To determine the type of function depicted in the table, let’s examine the pattern of $f(x)$ values with respect to the changes in $x$.

Observing the Change

1. When $x$ goes from -1 to 0, $f(x)$ changes from -4 to -6.
   • Change in $f(x)$: $-6 - (-4) = -2$
2. When $x$ goes from 0 to 1, $f(x)$ changes from -6 to -14.
   • Change in $f(x)$: $-14 - (-6) = -8$
3. When $x$ goes from 1 to 2, $f(x)$ changes from -14 to -50.
   • Change in $f(x)$: $-50 - (-14) = -36$
4. When $x$ goes from 2 to 3, $f(x)$ changes from -50 to -242.
   • Change in $f(x)$: $-242 - (-50) = -192$

The differences are not consistent, indicating this is not a linear function. Furthermore, the differences increase significantly, suggesting an exponential or polynomial growth.

Further Analysis

To determine if it's exponential, let's observe if there’s a common ratio. Dividing successive terms:

• $\frac{-6}{-4} = 1.5$
• $\frac{-14}{-6} \approx 2.33$
• $\frac{-50}{-14} \approx 3.57$
• $\frac{-242}{-50} \approx 4.84$

Since there is no consistent ratio, this function is likely a polynomial rather than exponential.

Conclusion

The function appears to be polynomial, possibly of a higher degree (such as cubic or quartic), due to the accelerating rate of change in values.

Would you like further details on finding the exact polynomial, or do you have any specific questions?

Related Questions

1. How can we determine the exact degree of this polynomial?
2. What methods can be used to determine a polynomial function from a table of values?
3. How does exponential growth differ from polynomial growth in tables of values?
4. What are common indicators of polynomial versus exponential functions?
5. Can you use finite differences to determine the degree of a polynomial function?

Tip

When faced with a table of values, calculate finite differences (first, second, etc.) to determine if a function is polynomial and to identify its degree.