Given the following function, determine the difference quotient, f(x+h)−f(x)h
.

f(x)=3x2+7x−8

The difference quotient is defined as:

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

where \( f(x) = 3x^2 + 7x - 8 \).

Let's calculate this step by step.

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

\[
f(x+h) = 3(x+h)^2 + 7(x+h) - 8
\]

Expand the terms:

\[
f(x+h) = 3(x^2 + 2xh + h^2) + 7(x + h) - 8
\]
\[
f(x+h) = 3x^2 + 6xh + 3h^2 + 7x + 7h - 8
\]

### Step 2: Compute \( f(x+h) - f(x) \)
Now subtract \( f(x) = 3x^2 + 7x - 8 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = \left( 3x^2 + 6xh + 3h^2 + 7x + 7h - 8 \right) - \left( 3x^2 + 7x - 8 \right)
\]

Simplify by canceling out common terms:

\[
f(x+h) - f(x) = 6xh + 3h^2 + 7h
\]

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

\[
\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 7h}{h}
\]

Simplify:

\[
\frac{f(x+h) - f(x)}{h} = 6x + 3h + 7
\]

### Final Answer:
The difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = 6x + 3h + 7
\]

Let me know if you'd like more details or have any questions!

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Here are 5 related questions:
1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as \( h \to 0 \)?
3. How would you find the derivative of \( f(x) = 3x^2 + 7x - 8 \)?
4. Can you calculate the difference quotient for a cubic function?
5. What is the geometric interpretation of the difference quotient?

**Tip:** The difference quotient is the foundation for finding derivatives, which describe the slope of a curve at any given point.

Given the following function, determine the difference quotient, f(x+h)−f(x)/h. f(x) = 3x^2 + 7x - 8.

Calculate the difference quotient for the quadratic function f(x) = 3x^2 + 7x - 8 with a step-by-step solution. This problem covers the algebraic expansion of functions and the fundamentals of the difference quotient, crucial for understanding derivatives. Suitable for high school students in Grades 10-12.

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