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Section4.3Exploring Two-Variable Data and Rate of Change

This section is about examining data that has been plotted on a Cartesian coordinate system, and then making observations. In some cases, we'll be able to turn those observations into useful mathematical calculations.

Figure4.3.1Alternative Video Lesson

Subsection4.3.1Modeling data with two variables

Using mathematics, we can analyze real data from the world around us. We can use what we discover to better understand the world, and sometimes to make predictions. Here's an example of data about the economic situation in the US:

a scatter plot about share of all income held by the top 1% in the US
Figure4.3.2Share of all income held by the top 1 %, United States, 1990–2013 (www.epi.org)

If this trend continues, what percentage of all income will the top 1 % have in the year 2030? If we model data in the chart with the trend line, we can estimate the value to be 28.6 %. This is one way math is used in real life.

Does that trend line have an equation like those we looked at in Section 4.2? Is it even correct to look at this data set and decide that a straight line is a good model? These are some of the questions we want to consider as we begin this section. The answers will evolve through the next several sections.

Subsection4.3.2Patterns in Tables

Example4.3.3

Find a pattern in each table. What is the missing entry in each table? Can you describe each pattern in words and/or mathematics?

black white
big small
short tall
few
USA Washington
UK London
France Paris
Mexico
1 2
2 4
3 6
5
Figure4.3.4Patterns in 3 tables
Solution
black white
big small
short tall
few many
USA Washington
UK London
France Paris
Mexico Mexico City
1 2
2 4
3 6
5 10
Figure4.3.5Patterns in 3 tables

First table

Each word on the right has the opposite meaning of the word to its left.

Second table

Each city on the right is the capital of the country to its left.

Third table

Each number on the right is double the number to its left.

We can view each table as assigning each input in the left column a corresponding output in the right column. In the first table, for example, when the input “big” is on the left, the output “small” is on the right. The first table's function is to output a word with the opposite meaning of each input word. (This is not a numerical example.)

The third table is numerical. And its function is to take a number as input, and give twice that number as its output. Mathematically, we can describe the pattern as “\(y=2x\text{,}\)” where \(x\) represents the input, and \(y\) represents the output. Labeling the table mathematically, we have Table 4.3.6.

\(x\)
(input)
\(y\)
(output)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(6\)
\(5\) \(10\)
\(10\) \(20\)
Pattern: \(y=2x\)
Table4.3.6Table with a mathematical pattern

The equation \(y=2x\) summarizes the pattern in the table. For each of the following tables, find an equation that describes the pattern you see. Numerical pattern recognition may or may not come naturally for you. Either way, pattern recognition is an important mathematical skill that anyone can develop. Solutions for these exercises provide some ideas for recognizing patterns.

Exercise4.3.7
Exercise4.3.8
Exercise4.3.9

Subsection4.3.3Rate of Change

For an hourly wage-earner, the amount of money they earn depends on how many hours they work. If a worker earns \(\$15\) per hour, then \(10\) hours of work corresponds to \(\$150\) of pay. Working one additional hour will change \(10\) hours to \(11\) hours; and this will cause the \(\$150\) in pay to rise by fifteen dollars to \(\$165\) in pay. Any time we compare how one amount changes (dollars earned) as a consequence of another amount changing (hours worked), we are talking about a rate of change.

Given a table of two-variable data, between any two rows we can compute a rate of change.

Example4.3.10

The following data, given in both table and graphed form, gives the counts of invasive cancer diagnoses in Oregon over a period of time. (wonder.cdc.gov)

Year Invasive Cancer
Incidents
1999 17,599
2000 17,446
2001 17,847
2002 17,887
2003 17,559
2004 18,499
2005 18,682
2006 19,112
2007 19,376
2008 20,370
2009 19,909
2010 19,727
2011 20,636
2012 20,035
2013 20,458

