To find the average rate of change of the quadratic function over the interval from [tex]\( x = 9 \)[/tex] to [tex]\( x = 10 \)[/tex], we need to follow these steps:
1. Determine the values of the function at [tex]\( x = 9 \)[/tex] and [tex]\( x = 10 \)[/tex].
2. Calculate the difference in the function values.
3. Calculate the difference in the [tex]\( x \)[/tex]-values.
4. Use these differences to find the average rate of change.
Step 1: Determine the values of the function at [tex]\( x = 9 \)[/tex] and [tex]\( x = 10 \)[/tex].
Given the function [tex]\( f(x) = x^2 - 1 \)[/tex],
- For [tex]\( x = 9 \)[/tex], we have [tex]\( f(9) = 9^2 - 1 \)[/tex]. Therefore, [tex]\( f(9) = 81 - 1 = 80 \)[/tex].
- For [tex]\( x = 10 \)[/tex], we have [tex]\( f(10) = 10^2 - 1 \)[/tex]. Therefore, [tex]\( f(10) = 100 - 1 = 99 \)[/tex].
Step 2: Calculate the difference in the function values.
[tex]\[ f(10) - f(9) = 99 - 80 = 19 \][/tex]
Step 3: Calculate the difference in the [tex]\( x \)[/tex]-values.
[tex]\[ 10 - 9 = 1 \][/tex]
Step 4: Use these differences to find the average rate of change.
[tex]\[ \text{Average rate of change} = \frac{f(10) - f(9)}{10 - 9} = \frac{19}{1} = 19 \][/tex]
Thus, the average rate of change for the interval from [tex]\( x = 9 \)[/tex] to [tex]\( x = 10 \)[/tex] is [tex]\( 19 \)[/tex].
The correct answer is:
C. -19
