f(x) = 1 + 2x − x2,    
f(3 + h) − f(3)
h


To evaluate the given expression, we'll follow these steps:

### 1. Calculate \( f(3 + h) \):
Given \( f(x) = 1 + 2x - x^2 \), substitute \( x = 3 + h \) into the function.

\[
f(3 + h) = 1 + 2(3 + h) - (3 + h)^2
\]

Simplify:

\[
f(3 + h) = 1 + 6 + 2h - (9 + 6h + h^2)
\]

\[
f(3 + h) = 7 + 2h - 9 - 6h - h^2
\]

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

### 2. Calculate \( f(3) \):
Substitute \( x = 3 \) into the function \( f(x) \):

\[
f(3) = 1 + 2(3) - 3^2
\]

\[
f(3) = 1 + 6 - 9 = -2
\]

### 3. Calculate \( \frac{f(3 + h) - f(3)}{h} \):
Now, substitute \( f(3 + h) \) and \( f(3) \) into the expression:

\[
\frac{f(3 + h) - f(3)}{h} = \frac{(-2h - h^2 - 2) - (-2)}{h}
\]

Simplify the expression:

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

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

### Final Answer:
\[
\frac{f(3 + h) - f(3)}{h} = -2 - h
\]

Would you like any further details or have any questions?

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Here are 5 related questions you might find interesting:
1. What is the derivative of \( f(x) = 1 + 2x - x^2 \)?
2. How does the expression \( \frac{f(3 + h) - f(3)}{h} \) relate to the derivative of \( f(x) \)?
3. What would the expression \( \frac{f(a + h) - f(a)}{h} \) look like for any \( a \)?
4. Can you find the value of the derivative at \( x = 3 \) using the first principles definition of a derivative?
5. How would the function \( f(x) = 1 + 2x - x^2 \) change if the coefficients were different?

**Tip:** When simplifying expressions, carefully distribute negative signs and combine like terms to avoid common errors.

f(x) = 1 + 2x − x^2, f(3 + h) − f(3)/h

Explore the process of evaluating the difference quotient for the quadratic function f(x) = 1 + 2x − x^2. This detailed step-by-step guide demonstrates key algebraic concepts, including the first principles definition of a derivative, and is suitable for students in Grades 10-12.

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