Show that the following data could not possibly be fit by an exponential function
x y
3 72
5 200
7 392
To determine whether the given data can be fit by an exponential function, we need to check if there is a constant ratio between consecutive y-values for different x-values. If the ratio is not constant, then the data cannot be modeled by an exponential function.
Let's calculate the ratios for the given data:
Ratio between y-values for x=3 and x=5: 200 / 72 ≈ 2.778
Ratio between y-values for x=5 and x=7: 392 / 200 = 1.96
As the ratios are not equal, we can see that there is no constant ratio between consecutive y-values. Hence, the data cannot be fit by an exponential function.
To check for the possibility of fitting data with an exponential function in general, you can follow these steps:
1. Plot the given data points on a graph.
2. Draw a smooth curve that you think might fit the data based on its shape.
3. Calculate the ratios between consecutive y-values for different x-values.
4. If the ratios are constant, then the data can be modeled by an exponential function. However, if the ratios vary significantly, as in this case, the data cannot be fit by an exponential function.
Please note that this is a visual approach to determine whether data can be fit by an exponential function. For a more rigorous mathematical analysis, you can examine the residuals and perform regression analysis to find the best-fitting function.