13 Data tables

Data tables are an efficient way to organize and present information. In this class, we use data tables that relate two different quantities (also called variables).

Example 13.1 (Activity: reading a data table) The following table shows the relationship between the variables

$x$ is the time, measured in years,
$y$ is the population of Mora County, measured in people.

$x$ time	$y$ population
1980	4,205
1990	4,264
2000	5,180
2010	4,881
2020	4,189

Data source: Wikipedia

Answer these questions based on the table:

When $x = 1980$, what is $y$?
When $x = 2000$, what is $y$?
For what value of $x$ is the value of $y$ greater than $4200$?

Solutions

When $x = 1980$, $y = 4{,}205$.
When $x = 2000$, $y = 5{,}180$.
$y > 4200$ for $x = 1980, 1990, 2000, 2010$. (In 2020 the population was $4{,}189 < 4{,}200$.)

In the next part of this section, we explore three ways to construct data tables.

13.1 Data tables from formulas

One way to construct a data table is using a formula.

Example 13.2 (Activity: data table from a formula) We use the variables

$x$ is the width of the square, and
$y$ is the area of the square.

Use this formula to complete the following table.

$x$ width	$y$ area
$1$	$1^2 = 1$
$2$	$2^2 = 4$
$3$
$4$
$5$

Then answer the following questions:

If $x = 3$, what is the value of $y$?
If $x = 5$, what is the value of $y$?
If $y = 49$, what is the value of $x$?

Solutions

$x$ width	$y$ area
$1$	$1^2 = 1$
$2$	$2^2 = 4$
$3$	$3^2 = 9$
$4$	$4^2 = 16$
$5$	$5^2 = 25$

If $x=3$, $y = 3^2 = 9$.
If $x=5$, $y = 5^2 = 25$.
If $y=49$, $x = \sqrt{49} = 7$.

Example 13.3 (Activity: data table from a constraint) There are many possible rectangles that have an area of 48 square inches. Complete the following data table giving the length and width of possible rectangles.

$x$ length	$y$ width
1	48
2
3
4
5
6
7
8

Answer the following questions:

If $x = 3$, what is $y$?
If $x = 6$, what is $y$?
If $y = 12$, what is $x$?
Is it possible for $x$ and $y$ to have the same value?

Solutions

$x$ length	$y$ width
1	48
2	24
3	16
4	12
5	9.6
6	8
7	$\approx 6.86$
8	6

If $x=3$, $y = 48 \div 3 = 16$.
If $x=6$, $y = 48 \div 6 = 8$.
If $y=12$, $x = 48 \div 12 = 4$.
Yes — when $x = y$, we need $x^2 = 48$, so $x = \sqrt{48} \approx 6.93$.

13.2 Data tables from base values and rates

Another way to get data tables is from situations involving base values and rates.

Example 13.4 (Activity: data table from base value and rate) Amanda works the opening shift at the Ranchos Coffee Company. She gets a $20 bonus for opening early (at 5:30am!) and then earns $13.50 per hour.

Complete the following table indicating how much she earns.

$x$ hours worked	$y$ pay for the day
1	$20 + $13.50 = $33.50
2
3
4
5

Now answer the following questions:

If $x=5$, what is the value of $y$?
If Amanda works a 10 hour shift, how much money will she earn? Express this using the variables $x$ and $y$.
How large does $x$ need to be in order for $y$ to be larger than $100?

Solutions

$x$ hours worked	$y$ pay for the day
1	$33.50
2	$47.00
3	$60.50
4	$74.00
5	$87.50

If $x=5$, $y = 20 + 13.50(5) = \$87.50$.
A 10-hour shift: $x=10$, $y = 20 + 13.50(10) = \$155.00$.
$y > 100$ when $20 + 13.50x > 100$, so $13.50x > 80$, so $x > 5.93$. Amanda needs to work at least $6$ hours.

13.3 Data tables from percents

Finally, we can compute data tables from percent changes.

Example 13.5 (Activity: data table from percent change) Prices are always going up, and the Ranchos Coffee Company is no exception. In 2015, when they opened, a cup of Talpa House Coffee cost $2.00. Each year since then the price has risen by 5%.

The following table uses the variables:

$x$ is the number of years since 2015,
$y$ is the price of a cup of Talpa House Coffee.

Complete the table:

$x$ years since 2015	$y$ price
0	$2.00
1	$2.10
2	$2.21
3
4
5
6

Answer the following questions:

How did we compute that when $x=2$ then $y = \$2.21$?
In what year did the price exceed $2.50?

Solutions

$x$ years since 2015	$y$ price
0	$2.00
1	$2.10
2	$2.21
3	$\approx \$2.32$
4	$\approx \$2.43$
5	$\approx \$2.55$
6	$\approx \$2.68$

$y = 2.10 \cdot 1.05 = 2.205 \approx \$2.21$ (multiply the previous year’s price by $1.05$).
The price first exceeds $\$2.50$ at $x=5$ (year 2020), when $y \approx \$2.55$.

