To find and simplify the difference quotient for the function [tex]\( f(x) = -3x^2 + 5x + 5 \)[/tex], follow these steps:
1. Write the expression for the difference quotient:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute [tex]\( f(x) = -3x^2 + 5x + 5 \)[/tex] into the quotient:
[tex]\[
f(x) = -3x^2 + 5x + 5
\][/tex]
3. Find [tex]\( f(x+h) \)[/tex]:
Substitute [tex]\( x+h \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[
f(x+h) = -3(x+h)^2 + 5(x+h) + 5
\][/tex]
Expand [tex]\( (x+h)^2 \)[/tex]:
[tex]\[
(x+h)^2 = x^2 + 2xh + h^2
\][/tex]
Then,
[tex]\[
f(x+h) = -3(x^2 + 2xh + h^2) + 5(x+h) + 5
\][/tex]
Simplify:
[tex]\[
f(x+h) = -3x^2 - 6xh - 3h^2 + 5x + 5h + 5
\][/tex]
4. Compute [tex]\( f(x+h) - f(x) \)[/tex]:
[tex]\[
f(x+h) - f(x) = (-3x^2 - 6xh - 3h^2 + 5x + 5h + 5) - (-3x^2 + 5x + 5)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
f(x+h) - f(x) = -3x^2 - 6xh - 3h^2 + 5x + 5h + 5 + 3x^2 - 5x - 5
\][/tex]
Simplify:
[tex]\[
f(x+h) - f(x) = -6xh - 3h^2 + 5h
\][/tex]
5. Divide by [tex]\( h \)[/tex]:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{-6xh - 3h^2 + 5h}{h}
\][/tex]
6. Simplify the resulting expression:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{h(-6x - 3h + 5)}{h}
\][/tex]
Cancel [tex]\( h \)[/tex] in the numerator and denominator (since [tex]\( h \neq 0 \)[/tex]):
[tex]\[
\frac{f(x+h) - f(x)}{h} = -6x - 3h + 5
\][/tex]
Therefore, the simplified difference quotient for the function [tex]\( f(x) = -3x^2 + 5x + 5 \)[/tex] is:
[tex]\[
-6x - 3h + 5
\][/tex]
