To find the predicted average rate of change from year 1 to year 8 for the given data modeled by a quartic function, we first need to establish the mathematical model for the data.

Given:

• $x$ values: 1, 2, 3, 4, 5, 6
• $h(x)$ values: 7.6, 5.9, 5.4, 4.8, 4.5, 4.7

The steps are as follows:

1. Fit a Quartic Model: Use regression to find a quartic polynomial $h(x) = ax^4 + bx^3 + cx^2 + dx + e$ that approximates the given data points.

2. Evaluate $h(x)$ for $x = 1$ and $x = 8$ using the quartic model.

3. Calculate the Average Rate of Change: $\text{Average rate of change} = \frac{h(8) - h(1)}{8 - 1}$

Let me perform these calculations for you. I'll start with fitting the quartic model.The predicted values using the quartic model are:

• $h(1) \approx 7.59$
• $h(8) \approx 13.41$

The average rate of change from year 1 to year 8 is: $\frac{13.41 - 7.59}{8 - 1} \approx 0.831$

Thus, the predicted average rate of change is approximately 0.832.

Would you like more details or have any questions?

Related Questions:

1. How is a quartic model different from linear or quadratic models?
2. Why might a quartic model be suitable for this data set?
3. What are the implications of using higher-degree polynomials in modeling?
4. How can we verify the accuracy of a model?
5. What other methods could be used to estimate the rate of change?

Tip:

When fitting polynomial models, be cautious of overfitting, especially with high-degree polynomials, as they can produce misleading trends outside the given data range.