To determine which function best fits the data provided, let's analyze the x-values and corresponding y-values:
- \( f(0) = 5 \)
- \( f(1) = 15 \)
- \( f(2) = 45 \)
- \( f(3) = 135 \)
- \( f(4) = 405 \)
Looking at the y-values, they appear to increase quite rapidly, suggesting a possible exponential or polynomial relationship.
Let's evaluate each of the options given.
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Option 1: \( f(x) = 2.23 \cdot (1.40)^x \)
- \( f(-1) = 2.23 \cdot (1.40)^{-1} = 2.23 / 1.40 \approx 1.59 \)
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Option 2: \( f(x) = 5 \cdot 3^x \)
- \( f(-1) = 5 \cdot 3^{-1} = 5 / 3 \approx 1.67 \)
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Option 3: \( f(x) = 92x - 63 \)
- \( f(-1) = 92(-1) - 63 = -92 - 63 = -155 \)
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Option 4: \( f(x) = 41.43x^2 - 73.71x + 19.86 \)
- \( f(-1) = 41.43(-1)^2 - 73.71(-1) + 19.86 = 41.43 + 73.71 + 19.86 = 135 \)
Among these options, only the fourth function accurately produces a value for \( f(-1) \) that matches the provided output of 135. However, we need a function that truly describes the behavior of the provided table of values.
Since the sequence of y-values appears to follow a pattern of multiplication by three for increasing x-values (with some initial adjustment), let's check the polynomial:
- The actual values are growing more like a polynomial than an exponential. The resulting \( f(-1) = 135 \) suggests it might create positive values through higher order terms, thus potentially serving as an appropriate approximation for the dataset.
Given all of these evaluations, the best answer is as follows:
The function that best models the data is:
\( f(x) = 41.43x^2 - 73.71x + 19.86 \)
And \( f(-1) = 135 \).
3 answers