To determine which function best models the given data, we need to analyze the pattern in the provided \( x \)-Values and \( y \)-Values. Let's look at the data:
| x-Values | y-Values | |----------|----------| | 0 | 5 | | 1 | 15 | | 2 | 45 | | 3 | 135 | | 4 | 405 |
Observing the \( y \)-Values, we can see that:
- Each \( y \)-Value seems to be multiplied by 3 as we move from one \( x \)-Value to the next:
- \( 5 \times 3 = 15 \)
- \( 15 \times 3 = 45 \)
- \( 45 \times 3 = 135 \)
- \( 135 \times 3 = 405 \)
This suggests that the function may be exponential.
The second response suggests a function of the form:
\( f(x) = 5 \cdot 3^x \)
Let's evaluate this function at \( x = -1 \):
\[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
The other responses can be ruled out based on the multipliers or degree of the polynomial when analyzed correctly against the generated \( y \)-values.
Thus, the function that best models the data is:
f(x) = 5 ⋅ 3^x; f(−1) = \frac{5}{3}
Final Answer:
f(x) = 5⋅3^x; f(−1) = 5/3