Find the difference quotient of​ f; that​ is, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction comma
h not equals 0 comma for the following function.  Be sure to simplify.
f left parenthesis x right parenthesis equals x squared minus 6 x plus 4

We are asked to find the difference quotient of the function \( f(x) = x^2 - 6x + 4 \). The difference quotient is given by:

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

### Step 1: Find \( f(x+h) \)
To do this, replace every occurrence of \( x \) in \( f(x) \) with \( x+h \).

\[
f(x+h) = (x+h)^2 - 6(x+h) + 4
\]

Now expand and simplify each term:

\[
(x+h)^2 = x^2 + 2xh + h^2
\]
\[
-6(x+h) = -6x - 6h
\]

Thus,

\[
f(x+h) = (x^2 + 2xh + h^2) - (6x + 6h) + 4
\]

Simplifying:

\[
f(x+h) = x^2 + 2xh + h^2 - 6x - 6h + 4
\]

### Step 2: Subtract \( f(x) \) from \( f(x+h) \)
Now subtract \( f(x) = x^2 - 6x + 4 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = \left(x^2 + 2xh + h^2 - 6x - 6h + 4\right) - \left(x^2 - 6x + 4\right)
\]

Distribute the negative sign and subtract:

\[
f(x+h) - f(x) = x^2 + 2xh + h^2 - 6x - 6h + 4 - x^2 + 6x - 4
\]

Now cancel out the like terms:

- \( x^2 \) cancels with \( -x^2 \)
- \( -6x \) cancels with \( +6x \)
- \( 4 \) cancels with \( -4 \)

This leaves:

\[
f(x+h) - f(x) = 2xh + h^2 - 6h
\]

### Step 3: Divide by \( h \)
Now, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 6h}{h}
\]

Simplify each term:

\[
\frac{f(x+h) - f(x)}{h} = 2x + h - 6
\]

### Final Answer
Thus, the simplified difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = 2x + h - 6, \quad h \neq 0
\]

Would you like more details on this process, or do you have any questions?

Here are five related questions to expand on this topic:
1. How does the difference quotient relate to the derivative of a function?
2. Can you compute the difference quotient for \( f(x) = 3x^2 - 2x + 1 \)?
3. What happens to the difference quotient as \( h \) approaches zero?
4. How would the result change if \( f(x) \) were a cubic function?
5. How can the difference quotient be interpreted geometrically?

**Tip:** The difference quotient provides an average rate of change of the function over an interval and is a crucial step in finding the derivative.

Find the difference quotient of​ f; that​ is, find StartFraction f(x + h) - f(x) Over h, h ≠ 0, for the following function. Be sure to simplify: f(x) = x^2 - 6x + 4.

Learn how to calculate the difference quotient for the quadratic function f(x) = x^2 - 6x + 4. This step-by-step solution explores key concepts in algebra and quadratic functions, including how to simplify the difference quotient. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation