To determine the nature of the function model that fits the given dataset, we can analyze the growth of the y-values as the x-values increase. Here are the y-values corresponding to each x-value:
- For \( x = 0 \): \( y = 5 \)
- For \( x = 1 \): \( y = 35 \)
- For \( x = 2 \): \( y = 245 \)
- For \( x = 3 \): \( y = 1715 \)
- For \( x = 4 \): \( y = 12005 \)
- For \( x = 5 \): \( y = 84035 \)
Step 1: Determine the type of function
Linear Function
A linear function has constant increases in y-values. The differences between successive y-values are:
- \( 35 - 5 = 30 \)
- \( 245 - 35 = 210 \)
- \( 1715 - 245 = 1470 \)
- \( 12005 - 1715 = 10290 \)
- \( 84035 - 12005 = 72030 \)
These increases are not constant, so this is not a linear function.
Quadratic Function
A quadratic function has constant second differences. Let's find the first differences (as above) and then find second differences:
First differences:
- \( 30, 210, 1470, 10290, 72030 \)
Second differences:
- \( 210 - 30 = 180 \)
- \( 1470 - 210 = 1260 \)
- \( 10290 - 1470 = 8840 \)
- \( 72030 - 10290 = 61740 \)
These differences are not constant, indicating it is not a quadratic function.
Exponential Function
An exponential function has y-values that increase by a constant factor. Let's see if the ratios between successive y-values are constant:
- \( \frac{35}{5} = 7 \)
- \( \frac{245}{35} = 7 \)
- \( \frac{1715}{245} = 7 \)
- \( \frac{12005}{1715} = 7 \)
- \( \frac{84035}{12005} = 7 \)
All of these ratios equal \( 7 \), showing that the y-values are increasing by a constant factor as x increases.
Conclusion
The dataset represents an exponential function.
The function can be modeled as: \[ f(x) = 5 \cdot 7^x \] This confirms the response: \[ f(x) = 5(7^x) \]