To find the function that best models the given data, let's analyze the provided points in pairs:
- (0, 0)
- (1, 1)
- (2, 15)
- (3, 135)
- (4, 405)
Looking at the y-values, we can observe that as x increases, the y-values grow rapidly, suggesting that this could be a polynomial or exponential function.
Now, let’s see if we can find a function that fits the data. From the analysis of the growth in y-values, we see that it appears to grow quadratically but we will need to do a closer inspection.
Plugging the x-values into possible functions gives us a chance to find which matches well.
- For the polynomial:
- If we analyze the pattern:
- At \( x=0 \), \( y=0 \)
- At \( x=1 \), \( y=1 \)
- At \( x=2 \), \( y=15 \) (difference suggests from 1 to 15)
- At \( x=3 \), \( y=135 \) (huge increase)
- At \( x=4 \), \( y=405 \)
- If we analyze the pattern:
The growth can be shifted explored by using polynomial regression to find the curve that matches.
Testing the options:
-
Calculating f(-1) using each potential function:
-
f(x) = 92x - 63
- \( f(-1) = 92(-1) - 63 = -92 - 63 = -155 \)
-
f(x) = 5.32
- \( f(-1) = 5.32 \) (this function is constant)
-
f(x) = 2.23(1.4)^x
- \( f(-1) = 2.23(1.4)^{-1} = 2.23 / 1.4 = ~1.59 \)
-
f(x) = 41.43x^2 - 73.71x + 19.86
- Using \( x = -1 \) \[ f(-1) = 41.43(-1)^2 - 73.71(-1) + 19.86 = 41.43 + 73.71 + 19.86 = 135 \]
-
Conclusion:
The value of the function that best describes the data appears to be \( f(x) = 41.43x^2 - 73.71x + 19.86 \) based on the rapid growth in \( y \).
Thus, for this function, the value when \( r = -1 \) is:
(
f(-1) = 135 \
)
The correct answer is: f(x) = 41.43x^2 - 73.71x + 19.86; f(-1) = 135
1 answer