- Calculate ant population at 3, 6, and 9 months by repeatedly doubling: 44, 88, 176.
- Solve the linear equation \(y = \frac{3}{2}x + 3\) for \(x\) given \(y \in \{9, 6, 0, -3\}\) to obtain \(x \in \{4, 2, -2, -4\}\).
- Confirm the relation is a function because each \(x\) yields one \(y\), and classify it as linear due to the form \(y = mx + b\).
- Final categorization: The relation is a \(\boxed{\text{linear function}}\).
Explanation
- Ant Farm Population Growth Analysis
The ant farm problem follows an exponential growth pattern where the population doubles every 3 months. Given an initial population of 22 at 0 months, we can determine the population at later intervals by repeatedly multiplying by 2.
At 3 months, the population is \(22 \times 2 = 44\).
At 6 months, the population is \(44 \times 2 = 88\).
At 9 months, the population is \(88 \times 2 = 176\).
The completed table is as follows:
| Number of months | 0 | 3 | 6 | 9 \) Population | 22 | 44 | 88 | 176 \) |
- Completing the Function Table
To complete the table for the relation \(y = \frac{3}{2}x + 3\), we solve the equation for \(x\) using the given values of \(y\):
- For \(y = 9\): \(9 = \frac{3}{2}x + 3 \implies 6 = \frac{3}{2}x \implies x = 6 \times \frac{2}{3} = 4\).
- For \(y = 6\): \(6 = \frac{3}{2}x + 3 \implies 3 = \frac{3}{2}x \implies x = 3 \times \frac{2}{3} = 2\).
- For \(y = 0\): \(0 = \frac{3}{2}x + 3 \implies -3 = \frac{3}{2}x \implies x = -3 \times \frac{2}{3} = -2\).
- For \(y = -3\): \(-3 = \frac{3}{2}x + 3 \implies -6 = \frac{3}{2}x \implies x = -6 \times \frac{2}{3} = -4\).
The completed table is:
| \(x\) | 4 | 2 | -2 | -4 \) \(y\) | 9 | 6 | 0 | -3 \) |
- Classifying the Relation
A relation is a function if every input \(x\) corresponds to exactly one output \(y\). Since the equation \(y = \frac{3}{2}x + 3\) provides a single, unique result for any value of \(x\) we plug into it, this relation is a function.
Furthermore, a linear function is a function that can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. The given equation, \(y = \frac{3}{2}x + 3\), matches this form perfectly with \(m = \frac{3}{2}\) and \(b = 3\). Therefore, it is a linear function.
Examples
Understanding exponential growth, like the doubling ant population, helps predict resource needs over time in biology or business. Similarly, linear functions are essential in everyday life, such as calculating the total cost of a service that has a fixed fee plus an hourly rate. For example, if a repair technician charges a flat fee of \(\$3\) plus \(\$1.50\) per hour, the total cost \(y\) for \(x\) hours of work is modeled by the linear equation \(y = 1.5x + 3\). Identifying whether a situation is linear or exponential allows you to better plan for the future.