Have you ever studied bacteria in your science classes? The rate of bacterial growth depends on many factors, including the temperature of the environment. You can use the quadratic function below to calculate how many bacteria you will have after one week. For this function, the independent variable is the temperature of the environment, and the dependent variable is the number of bacteria after one week.
For Example
\( b(t) = -t^{2} + 40t + 500 \)
What does the graph of this function look like?
Let's review some key concepts of graphical analysis using the tabs below.
The graph of this function shows you what temperatures encourage the bacteria to grow and what temperatures cause the number of bacteria to remain the same or even decrease. The table below shows how the bacteria react to certain temperatures
| Temperature (t) in Celsius |
Number of Bacteria (after 1 week) |
|---|---|
| -10 | 0 |
| 0 | 500 |
| 10 | 800 |
| 20 | 900 |
| 30 | 800 |
| 40 | 500 |
| 50 | 0 |
Plot these points on a coordinate plane and connect them using a smooth curve. Notice that the axes are labeled for the specific variables they each represent in this problem. The vertical axis is “Number of Bacteria” and the horizontal axis is “Temperature".
Recall that the graph of a quadratic function is a curve called a parabola. A parabola has some key features:
This is the graph of the function \( b(t) = -t^{2} + 40t + 500 \).
Maximum
When a parabola opens down, it is said to have a maximum. This maximum occurs at the highest point on the parabola.
Minimum
When a parabola opens up, it is said to have a minimum. The minimum occurs at the lowest point on the parabola.
Intercepts
The intercepts of a graph occur where the parabola crosses the horizontal and vertical axes.
These terms will be important as we start to analyze our function.
It's time to take a closer look at our bacterial growth to see what you can say about the growth of bacteria based on the graph. Use this graph to answer the following questions. When you think you've got it, click on the question to show the answer.
What can you say about the bacterial growth at certain temperatures? Use the graph to analyze the function. Answer each of the questions. Click the question to check your answer.
