If f(x) = x2 − 4x, then f(x + h) − f(x) =

Given the function \( f(x) = x^2 - 4x \), let's find \( f(x + h) - f(x) \).

1. First, substitute \( x + h \) into \( f(x) \):

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

Expand both terms:

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

So,

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

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

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

Simplify:

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

Cancel out \( x^2 \) and \( -4x \):

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

Thus, the result is:

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

Would you like further details on this? Here are five related questions you could explore:

1. What is the difference quotient \( \frac{f(x+h) - f(x)}{h} \) for this function?
2. How does this result relate to the derivative of \( f(x) \)?
3. What is the general form for \( f(x+h) - f(x) \) for a quadratic function?
4. Can you find the second derivative of \( f(x) = x^2 - 4x \)?
5. How does the value of \( h \) affect the expression \( f(x+h) - f(x) \)?

**Tip:** When simplifying expressions, always look for common terms to cancel early to reduce complexity.

If f(x) = x^2 − 4x, then f(x + h) − f(x) = ?

Explore the algebraic expansion of the function f(x) = x^2 − 4x and learn how to compute f(x + h) − f(x). This solution breaks down key algebraic concepts like quadratic function expansion and simplification, suitable for students in Grades 9-11.

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