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what is the slope as the difference quotient

The function shown in the image is:

\[
f(x) = 4x^2 - 5x + 7
\]

To find the slope of the function using the **difference quotient**, we need to compute the expression:

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

### Steps to compute the difference quotient:

1. **Find** \( f(x+h) \):
   \[
   f(x+h) = 4(x+h)^2 - 5(x+h) + 7
   \]
   Expanding this:
   \[
   f(x+h) = 4(x^2 + 2xh + h^2) - 5(x + h) + 7
   \]
   Simplify further:
   \[
   f(x+h) = 4x^2 + 8xh + 4h^2 - 5x - 5h + 7
   \]

2. **Subtract** \( f(x) \) from \( f(x+h) \):
   \[
   f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 5x - 5h + 7) - (4x^2 - 5x + 7)
   \]
   Simplifying:
   \[
   f(x+h) - f(x) = 8xh + 4h^2 - 5h
   \]

3. **Divide by** \( h \):
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 - 5h}{h}
   \]
   Factor out \( h \):
   \[
   \frac{f(x+h) - f(x)}{h} = 8x + 4h - 5
   \]

### Conclusion:

The **difference quotient** is:

\[
8x + 4h - 5
\]

As \( h \to 0 \), the slope of the function (the derivative) is:

\[
f'(x) = 8x - 5
\]

Would you like further explanation or more details on this process?

Here are some related questions:
1. What is the interpretation of the derivative in this context?
2. How would you calculate the slope at a specific point using this derivative?
3. How does the difference quotient relate to the concept of limits?
4. Can the difference quotient be applied to non-polynomial functions?
5. What role does \( h \) play in the difference quotient?

**Tip:** The difference quotient is a fundamental concept in calculus as it represents the average rate of change of the function, and in the limit, it gives the instantaneous rate of change.

What is the slope as the difference quotient for f(x) = 4x^2 - 5x + 7?

Learn how to find the slope of the quadratic function f(x) = 4x^2 - 5x + 7 using the difference quotient formula. This problem explores the concepts of derivatives and limits, suitable for high school students in Grades 10-12.

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