Evaluate the difference quotient for the given function. Simplify your answer.

*f*(*x*) = 4 + 3*x* − *x*2,    

*f*(4 + *h*) − *f*(4)

*h*

The difference quotient is given by:

\[
\frac{f(4 + h) - f(4)}{h}
\]

Given the function \( f(x) = 4 + 3x - x^2 \), we'll first find \( f(4 + h) \) and \( f(4) \).

### Step 1: Calculate \( f(4 + h) \)
Substitute \( 4 + h \) into the function \( f(x) = 4 + 3x - x^2 \):

\[
f(4 + h) = 4 + 3(4 + h) - (4 + h)^2
\]

First, simplify \( 3(4 + h) \):

\[
3(4 + h) = 12 + 3h
\]

Next, expand \( (4 + h)^2 \):

\[
(4 + h)^2 = 16 + 8h + h^2
\]

So, the expression for \( f(4 + h) \) becomes:

\[
f(4 + h) = 4 + 12 + 3h - (16 + 8h + h^2)
\]

Simplify the expression:

\[
f(4 + h) = 16 + 3h - 16 - 8h - h^2 = -5h - h^2
\]

### Step 2: Calculate \( f(4) \)
Substitute \( x = 4 \) into the function:

\[
f(4) = 4 + 3(4) - 4^2
\]

Simplify:

\[
f(4) = 4 + 12 - 16 = 0
\]

### Step 3: Compute the Difference Quotient
Now, substitute \( f(4 + h) \) and \( f(4) \) into the difference quotient formula:

\[
\frac{f(4 + h) - f(4)}{h} = \frac{-5h - h^2 - 0}{h}
\]

Simplify the expression:

\[
\frac{-5h - h^2}{h} = -5 - h
\]

Thus, the simplified difference quotient is:

\[
-5 - h
\]

Do you have any more questions or need further clarification on any steps?

Here are 5 related questions:

1. What is the difference quotient for \( f(x) = x^2 - 3x + 2 \)?
2. How would the difference quotient change if \( f(x) \) were a linear function?
3. What is the geometric interpretation of the difference quotient?
4. How does the difference quotient relate to the derivative of a function?
5. Evaluate the difference quotient for \( f(x) = \frac{1}{x} \) at a different point.

**Tip:** The difference quotient is a foundational concept in calculus, leading directly to the definition of the derivative. Practice simplifying these expressions to build your skills!

Learn how to evaluate the difference quotient for the function f(x) = 4 + 3x - x^2 at a specific point. This problem involves polynomial functions and requires understanding of function evaluation and simplification. Suitable for advanced high school students.

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