What is the rate of change in Oregon invasive cancer diagnoses between 2000 and 2010? The total (net) change in diagnoses over that timespan is

\begin{equation*} 19727-17446=2281\text{.} \end{equation*}

Since \(10\) years passed (which you can calculate as \(2010-2000\)), the rate of change is \(2281\) diagnoses per \(10\) years, or

\begin{equation*} \frac{2281\,\text{diagnoses}}{10\,\text{year}}=228.1\,\frac{\text{diagnoses}}{\text{year}}\text{.} \end{equation*}

We read that last quantity as “\(228.1\) diagnoses per year”. This rate of change means that between the years \(2000\) and \(2010\text{,}\) there were \(228.1\) more diagnoses each year, on average. (Notice that there was no single year in that span when diagnoses increased by \(228.1\text{.}\))

Let's practice calculating rates of change over different timespans:

Exercise4.3.11

We are ready to give a formal defintion for rate of change. Considering our work from Example 4.3.10 and Exercise 4.3.11, we settle on:

Definition4.3.12Rate of Change

If \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\) are two data points from a set of two-variable data, then the rate of change between them is

\begin{equation*} \frac{\text{change in $y$}}{\text{change in $x$}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\text{.} \end{equation*}

(The Greek letter delta, \(\Delta\text{,}\) is used to represent “change in” since it is the first letter of the Greek word for “difference”.)

In Example 4.3.10 and Exercise 4.3.11 we found three rates of change. Figure 4.3.13 highlights the three pairs of points that were used to make these calculations.

Figure4.3.13

Note how the larger the numerical rate of change between two points, the steeper the line is that connects them. This is such an important observation, we'll put it in an official remark.

Remark4.3.14

The rate of change between two data points is intimately related to the steepness of the line segment that connects those points.

  1. The steeper the line, the larger the rate of change, and vice versa.

  2. If one rate of change between two data points equals another rate of change between two different data points, then the corresponding line segments will have the same steepness.

  3. When a line segment between two data points slants down from left to right, the rate of change between those points will be negative.

In the solution to Exercise 4.3.8, the key observation was that the rate of change from one row to the next was constant: \(3\) units of increase in \(y\) for every \(1\) unit of increase in \(x\text{.}\) Graphing this pattern in Figure 4.3.15, we see that every line segment here has the same steepness, so the entire graph is a line.

Figure4.3.15

Whenever the rate of change is constant no matter which two \((x,y)\)-pairs (or data pairs) are chosen from a data set, then you can conclude the graph will be a straight line even without making the graph. For obvious reasons, we call this kind of relationship a linear relationship. We'll study linear relationships in more detail throughout this chapter. Right now in this section, we feel it is important to simply identify if data has a linear relationship or not.

Identify if each pattern below describes a linear relationship or not.

Exercise4.3.16
Exercise4.3.17
Exercise4.3.18

Let's return to the data that we opened the section with, in Figure 4.3.2. Is that data linear? Well, yes and no. To be completely honest, it's not linear. It's easy to pick out pairs of points where the steepness changes from one pair to the next. In other words, the points do not all fall into a single line.

However if we stand back, there does seem to be an overall upward trend that is captured by the line someone has drawn over the data. Points on this line do have a linear pattern. Let's estimate the rate of change between some points on this line. We are free to use any points to do this, so let's make this calculation easier by choosing points we can clearly identify on the graph: \((1991,15)\) and \((2020,25)\text{.}\)

a scatter plot about share of all income held by the top 1% in the US
Figure4.3.19Share of all income held by the top 1 %, United States, 1990–2013 (www.epi.org)

The rate of change between those two points is

\begin{equation*} \frac{25-15}{2020-1991}=\frac{10}{29}\approx0.3448\text{.} \end{equation*}

So we might say that on average the rate of change expressed by this data is 0.3448 %yr.

Subsection4.3.4Exercises

Find a formula relating \(y\) to \(x\) using a table. The pattern should be relatively easy to spot. Later sections will address more specific ways to approach this task in general.

