Evaluate the difference quotient for f left parenthesis x right parenthesis equals 4 x squared minus 3 x plus 5.

To evaluate the **difference quotient** for the function \( f(x) = 4x^2 - 3x + 5 \), we use the formula for the difference quotient:

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

### Step 1: Find \( f(x+h) \)
Substitute \( x+h \) into the function \( f(x) = 4x^2 - 3x + 5 \):

\[
f(x+h) = 4(x+h)^2 - 3(x+h) + 5
\]

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

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

Now substitute this back into the function:

\[
f(x+h) = 4(x^2 + 2xh + h^2) - 3(x+h) + 5
\]

Expand the terms:

\[
f(x+h) = 4x^2 + 8xh + 4h^2 - 3x - 3h + 5
\]

### Step 2: Calculate the Difference Quotient
Now, subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3x - 3h + 5) - (4x^2 - 3x + 5)
\]

Simplify the expression:

\[
f(x+h) - f(x) = 8xh + 4h^2 - 3h
\]

Now, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 - 3h}{h}
\]

Simplify by canceling out \( h \) from each term:

\[
\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3
\]

### Final Answer:
The difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3
\]

Do you want more details on this process or have any questions?

Here are 5 related questions for further exploration:
1. What happens to the difference quotient as \( h \) approaches 0?
2. How do you interpret the difference quotient geometrically?
3. How would the difference quotient change if the function had a cubic term?
4. What is the derivative of \( f(x) = 4x^2 - 3x + 5 \) using limits?
5. Can you evaluate the difference quotient for \( f(x) = x^3 + 2x \)?

**Tip:** The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval.

Evaluate the difference quotient for f(x) = 4x^2 - 3x + 5.

Learn how to evaluate the difference quotient for the quadratic function f(x) = 4x^2 - 3x + 5. This step-by-step guide covers the key concepts of algebra and calculus, including function expansion and difference quotients. Suitable for high school students in Grades 11-12.

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