specifies the shortest time interval. If we look more closely at the four options, we can see that choice A describes the most accurate method of graph approximation (curve fitting). Vertical bar graphs (choice B) and point-to-point graphs (choice C) are less precise than curve fitting for any particular time interval. Choice D doesn't apply in this situation at all. No good mathematician or scientist would try to make a pie graph showing a function of temperature versus time. A pie graph can effectively show relative proportions, but it can't portray a function between two variables. Of the four options given here, choice A represents the best answer.

4. When we look at Figs. 7-22 and 7-23, we can see that the average monthly rainfall for Happyton (dark bars in Fig. 7-22) trends generally upward from January through August. The same thing holds true for Happyton's average monthly temperature (light bars in Fig. 7-22), and for Blissville's monthly temperature (light bars in Fig. 7-23). All three choices A, B, and C will work okay here, so we should choose D, "All of the above," as the correct answer. We might do well to note in passing, however, the difference between the criteria for functions that trend generally upward and the criteria for strictly nondecreasing functions. If this question had asked "Which of the phenomena are strictly nondecreasing from January through August?" instead of "Which of the phenomena trend generally upward from January through August?" then choices B and C would hold true, but choice A would not. The average monthly rainfall for Happyton goes down slightly from January to February, disqualifying it from the standard for a strictly nondecreasing function, even though, over the period from January through August, the function clearly trends upward.

5. In a horizontal bar graph, we plot the independent variable along the vertical axis, usually increasing from bottom to top; we plot the dependent variable along the horizontal axis, increasing from left to right. The left-to-right widths of the bars therefore depend on the value of the dependent variable. The correct choice is B.

6. In a vertical bar graph, we plot the independent variable along the horizontal axis, increasing as we move toward the right. We plot the dependent variable along the vertical axis, increasing as we move upward. We choose the left-to-right bar width (or "thickness") for the best illustration clarity, preferably making all of the bars equally wide. The left-to-right "bar thickness" doesn't depend on any particular characteristic of the phenomenon in question. The right answer is D.

7. For any particular function, a continuous-curve graph offers precision at least as good, and usually better, than a point-to-point graph does. A point-to-point graph is never more precise than continuous-curve graph. The statement of choice C is false, so it's the right answer here. The statements of choices A and B are usually true; the statement of choice D is sometimes true.

8. In a pie graph, the sample proportions always add up to 100%. In effect, the samples, which look like "pie slices," can never add up to anything more or less than the "whole pie." The correct choice is A. Continuous-curve graphs don't portray sample proportions; they show specific values of the dependent variable, so choice B is not only incorrect but irrelevant. In bar graphs, the proportions sometimes add up to more or less than 100%, so choices C and D are both wrong.

9. In Fig. 7-25, we have a continuous-curve graph for the stock price between 10:00 and 11:00, and another continuous-curve plot for the stock price from 12:00 to 2:00. The dashed line marked L shows an example of linear interpolation. It provides an "intermediate-value guess" for the stock price during the "time gap" between 11:00 and 12:00. The correct choice is C.

10. In Fig. 7-25, the dashed line marked M shows an example of linear extrapolation. It provides an "extension" or "forecast" for the stock price after 2:00. The correct choice is B. In passing, we might mention that if we had a stock-price data point for 3:00, at the extreme right-hand end of the graph, and if this point corresponded with the right-hand end of the dashed line marked M, then that dashed line would constitute linear interpolation, and we'd pick C rather than B! However, the graph lacks a data point for the stock price at 3:00, so we have to "forecast" the price for any time point later than 2:00. Once again, the answer is B.