To find the average rate of change of a quadratic function over a specified interval, we can use the formula for the average rate of change, which is simply the change in the function values divided by the change in the input values (x-values).
The formula for average rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given the interval from [tex]\(x = 4\)[/tex] to [tex]\(x = 6\)[/tex]:
1. Identify the function values [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] at the given [tex]\(x\)[/tex]-values [tex]\(x_1 = 4\)[/tex] and [tex]\(x_2 = 6\)[/tex].
From the table:
- When [tex]\(x = 4\)[/tex], [tex]\(y = -17\)[/tex], so [tex]\(y_1 = -17\)[/tex].
- When [tex]\(x = 6\)[/tex], [tex]\(y = -37\)[/tex], so [tex]\(y_2 = -37\)[/tex].
2. Apply the values to the average rate of change formula:
[tex]\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-37 - (-17)}{6 - 4}
\][/tex]
3. Calculate the difference in [tex]\(y\)[/tex]-values and [tex]\(x\)[/tex]-values:
[tex]\[
y_2 - y_1 = -37 - (-17) = -37 + 17 = -20
\][/tex]
[tex]\[
x_2 - x_1 = 6 - 4 = 2
\][/tex]
4. Divide the change in [tex]\(y\)[/tex]-values by the change in [tex]\(x\)[/tex]-values:
[tex]\[
\text{Average Rate of Change} = \frac{-20}{2} = -10
\][/tex]
Therefore, the average rate of change for the interval from [tex]\(x=4\)[/tex] to [tex]\(x=6\)[/tex] is [tex]\(-10\)[/tex].
Thus, the correct answer is:
A. -10
