To solve for the difference quotient [tex]\(\frac{f(x+h)-f(x)}{h}\)[/tex] where [tex]\(h \neq 0\)[/tex] for the function [tex]\( f(x) = -4x^2 - 2x + 6 \)[/tex], follow these steps:
1. Substitute [tex]\(x + h\)[/tex] into the function [tex]\(f(x)\)[/tex]:
Given the function [tex]\(f(x) = -4x^2 - 2x + 6\)[/tex], calculate [tex]\(f(x + h)\)[/tex]:
[tex]\[
f(x + h) = -4(x + h)^2 - 2(x + h) + 6
\][/tex]
2. Expand [tex]\((x + h)^2\)[/tex]:
[tex]\((x + h)^2 = x^2 + 2xh + h^2\)[/tex].
Substitute into [tex]\(f(x + h)\)[/tex]:
[tex]\[
f(x + h) = -4(x^2 + 2xh + h^2) - 2(x + h) + 6
\][/tex]
3. Distribute and simplify:
Distribute the [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex] in the expression:
[tex]\[
f(x + h) = -4x^2 - 8xh - 4h^2 - 2x - 2h + 6
\][/tex]
4. Compute the difference [tex]\(f(x + h) - f(x)\)[/tex]:
Subtract [tex]\(f(x) = -4x^2 - 2x + 6\)[/tex] from [tex]\(f(x + h)\)[/tex]:
[tex]\[
f(x + h) - f(x) = (-4x^2 - 8xh - 4h^2 - 2x - 2h + 6) - (-4x^2 - 2x + 6)
\][/tex]
Simplify the expression by combining like terms:
[tex]\[
f(x + h) - f(x) = -4x^2 - 8xh - 4h^2 - 2x - 2h + 6 + 4x^2 + 2x - 6
\][/tex]
After canceling terms, this simplifies to:
[tex]\[
f(x + h) - f(x) = -8xh - 4h^2 - 2h
\][/tex]
5. Form the difference quotient:
The difference quotient is given by:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{-8xh - 4h^2 - 2h}{h}
\][/tex]
6. Simplify the quotient:
Factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[
\frac{-8xh - 4h^2 - 2h}{h} = \frac{h(-8x - 4h - 2)}{h}
\][/tex]
Cancel [tex]\(h\)[/tex] from the numerator and the denominator:
[tex]\[
-8x - 4h - 2
\][/tex]
Therefore, the simplified difference quotient is:
[tex]\[
\frac{f(x+h)-f(x)}{h} = -8x - 4h - 2
\][/tex]
