Question 4·Easy·Two-Variable Data: Models and Scatterplots
The table shows selected values from a function .
Which of the following is the best description of the function ?
For tables asking you to choose between linear and exponential (increasing or decreasing), first decide the direction: check quickly if the outputs go up or down as x increases to eliminate two choices. Next, test for linearity: compute a few differences between consecutive outputs—if they’re constant, it’s linear; if not, check the ratio of consecutive outputs—if that ratio is constant, it’s exponential. Finally, note whether the constant change is positive or negative (for linear) or whether the constant multiplier is greater than 1 or less than 1 (for exponential) to decide between increasing and decreasing.
Hints
Look at the direction of change
Compare , , , and . Are the values getting larger or smaller as increases?
Test for a linear pattern
For a linear function, when increases by 1, the change in should be the same each time. Subtract consecutive values and see if these differences match.
If it’s not linear, check multiplicative change
If the differences between values are not constant, divide each by the previous one. Is there a constant factor you multiply by each time?
Desmos Guide
Enter the data as a table
In Desmos, create a table and enter the x-values 0, 1, 2, 3 in the first column and the corresponding values 100, 90, 81, 72.9 in the second column. You should see four plotted points.
Compare linear and exponential fits
Add a linear regression by typing y1 ~ m x1 + b and an exponential regression by typing y1 ~ a b^x1. Check which model goes exactly through all four points and note whether the base is greater than 1 or less than 1.
Use the graph shape and base to match the description
Look at how the model’s curve moves as increases (upward or downward) and whether the base is above or below 1. Use this information to choose the option that correctly describes both the type of function (linear vs. exponential) and whether it is increasing or decreasing.
Step-by-step Explanation
Decide if the function is increasing or decreasing
Look at the values of as increases:
Each new value is smaller than the previous one, so the function is decreasing, not increasing. This immediately rules out the two "increasing" options.
Check if the pattern is linear (constant difference)
For a linear function, the difference between consecutive outputs is constant when increases by the same amount.
Compute the differences:
- From to : (so the change is )
- From to : (change is )
- From to : (change is )
These differences are not the same, so the function is not linear. That rules out both linear options.
Check for an exponential pattern (constant ratio) and describe it
For an exponential function, each output is multiplied by the same constant factor when increases by 1.
Compute the ratios of consecutive values:
The ratio is always , so the function is exponential with a constant factor less than 1, which makes it a decreasing exponential function. Therefore, the correct choice is D) Decreasing exponential.