To determine the function that best models the data in the table and find \( f(-1) \), we can start by analyzing the pattern of the y-values corresponding to the x-values.
Here are the x and y values again for reference:
| x | y | |---|----| | 0 | 5 | | 1 | 15 | | 2 | 45 | | 3 | 135| | 4 | 405|
Notice the pattern:
- \( 5 \) at \( x=0 \)
- \( 15 = 5 \cdot 3 \) at \( x=1 \)
- \( 45 = 15 \cdot 3 \) at \( x=2 \)
- \( 135 = 45 \cdot 3 \) at \( x=3 \)
- \( 405 = 135 \cdot 3 \) at \( x=4 \)
The pattern shows that y-values can be generated by multiplying the previous y-value by \( 3 \). Thus, it appears that the function is an exponential function.
The closest fitting function appears to be: \[ f(x) = 5 \cdot 3^x \]
Now, let's calculate \( f(-1) \): \[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
Now, let's look at the responses given:
- \( f(x) = 92x - 63; f(-1) = -155 \)
- \( f(x) = 2.23 \cdot (1.40)^x; f(-1) = 1.59 \)
- \( f(x) = 5 \cdot 3^x; f(-1) = \frac{5}{3} \)
- \( f(x) = 41.43x^2 - 73.71x + 19.86; f(-1) = 135 \)
It seems the third option accurately reflects the data and calculated value. Therefore, the correct answer is:
Response: \( f(x) = 5 \cdot 3^x; f(-1) = \frac{5}{3} \)