So far we have looked at linear models. We will add quadratic, exponential, and some variations in later sections. One of the ways we distinguish between models is by the rate at which they grow. Often the rate at which something is happening is more important than how much there currently is. This section presents two methods for identifying rates from tables of data. How to identify each type by graph is presented in the appropriate chapter and section.
In order to see how these differences can help us distinguish between linear and other models, consider Vasya’s salary in Example 3.2.20. We know that the difference between each year’s salary is $5000.00, because we are told that was the raise each year. This is linear model. In contrast Tien’s raises are different each year (they grow year to year). This means his salary does not grow linearly.
Example3.3.2.Differences for Atmospheric Pressure Model.
Consider the model in Example 3.2.35. Table 3.3.3 calculates the differences every 2000 ft. Notice that the differences are all the same namely -2.00. This model is linear.
Table 3.3.4 calculates the differences every 1000 ft. In this table the differences are all -1.00. This still indicates that the model is linear. It does not matter what interval we choose. If the differences over evenly spaced intervals are the same, then the model is linear.
This states that a linear model grows by the same amount from one step to the next (rate of change or difference). This equal growth results from the ratio \(m\) in the form \(y=mx+b\text{.}\) Consider the specific case \(y=3x+2\text{.}\) In the table below notice that the differences are all the same and that the difference is 3. 3 is the ratio from the equation. This is always the case. The slope is how fast the line grows.
Notice we used this constant addition property when working with ratio problems like Example 2.3.4. Relations defined by fixed ratios like these are linear.
For linear data consecutive differences are always the same. The next examples (Example 3.3.7 to Example 3.3.10) illustrate known, non-linear data and how the differences for those look.
However, the first differences increase in a suspiciously simple pattern (odd numbers). Checking the second differences (the differences of the 1st differences) we see a linear pattern. This turns out to be the pattern for all quadratic data.
Consider Table 3.3.11. The differences are not the same nor do they show the same pattern as quadratics. However, there is a pattern in the differences. Notice that the differences are exactly equal to the original data. This means that the rate of increase is determined by the current scale. In other words, the bigger it is, the faster it grows. This is the pattern of data that varies exponentially.
Consider Table 3.3.13. The differences are not exactly equal to the original numbers. However, notice that \(6=2 \cdot 3\text{,}\)\(18=2 \cdot 9\text{,}\) and \(54=2 \cdot 27\text{.}\) The differences are double the original numbers. In general for exponential data the differences will be the original data scaled by some number.
The previous section analyzed change as the difference (subtraction) of consecutive numbers (salaries in these examples). This section analyzes change using the percent increase for each pair of consecutive numbers.
We will first calculate the percent increase of salary each year for Tien and Vasya. Because salary numbers are exact, we will not use significant digits. Rather we will round to the nearest percent. This is in Table 3.3.16 and Table 3.3.17.
Notice that for Tien the percent increase is the same each year. It is 5%. For Vasya, the percent increase is not the same each year. How does the percent increase change for her?
The table below gives an amount of caffeine in the blood stream. This data is exponential with a ratio of 0.87. The ratio in this example means there is a 13% decrease per hour of caffeine in the blood stream. This is in contrast to the previous example which was an increasing exponential.
Although Definition 3.3.14 and Definition 3.3.18 are phrased differently they both accurately describe exponential relations. Generally it is easier to test if data is exponential by testing the ratios of terms rather than the differences. Table 3.3.20 shows an example of analyzing data using both differences and ratios. Notice that the differences are a scaled version of the original data (scaled by \(1/3\)). The ratio from the quotient is \(4/3\) which gives about a 33% increase. For the curious the data was generate by \(5\left(\frac{4}{3}\right)^n\text{.}\)
In Example 3.2.6 we found a value between two entries in a table. That is interpolation. In other cases we want to find a value past the end of a table. This is called extrapolation.
From Example 3.3.15 we know each entry is 1.05 times the previous year’s salary. Because that was the pattern every year, we might safely suppose it will occur again. Thus we expect his 2022 salary to be \(1.05 \cdot \$73,773.33 \approx \$77,462.00\text{.}\)
From Example 3.2.20 we know that Vasya has received a $5,000 raise each year. Because that has been the pattern, we might safely suppose it will continue. Thus we expect her 2025 salary to be
