Find the difference quotient for the function f(x)=-2x+5

The difference quotient for a function \( f(x) \) is given by the formula:

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

where \( h \) is a small increment in \( x \).

For the function \( f(x) = -2x + 5 \), we will first find \( f(x+h) \):

\[
f(x+h) = -2(x+h) + 5 = -2x - 2h + 5
\]

Now, substitute this into the difference quotient formula:

\[
\frac{f(x+h) - f(x)}{h} = \frac{(-2x - 2h + 5) - (-2x + 5)}{h}
\]

Simplify the expression:

\[
= \frac{-2x - 2h + 5 + 2x - 5}{h}
= \frac{-2h}{h}
= -2
\]

Thus, the difference quotient for the function \( f(x) = -2x + 5 \) is \( -2 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What does the difference quotient represent geometrically?
2. How does the difference quotient relate to the derivative of a function?
3. What is the difference quotient for the function \( f(x) = x^2 \)?
4. How does the difference quotient change for a quadratic function?
5. Can you interpret the result \( -2 \) in terms of the slope of a linear function?

**Tip:** The difference quotient is a fundamental concept for finding the derivative, which measures the rate of change of a function.

Find the difference quotient for the function f(x) = -2x + 5.

Learn how to calculate the difference quotient for the linear function f(x) = -2x + 5 with a detailed solution. This problem introduces the concept of difference quotients and how it relates to the slope of linear functions. Suitable for students in Grades 10-12.

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