Question

x-Values y-Values

0

1

5

2

15

45

3

135

4

405

Select the function that best models the data and determine the value of the function at r = - 1

(1 point)

f(x)=92x-63. f(- 1) = - 155

f(x) = 5.32 f(-1) =

f(x)=2.23.(1.40) f(- 1) = 1.59

f(x) = 41.43x ^ 2 73.71x + 19.86; f(-1) = 135

Answer 1

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.

1. **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 \( x=4 \), \( y=405 \)

The growth can be shifted explored by using polynomial regression to find the curve that matches.

### Testing the options:
1. **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**