1

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(-2\) \({-8}\)
\(-1\) \({-4}\)
\(0\) \({0}\)
\(1\) \({4}\)
\(2\) \({8}\)

2

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(3\) \({12}\)
\(4\) \({16}\)
\(5\) \({20}\)
\(6\) \({24}\)
\(7\) \({28}\)

3

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(6\) \({11}\)
\(7\) \({12}\)
\(8\) \({13}\)
\(9\) \({14}\)
\(10\) \({15}\)

4

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(7\) \({16}\)
\(8\) \({17}\)
\(9\) \({18}\)
\(10\) \({19}\)
\(11\) \({20}\)

5

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(18\) \({11}\)
\(12\) \({5}\)
\(14\) \({7}\)
\(19\) \({12}\)
\(7\) \({0}\)

6

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(0\) \({2}\)
\(6\) \({8}\)
\(3\) \({5}\)
\(14\) \({16}\)
\(10\) \({12}\)

7

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(1\) \({1}\)
\(25\) \({5}\)
\(9\) \({3}\)
\(4\) \({2}\)
\(16\) \({4}\)

8

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(-4\) \({4}\)
\(-2\) \({2}\)
\(-1\) \({1}\)
\(-4\) \({4}\)
\(-2\) \({2}\)

9

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(0.1\) \({0.01}\)
\(0.3\) \({0.09}\)
\(0.5\) \({0.25}\)
\(0.7\) \({0.49}\)
\(0.9\) \({0.81}\)

10

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(0.7\) \({0.49}\)
\(1\) \({1}\)
\(1.3\) \({1.69}\)
\(1.6\) \({2.56}\)
\(1.9\) \({3.61}\)

11

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(59\) \({{\frac{1}{59}}}\)
\(54\) \({{\frac{1}{54}}}\)
\(24\) \({{\frac{1}{24}}}\)
\(31\) \({{\frac{1}{31}}}\)
\(41\) \({{\frac{1}{41}}}\)

12

Write an equation in the form \(y=\ldots\) suggested by the pattern in the table.

\(x\) \(y\)
\(70\) \({{\frac{1}{70}}}\)
\(20\) \({{\frac{1}{20}}}\)
\(65\) \({{\frac{1}{65}}}\)
\(4\) \({{\frac{1}{4}}}\)
\(40\) \({{\frac{1}{40}}}\)

Calculate some rates of change.

13

This table gives population estimates for Portland, Oregon from 1990 through 2014 (www.google.com/publicdata as of Oct 11, 2016; Google may update with better estimates).

Year Population Year Population
1990 487849 2003 539546
1991 491064 2004 533120
1992 493754 2005 534112
1993 497432 2006 538091
1994 497659 2007 546747
1995 498396 2008 556442
1996 501646 2009 566143
1997 503205 2010 585261
1998 502945 2011 593859
1999 503637 2012 602954
2000 529922 2013 609520
2001 535185 2014 619360
2002 538803

Find the rate of change in Portand population between 1991 and 2007. Just give the numerical value; the units are provided.

\(\,\frac{\text{people}}{\text{year}}\)

And what was the rate of change between 2010 and 2011?

\(\,\frac{\text{people}}{\text{year}}\)

List all the years where there is a negative rate of change between that year and the next year.

14

This table and graph gives population estimates for Portland, Oregon from 1990 through 2014 (www.google.com/publicdata as of Oct 11, 2016; Google may update with better estimates).

Year Population Year Population
1990 487849 2003 539546
1991 491064 2004 533120
1992 493754 2005 534112
1993 497432 2006 538091
1994 497659 2007 546747
1995 498396 2008 556442
1996 501646 2009 566143
1997 503205 2010 585261
1998 502945 2011 593859
1999 503637 2012 602954
2000 529922 2013 609520
2001 535185 2014 619360
2002 538803

Between what two years that are two years apart was the rate of change highest?

What was that rate of change? Just give the numerical value; the units are provided.

\(\,\frac{\text{people}}{\text{year}}\)