Example 12 from Precalculus, 2.1 Linear Functions

Using Tabular Form to Write an Equation for a Linear Function

Table 1 relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.

w, number of weeks	0	2	4	6
P(w), number of rats	1000	1080	1160	1240

Table 1

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can see from the table that the initial value for the number of rats is 1000, so $b = 1000.$

Rather than solving for $m,$ we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.

$P (w) = 40 w + 1000$

If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using $(2, 1080)$ and $(6, 1240)$

$\begin{array}{l} m = \frac{1240 - 1080}{6 - 2} \\ = \frac{160}{4} \\ = 40 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A new plant food was introduced to a young tree to test its effect on the height of the tree. Table 2 shows the height of the tree, in feet, $x$ months since the measurements began. Write a linear function, $H (x),$ where $x$ is the number of months since the start of the experiment.

$x$	0	2	4	8	12
$H (x)$	12.5	13.5	14.5	16.5	18.5

Table 2

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$x$	0	2	4	8	12
$H (x)$	12.5	13.5	14.5	16.5	18.5

Table 2

Solution

Your turn

Solution

Your turn