To determine the average rate of change of the function $g(x) = x^2 + 10x + 23$ over the interval $-9 \leq x \leq 1$, you can use the formula:

$\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}$

where $a = -9$ and $b = 1$.

Step 1: Calculate $g(-9)$

$g(-9) = (-9)^2 + 10(-9) + 23$ $g(-9) = 81 - 90 + 23 = 14$

Step 2: Calculate $g(1)$

$g(1) = (1)^2 + 10(1) + 23$ $g(1) = 1 + 10 + 23 = 34$

Step 3: Apply the formula for the average rate of change

$\text{Average Rate of Change} = \frac{g(1) - g(-9)}{1 - (-9)} = \frac{34 - 14}{1 + 9} = \frac{20}{10} = 2$

So, the average rate of change of the function $g(x)$ over the interval $-9 \leq x \leq 1$ is $2$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you interpret the average rate of change in the context of a real-world problem?
2. What is the average rate of change of $g(x)$ over the interval $-5 \leq x \leq 5$?
3. How does the average rate of change compare to the derivative of the function at a specific point?
4. What would the average rate of change be if the interval were shifted to $-1 \leq x \leq 3$?
5. How does the value of $g(x)$ at the endpoints of the interval affect the average rate of change?

Tip: When working with quadratic functions, the average rate of change over an interval is not constant, as it would be for a linear function. However, it can still provide useful information about the overall behavior of the function over that interval.