https://file.aiquickdraw.com/mathbot/file/1726710061543a8lpgp27.jpg

The problem in the image asks to find and simplify the difference quotient for the given function:

\[
f(x) = -x^2 + 7x + 4
\]

The difference quotient formula is:

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

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

We need to plug \( x+h \) into the function \( f(x) \):

\[
f(x+h) = -(x+h)^2 + 7(x+h) + 4
\]

Now expand this expression:

\[
f(x+h) = -(x^2 + 2xh + h^2) + 7(x+h) + 4
\]
\[
= -x^2 - 2xh - h^2 + 7x + 7h + 4
\]

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

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

\[
f(x+h) - f(x) = (-x^2 - 2xh - h^2 + 7x + 7h + 4) - (-x^2 + 7x + 4)
\]
\[
= -x^2 - 2xh - h^2 + 7x + 7h + 4 + x^2 - 7x - 4
\]
Simplify:
\[
= -2xh - h^2 + 7h
\]

### Step 3: Divide by \( h \)

Now divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2 + 7h}{h}
\]

Simplify each term:

\[
= -2x - h + 7
\]

### Final Result:

\[
\frac{f(x+h) - f(x)}{h} = -2x - h + 7
\]

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:
1. How is the difference quotient related to the derivative of a function?
2. What happens to the difference quotient as \( h \) approaches 0?
3. Why do we expand \( f(x+h) \) before subtracting \( f(x) \)?
4. Can you compute the derivative of \( f(x) = -x^2 + 7x + 4 \) using the power rule and compare it with the difference quotient?
5. How would the difference quotient change if we had a cubic function instead?

**Tip**: The difference quotient gives an approximation of the derivative, which represents the slope of the tangent line at any point on the curve.

Find and simplify the difference quotient for f(x) = -x^2 + 7x + 4 using f(x+h) - f(x)/h.

Learn how to simplify the difference quotient for the quadratic function f(x) = -x^2 + 7x + 4. This detailed solution walks through key algebraic concepts, making it perfect for high school students learning about derivatives and difference quotients.

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