https://file.aiquickdraw.com/mathbot/file/172669156373077q4qkba.jpg

Let's work through the problem step by step.

We are given:
\[
f(x) = 8 - x^2
\]
and need to simplify:
\[
f(x+h) - f(x)
\]

### Step 1: Find \( f(x+h) \)
Substitute \( x+h \) into the function \( f(x) \):
\[
f(x+h) = 8 - (x+h)^2
\]

### Step 2: Expand \( (x+h)^2 \)
\[
(x+h)^2 = x^2 + 2xh + h^2
\]
Thus,
\[
f(x+h) = 8 - (x^2 + 2xh + h^2) = 8 - x^2 - 2xh - h^2
\]

### Step 3: Subtract \( f(x) \) from \( f(x+h) \)
Now subtract \( f(x) = 8 - x^2 \) from \( f(x+h) \):
\[
f(x+h) - f(x) = \left(8 - x^2 - 2xh - h^2\right) - (8 - x^2)
\]

Simplify the expression:
\[
f(x+h) - f(x) = 8 - x^2 - 2xh - h^2 - 8 + x^2
\]

The \( 8 \) and \( -8 \) cancel, and the \( -x^2 \) and \( x^2 \) cancel, leaving:
\[
f(x+h) - f(x) = -2xh - h^2
\]

### Final Answer:
\[
f(x+h) - f(x) = -2xh - h^2
\]

Would you like more details on any part of the solution?

Here are 5 related questions:
1. How would this expression be used to find the difference quotient?
2. How does the expansion of \( (x+h)^2 \) affect the final expression?
3. What is the geometric interpretation of \( f(x+h) - f(x) \)?
4. How do we generalize this process for other types of functions?
5. How would this approach change for a cubic function?

**Tip:** When simplifying expressions like \( f(x+h) - f(x) \), be careful with distributing negative signs and combining like terms to avoid errors.

Given f(x) = 8 - x^2, find and simplify f(x + h) - f(x).

Learn how to simplify f(x + h) - f(x) for the quadratic function f(x) = 8 - x^2. This step-by-step guide covers expanding expressions and subtracting functions. Suitable for high school students.

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