https://file.aiquickdraw.com/mathbot/file/1727330802540mjkn9mcr.jpg

Let's solve the questions based on the function provided:

The function is:

\[
f(x) = x^2 + 2x + 1
\]

### a. \( f(a) \)

To find \( f(a) \), substitute \( a \) for \( x \) in the function \( f(x) \):

\[
f(a) = a^2 + 2a + 1
\]

### b. \( f(a+h) \)

To find \( f(a+h) \), substitute \( a + h \) for \( x \) in the function:

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

Now expand this expression:

\[
(a+h)^2 = a^2 + 2ah + h^2
\]
\[
2(a+h) = 2a + 2h
\]

Thus:

\[
f(a+h) = a^2 + 2ah + h^2 + 2a + 2h + 1
\]

### c. \( \frac{f(a+h) - f(a)}{h} \)

Now, calculate the difference quotient:

\[
\frac{f(a+h) - f(a)}{h} = \frac{(a^2 + 2ah + h^2 + 2a + 2h + 1) - (a^2 + 2a + 1)}{h}
\]

Simplify the expression in the numerator:

\[
= a^2 + 2ah + h^2 + 2a + 2h + 1 - a^2 - 2a - 1
\]

After cancellation, we are left with:

\[
= 2ah + h^2 + 2h
\]

Thus, the expression becomes:

\[
\frac{2ah + h^2 + 2h}{h}
\]

Now, factor out \( h \) from the numerator:

\[
= \frac{h(2a + h + 2)}{h}
\]

Cancel the \( h \):

\[
= 2a + h + 2
\]

So:

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

---

Would you like more details or have any questions about this solution? 

Here are 5 related questions to explore:

1. What is the meaning of the difference quotient in calculus?
2. How would you find the derivative of \( f(x) = x^2 + 2x + 1 \) using limits?
3. How does the expression \( \frac{f(a+h) - f(a)}{h} \) lead to the derivative when \( h \to 0 \)?
4. Can you apply the same method to find the difference quotient for a cubic function?
5. How does the difference quotient represent the slope of the tangent line at a point?

**Tip:** The difference quotient is an important step toward understanding the derivative of a function, which gives the rate of change at any point.

Consider the function f(x) = x^2 + 2x + 1 and evaluate the specified values: a. f(a), b. f(a+h), and c. (f(a+h)-f(a))/h.

Explore how to calculate f(a), f(a+h), and the difference quotient for the quadratic function f(x) = x^2 + 2x + 1. This problem involves applying algebraic techniques to expand and simplify expressions and is suited for students in Grades 10-12.

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