13.4 Homework exercises

Exercise 13.1 The table below shows the average temperature in Santa Fe for the first 12 days of January 2026. (Source: https://www.weather.gov/wrh/Climate?wfo=abq) The table uses the variables

$x$ the day of the month,
$y$ the average temperature that day (measured in degrees F).

Day $x$	Average temperature $y$
1	43
2	44
3	42
4	43
5	37
6	38
7	38
8	37
9	28
10	24
11	30
12	36

Use the table to answer the following questions:

When $x = 4$, what is $y$?
When $x = 9$, what is $y$?
What $x$ value corresponds to the largest $y$ value?
What $x$ value corresponds to the smallest $y$ value?

Solutions

When $x=4$, $y=43$.
When $x=9$, $y=28$.
$x=2$ corresponds to the largest $y$ value ($y=44$).
$x=10$ corresponds to the smallest $y$ value ($y=24$).

Exercise 13.2 Use the data from https://en.wikipedia.org/wiki/Taos_County,_New_Mexico to construct a table showing the population of Taos County for the years 1980 - 2020. Your table should look very similar to the table in Example 13.1.

Using your table, answer the following questions:

When $x=1980$, what is $y$?
When $x = 2000$, what is $y$?
For what $x$ value is $y$ the largest?
For what $x$ value is $y$ the smallest?

Exercise 13.3 Construct a table showing the relationship between the variables

$x$ the width of a square, and
$y$ the perimeter of the square.

Your table should have 5 rows in it.

Use your table to address these questions:

If $x=3$, what is $y$?
If $y = 16$, what is $x$?

Solutions

$x$ width	$y$ perimeter
1	4
2	8
3	12
4	16
5	20

If $x=3$, $y = 4(3) = 12$.
If $y=16$, $x = 16 \div 4 = 4$.

Exercise 13.4 Rosalie buys a 50 pound bag of dog food. Each day her dogs, named Bruno and Macho, eat a total 1/2 pound of food.

Make a table that relates the variables

$x$ the number of days since Rosalie bought the bag of food, and
$y$ the amount of dog food remaining in the bag.

Your table should have at least 7 rows in it.

Use your table to address these questions:

If $x=2$, what is $y$?
If $x=6$, what is $y$?
If $y = 48$, what is $x$?

Challenge: what value of $x$ will correspond to $y=0$? What does this mean about the bag of food?

Solutions

$x$ days	$y$ food remaining (lbs)
0	50
1	49.5
2	49
3	48.5
4	48
5	47.5
6	47
7	46.5

If $x=2$, $y = 50 - \tfrac{1}{2}(2) = 49$ lbs.
If $x=6$, $y = 50 - \tfrac{1}{2}(6) = 47$ lbs.
If $y=48$, then $48 = 50 - \tfrac{1}{2}x$, so $x = 4$ days.
Challenge: $y=0$ when $x=100$. The bag of food will last 100 days.

Exercise 13.5 Santiago buys a used pickup truck for $20,000. Each year, the truck loses 5% of its value. (In business-speak this is called depreciation.) Make a table that relates these variables:

$x$ is the number of years since Santiago bought the truck, and
$y$ the value of the truck.

Your table should have at least 5 rows in it. The first row should have $x=0$ and $y = 20,000$.

Answer the following questions:

When $x =1$, you should have $y=19,000$. How did this get computed?
When $x=2$, what is $y$?
When $x=5$, what is $y$?

Challenge: After 10 years, what will be the value of the truck?

Solutions

$x$ years	$y$ value
0	$20,000
1	$19,000
2	$18,050
3	$17,147.50
4	$16,290.13
5	$15,475.62

When $x=1$: $y = 20{,}000 \cdot 0.95 = 19{,}000$. (Multiply by $0.95$ to apply the $5\%$ decrease.)
When $x=2$: $y = 19{,}000 \cdot 0.95 = 18{,}050$.
When $x=5$: $y = 20{,}000 \cdot (0.95)^5 \approx \$15{,}476$.
Challenge: $y = 20{,}000 \cdot (0.95)^{10} \approx \$11{,}975$.

\(x\) width	\(y\) area
\(1\)	\(1^2 = 1\)
\(2\)	\(2^2 = 4\)
\(3\)
\(4\)
\(5\)

\(x\) width	\(y\) area
\(1\)	\(1^2 = 1\)
\(2\)	\(2^2 = 4\)
\(3\)	\(3^2 = 9\)
\(4\)	\(4^2 = 16\)
\(5\)	\(5^2 = 25\)

\(x\) years since 2015	\(y\) price
0	$2.00
1	$2.10
2	$2.21
3	\(\approx \$2.32\)
4	\(\approx \$2.43\)
5	\(\approx \$2.55\)
6	\(\approx \$2.68\)