The expression StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
 for hnot equals0 is called the difference quotient. Find and simplify the difference quotient for the following function.
​f(x)equalsnegative 9xsquaredplus2xplus8


We are tasked with finding and simplifying the difference quotient for the function:

\[
f(x) = -9x^2 + 2x + 8
\]

The difference quotient is given by the expression:

\[
\frac{f(x + h) - f(x)}{h} \quad \text{for} \quad h \neq 0
\]

### Step 1: Find \( f(x + h) \)

We start by substituting \( x + h \) into the function \( f(x) \):

\[
f(x + h) = -9(x + h)^2 + 2(x + h) + 8
\]

Now, let's expand and simplify:

\[
f(x + h) = -9[(x + h)^2] + 2(x + h) + 8
\]

First, expand \( (x + h)^2 \):

\[
(x + h)^2 = x^2 + 2xh + h^2
\]

So,

\[
f(x + h) = -9(x^2 + 2xh + h^2) + 2(x + h) + 8
\]
\[
= -9x^2 - 18xh - 9h^2 + 2x + 2h + 8
\]

### Step 2: Find \( f(x + h) - f(x) \)

Now subtract \( f(x) \) from \( f(x + h) \):

\[
f(x + h) - f(x) = (-9x^2 - 18xh - 9h^2 + 2x + 2h + 8) - (-9x^2 + 2x + 8)
\]

Distribute the negative sign:

\[
f(x + h) - f(x) = -9x^2 - 18xh - 9h^2 + 2x + 2h + 8 + 9x^2 - 2x - 8
\]

Now, combine like terms:

\[
f(x + h) - f(x) = (-18xh - 9h^2 + 2h)
\]

### Step 3: Simplify the difference quotient

Now, divide \( f(x + h) - f(x) \) by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{-18xh - 9h^2 + 2h}{h}
\]

Factor out \( h \) from the numerator:

\[
\frac{-18xh - 9h^2 + 2h}{h} = -18x - 9h + 2
\]

Thus, the simplified difference quotient is:

\[
\boxed{-18x - 9h + 2}
\]

This is the simplified form of the difference quotient for the given function.

Would you like any further details or have any questions?

Here are some related questions for further exploration:
1. What is the purpose of the difference quotient in calculus?
2. How does the difference quotient relate to the derivative of a function?
3. Can you apply the difference quotient to a linear function?
4. How does the result change if you modify the function to include a higher degree term?
5. How do you interpret the result of the difference quotient geometrically?

**Tip**: The difference quotient gives the average rate of change of a function over an interval. It becomes the derivative when the interval approaches zero.

The expression StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction for h not equals 0 is called the difference quotient. Find and simplify the difference quotient for the following function: f(x) equals negative 9x squared plus 2x plus 8

Learn how to find and simplify the difference quotient for the quadratic function f(x) = -9x² + 2x + 8. This step-by-step solution covers key algebraic techniques such as polynomial expansion and simplifies the difference quotient, a fundamental concept in calculus. Suitable for high school students in Grades 10-12.

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