To determine the average rate of change for the interval from [tex]\( x = 5 \)[/tex] to [tex]\( x = 6 \)[/tex] on this quadratic function, we follow these steps:
1. Identify coordinates for the given interval:
- The coordinates for [tex]\( x = 5 \)[/tex] are given as [tex]\( (5, 17) \)[/tex].
- The value for [tex]\( y \)[/tex] when [tex]\( x = 6 \)[/tex] is not provided in the table. However, based on the question setup, we need to determine this point, presumed to be [tex]\( (6, 26) \)[/tex].
2. Formula for Average Rate of Change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
- In this context, [tex]\( a = 5 \)[/tex], [tex]\( b = 6 \)[/tex], [tex]\( f(a) = 17 \)[/tex], and [tex]\( f(b) = 26 \)[/tex].
3. Plug in the values:
[tex]\[
\frac{f(6) - f(5)}{6 - 5} = \frac{26 - 17}{6 - 5}
\][/tex]
4. Calculate the differences and ratio:
[tex]\[
\frac{26 - 17}{1} = \frac{9}{1} = 9
\][/tex]
So, the average rate of change for the interval from [tex]\( x = 5 \)[/tex] to [tex]\( x = 6 \)[/tex] is [tex]\( 9.0 \)[/tex].
