Question
Miriam is studying the growth rate of bacteria. She puts the data in the following table, where x represents minutes and y represents the number of bacteria.
x|0|1|2|3
y|00.8|0.4|2|10
Which type of function is represented by the data in the table?
x|0|1|2|3
y|00.8|0.4|2|10
Which type of function is represented by the data in the table?
Answers
GPT-4o mini
To determine the type of function represented by the data, let's analyze the values given in the table:
```
x | 0 | 1 | 2 | 3
y | 0.8 | 0.4 | 2 | 10
```
The values of \(y\) do not seem to form a linear pattern or a simple polynomial function, so let's look at how they change:
- From \(x = 0\) to \(x = 1\): \(y\) goes from 0.8 to 0.4 (a decrease of 0.4)
- From \(x = 1\) to \(x = 2\): \(y\) goes from 0.4 to 2 (an increase of 1.6)
- From \(x = 2\) to \(x = 3\): \(y\) goes from 2 to 10 (an increase of 8)
The changes suggest a nonlinear relationship. Additionally, the values of \(y\) exhibit a rapid increase, which can often be indicative of exponential growth. To confirm this, let’s look at the ratio of successive \(y\) values:
- From \(0.8\) to \(0.4\): \(0.4 / 0.8 = 0.5\)
- From \(0.4\) to \(2\): \(2 / 0.4 = 5\)
- From \(2\) to \(10\): \(10 / 2 = 5\)
These ratios suggest that the growth of \(y\) is accelerating, which is characteristic of an exponential function.
Based on this analysis, it is likely that the data represents an **exponential growth function** of the form \(y = a \cdot b^x\), where \(a\) and \(b\) are constants.
To summarize, the data in the table likely represents **exponential growth** in the number of bacteria as a function of time (in minutes).
```
x | 0 | 1 | 2 | 3
y | 0.8 | 0.4 | 2 | 10
```
The values of \(y\) do not seem to form a linear pattern or a simple polynomial function, so let's look at how they change:
- From \(x = 0\) to \(x = 1\): \(y\) goes from 0.8 to 0.4 (a decrease of 0.4)
- From \(x = 1\) to \(x = 2\): \(y\) goes from 0.4 to 2 (an increase of 1.6)
- From \(x = 2\) to \(x = 3\): \(y\) goes from 2 to 10 (an increase of 8)
The changes suggest a nonlinear relationship. Additionally, the values of \(y\) exhibit a rapid increase, which can often be indicative of exponential growth. To confirm this, let’s look at the ratio of successive \(y\) values:
- From \(0.8\) to \(0.4\): \(0.4 / 0.8 = 0.5\)
- From \(0.4\) to \(2\): \(2 / 0.4 = 5\)
- From \(2\) to \(10\): \(10 / 2 = 5\)
These ratios suggest that the growth of \(y\) is accelerating, which is characteristic of an exponential function.
Based on this analysis, it is likely that the data represents an **exponential growth function** of the form \(y = a \cdot b^x\), where \(a\) and \(b\) are constants.
To summarize, the data in the table likely represents **exponential growth** in the number of bacteria as a function of time (in minutes).