To find the average rate of change of the quadratic function on the interval from [tex]\(x = 5\)[/tex] to [tex]\(x = 6\)[/tex], we proceed as follows:
1. Identify the function values at the given points:
- For [tex]\(x = 5\)[/tex], the table shows [tex]\(y = 18\)[/tex].
- For [tex]\(x = 6\)[/tex], based on the quadratic function given and its vertex at [tex]\((1,2)\)[/tex], the value is [tex]\(y = 27\)[/tex].
2. Determine the interval length:
- The interval length [tex]\(\Delta x\)[/tex] is [tex]\(6 - 5 = 1\)[/tex].
3. Calculate the average rate of change:
- The average rate of change is given by the formula:
[tex]\[
\text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y(6) - y(5)}{6 - 5}
\][/tex]
- Substituting the known values:
[tex]\[
\text{Average Rate of Change} = \frac{27 - 18}{1} = \frac{9}{1} = 9
\][/tex]
Therefore, the average rate of change for the interval from [tex]\(x=5\)[/tex] to [tex]\(x=6\)[/tex] is [tex]\(9\)[/tex].
So the correct answer is:
D. 9
