To find the average rate of change for the interval from [tex]\(x = 7\)[/tex] to [tex]\(x = 8\)[/tex] in a quadratic function with a given vertex at [tex]\((0, 3)\)[/tex], we can follow these steps:
1. Calculate [tex]\( y \)[/tex] values at [tex]\( x = 7 \)[/tex] and [tex]\( x = 8 \)[/tex]:
- A quadratic function of the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] will have specific [tex]\( y \)[/tex]-values at these points. For this problem, they are:
- When [tex]\( x = 7 \)[/tex]:
[tex]\[
y_1 = -(7^2) + 3 = -49 + 3 = -46
\][/tex]
- When [tex]\( x = 8 \)[/tex]:
[tex]\[
y_2 = -(8^2) + 3 = -64 + 3 = -61
\][/tex]
2. Apply the average rate of change formula:
The formula to calculate the average rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on the function is:
[tex]\[
\text{Average rate of change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values we have:
[tex]\[
\frac{y_2 - y_1}{x_2 - x_1} = \frac{-61 - (-46)}{8 - 7} = \frac{-61 + 46}{1} = \frac{-15}{1} = -15
\][/tex]
Thus, the average rate of change for the interval from [tex]\( x = 7 \)[/tex] to [tex]\( x = 8 \)[/tex] is:
[tex]\[
\boxed{-15}
\][/tex]
Option C. -15 is the correct answer.
