Question 55·Easy·Two-Variable Data: Models and Scatterplots
A patch of algae on a pond doubles in area every 3 days, starting at an initial area of 4 square centimeters. What type of function best models the relationship between the algae's area and time?
For questions about the type of function that models a situation, first decide whether the quantity is increasing or decreasing as the input (usually time) increases to eliminate half the choices quickly. Next, check whether the change is by a constant amount (suggesting a linear model) or by a constant factor (suggesting an exponential model). Match that pattern to the answer choice language (increasing/decreasing + linear/exponential) instead of trying to write full equations, which saves time and reduces errors.
Hints
Direction of change
Ask yourself: as time passes, does the area of the algae get bigger or smaller? This will help you decide between the options that say "increasing" and those that say "decreasing."
Type of change over equal time steps
Focus on the phrase "doubles in area every 3 days." Is the area going up by the same amount each time, or is it being multiplied by the same factor each time?
Connect the pattern to a function type
Functions where you add the same amount each step are one type; functions where you multiply by the same factor each step are another type. Think about which function type matches a "doubling" pattern, and then combine that with whether the values are increasing or decreasing.
Desmos Guide
Create a small table of values
In Desmos, add a table. In the first column (x1), enter times in days: 0, 3, 6, 9, 12. In the second column (y1), enter the corresponding areas following the doubling pattern starting at 4: 4, 8, 16, 32, 64. Look at how the points are positioned as x increases.
Compare with a straight-line (linear) fit
Type y1 ~ m x1 + b in Desmos to create a linear regression. Look at the graph: does a straight line fit all of your table points well, or do the points curve away from the line as x increases? This tells you whether a linear model is appropriate.
Compare with a curved (exponential) fit
Now type an exponential regression such as y1 ~ a * b^(x1/3). Observe whether this curve passes through or very near all of the data points and how the y-values behave as x increases. Use which model fits better (straight line vs curved growth) and whether the y-values go up or down to choose the function type in the answer choices.
Step-by-step Explanation
Decide if the algae’s area is increasing or decreasing
The problem says the algae doubles in area every 3 days. That means its area becomes larger and larger over time (for example: 4, then 8, then 16, etc.). So the relationship between area and time is increasing, not decreasing. That means any model that is "decreasing" can be eliminated.
Decide between linear change and multiplicative change
Now think about the pattern of change.
- Linear change means you add or subtract the same number each equal time step (like +5 every 3 days).
- In this problem, the area doubles every 3 days, which means you multiply by 2 each time (4 → 8 → 16 → 32 ...).
Because the change is by multiplying by the same factor (2) rather than adding the same amount, the relationship is not linear.
Name the function type that fits this pattern
When a quantity changes by being multiplied by the same factor over equal time intervals, that pattern is modeled by an exponential function. Here, the values are getting larger over time, so it is an increasing exponential relationship.
Therefore, the correct answer is Increasing exponential.