Let  f(x)=3x2−5x−9.  Find the difference quotient from  x  to  x+h. 

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

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

Given \( f(x) = 3x^2 - 5x - 9 \), let's compute the difference quotient step by step.

### Step 1: Find \( f(x+h) \)
Substitute \( x+h \) into \( f(x) \):

\[
f(x+h) = 3(x+h)^2 - 5(x+h) - 9
\]

First, expand \( (x+h)^2 \):

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

Now substitute back:

\[
f(x+h) = 3(x^2 + 2xh + h^2) - 5(x+h) - 9
\]

Expand the terms:

\[
f(x+h) = 3x^2 + 6xh + 3h^2 - 5x - 5h - 9
\]

### Step 2: Subtract \( f(x) \)
Now subtract \( f(x) = 3x^2 - 5x - 9 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 5x - 5h - 9) - (3x^2 - 5x - 9)
\]

Simplify the expression:

\[
f(x+h) - f(x) = 6xh + 3h^2 - 5h
\]

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

\[
\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 - 5h}{h}
\]

Simplify:

\[
\frac{f(x+h) - f(x)}{h} = 6x + 3h - 5
\]

Thus, the difference quotient is:

\[
6x + 3h - 5
\]

Would you like further details or have any questions?

### Related Questions:
1. How does the difference quotient relate to the derivative?
2. What is the significance of \( h \) in the difference quotient formula?
3. How would the difference quotient change if \( f(x) \) were a cubic function?
4. Can we simplify the difference quotient when \( h = 0 \)?
5. How can we use the difference quotient to find the slope of the tangent line?

### Tip:
The difference quotient is a key concept for understanding the derivative, as \( h \) approaches 0.

Let f(x)=3x^2−5x−9. Find the difference quotient from x to x+h.

Explore how to calculate the difference quotient for the function f(x) = 3x^2 - 5x - 9. This detailed step-by-step guide covers key calculus concepts, such as the difference quotient formula and polynomial expansions, suitable for high school students in Grades 11-12.

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