find the difference quotient of f. be sure to simplify f(x)=x^2-x+4

The difference quotient is used to approximate the derivative of a function \( f(x) \). It is given by the formula:

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

### Step 1: Find \( f(x+h) \)
Given \( f(x) = x^2 - x + 4 \), we substitute \( x+h \) into the function:

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

Expanding each term:

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

Thus, 

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

### Step 2: Compute the Difference Quotient

Now, plug \( f(x+h) \) and \( f(x) \) into the difference quotient formula:

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

Distribute and simplify the numerator:

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

### Step 3: Factor and Simplify

Now, factor out an \( h \) from the numerator:

\[
\frac{2xh + h^2 - h}{h} = \frac{h(2x + h - 1)}{h}
\]

Cancel the \( h \) from the numerator and denominator:

\[
= 2x + h - 1
\]

### Final Answer:

The simplified difference quotient is:

\[
2x + h - 1
\]

Would you like a detailed explanation of any specific step?

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Here are five related questions to expand your understanding:

1. What happens to the difference quotient as \( h \) approaches zero?
2. How would you find the derivative of \( f(x) = x^2 - x + 4 \) directly using differentiation rules?
3. Can you apply the difference quotient to a trigonometric function like \( f(x) = \sin(x) \)?
4. How does the difference quotient relate to the concept of a secant line on the graph of \( f(x) \)?
5. What role does the difference quotient play in understanding the slope of a curve?

**Tip:** The difference quotient is a fundamental concept for understanding how the derivative measures the rate of change of a function at a point.

Explore how to simplify the difference quotient for the quadratic function f(x)=x^2-x+4 with a detailed step-by-step solution. This guide covers key algebraic concepts, including the difference quotient formula and its connection to the derivative. Suitable for students in Grades 10-12.

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