A Problem Book in Algebra by V. A. Krechmar

By V. A. Krechmar

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3. The equation L = 4S is a linear model for the total number of legs L that S sheep have. Using the model, we find that 12 sheep have L = 4 * legs. = ٗ ٗ 4. Suppose digital cable service costs $49 a month with an initial installation fee of $110. We make a linear model for the total cost C of digital cable service for x months by writing the equation ______________. Think About It 5. What is the purpose of making a model? Support your answer by examples. 6. Explain how data, equations, and graphs work together to describe a real-world situation.

Let’s consider data sets whose inputs are evenly spaced. For instance, the inputs 0, 1, 2, 3, . . in Example 1 are evenly spaced—successive inputs are one unit apart. For data with evenly spaced inputs, there is a linear model for the data if the outputs increase (or decrease) by a constant amount between successive inputs. We can test whether data satisfy this condition by making a table of first differences. Each entry in the first difference column of the table is the difference between an output and the immediately preceding output.

The biologist wishes to determine whether there is a relationship between the levels of the different enzymes. (a) Make a scatter plot of the levels of the pairs of enzymes A and B. (b) Make a scatter plot of the levels of the pairs of enzymes A and C. (c) What relationships do you see in the data? 3 25 32 20 10 31 18 30 20 39 56 16 CHAPTER 1 ■ Data, Functions, and Models Solution Scatter plots for parts (a) and (b) are shown in Figures 3 and 4. 7) to represent this data pair. B 8 C 6 40 4 20 2 0 1 2 3 0 A 4 f i g u r e 3 Enzymes A and B 1 2 3 4 A f i g u r e 4 Enzymes A and C (c) From Figure 3 we see that there is no obvious relationship between the levels of enzymes A and B.

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