Let's go through the problem step by step.
1. Identify the function and values:
- The function given is [tex]\( f(x) = 6 - 8x \)[/tex].
- We are asked to evaluate the function at specific points [tex]\( a = 1 \)[/tex] and [tex]\( a + h = 2 \)[/tex] (considering [tex]\( h = 1 \)[/tex]).
2. Calculate [tex]\( f(a) \)[/tex]:
- Substitute [tex]\( a = 1 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(a) = f(1) = 6 - 8 \cdot 1 = 6 - 8 = -2
\][/tex]
So, [tex]\( f(a) = -2 \)[/tex].
3. Calculate [tex]\( f(a+h) \)[/tex]:
- Substitute [tex]\( a + h = 1 + 1 = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(a+h) = f(2) = 6 - 8 \cdot 2 = 6 - 16 = -10
\][/tex]
So, [tex]\( f(a+h) = -10 \)[/tex].
4. Compute the difference quotient:
- The difference quotient is given by [tex]\( \frac{f(a+h) - f(a)}{h} \)[/tex]:
[tex]\[
\frac{f(a+h) - f(a)}{h} = \frac{-10 - (-2)}{1} = \frac{-10 + 2}{1} = \frac{-8}{1} = -8
\][/tex]
The difference quotient is [tex]\( -8 \)[/tex].
To summarize, the function evaluations and the difference quotient are:
[tex]\[
f(a) = -2
\][/tex]
[tex]\[
f(a+h) = -10
\][/tex]
[tex]\[
\frac{f(a+h) - f(a)}{h} = -8
\][/tex]
