find the slope of the graph of y=f(x) at the designated point, f(x)=2x^2-7x + 2;(3, -1), the slope of the graph of y=f(x) at (3, -1) is?

To find the slope of the graph of \( y = f(x) \) at the designated point \( (3, -1) \), we need to calculate the derivative of \( f(x) = 2x^2 - 7x + 2 \) and then evaluate it at \( x = 3 \). The derivative represents the slope of the tangent line to the graph at a given point.

### Steps:

1. **Find the derivative of \( f(x) \):**

\[
f'(x) = \frac{d}{dx}(2x^2 - 7x + 2)
\]
Using basic differentiation rules:
\[
f'(x) = 4x - 7
\]

2. **Evaluate the derivative at \( x = 3 \):**

\[
f'(3) = 4(3) - 7 = 12 - 7 = 5
\]

### Conclusion:
The slope of the graph of \( y = f(x) \) at the point \( (3, -1) \) is **5**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How is the derivative of a polynomial function generally computed?
2. What does the derivative tell you about the behavior of a function?
3. How do you find the equation of the tangent line using the slope at a point?
4. How do we differentiate more complex functions, like products or quotients of functions?
5. What is the geometric interpretation of the second derivative?

**Tip**: Always verify whether the point given lies on the curve by substituting it into the original function!

Find the slope of the graph of y=f(x) at the designated point, f(x)=2x^2-7x + 2;(3, -1), the slope of the graph of y=f(x) at (3, -1) is?

Learn how to find the slope of the graph of a quadratic function at a specific point using derivatives. This example involves the function f(x) = 2x^2 - 7x + 2, and the calculation of its slope at point (3, -1). Suitable for high school students in Grades 10-12.

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