Find the slope of the graph of yequals​f(x) at the designated point.
​f(x)equals3 x squared minus 5 x plus 4​; left parenthesis 2 comma 6 right parenthesis
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Part 1
The slope of the graph of y equals f left parenthesis x right parenthesis at left parenthesis 2 comma 6 right parenthesis is 
  
enter your response here.

To find the slope of the graph of \( y = f(x) \) at the point \( (2, 6) \), we need to calculate the derivative of \( f(x) \) and evaluate it at \( x = 2 \).

Given the function:

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

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

   Differentiate the function with respect to \( x \):

   \[
   f'(x) = \frac{d}{dx} (3x^2) - \frac{d}{dx} (5x) + \frac{d}{dx} (4)
   \]

   Simplifying this, we get:

   \[
   f'(x) = 6x - 5
   \]

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

   Substitute \( x = 2 \) into the derivative:

   \[
   f'(2) = 6(2) - 5
   \]

   \[
   f'(2) = 12 - 5
   \]

   \[
   f'(2) = 7
   \]

Therefore, the slope of the graph of \( y = f(x) \) at the point \( (2, 6) \) is **7**.

Would you like more details on this solution, or do you have any questions? Here are some related questions that might interest you:

1. How do you find the slope of a curve at any given point?
2. What does the derivative of a function represent geometrically?
3. How can you verify that a point lies on a given curve?
4. What is the significance of a positive slope in a function's graph?
5. How do you find the equation of the tangent line to a curve at a given point?

**Tip:** The slope of the tangent line at a point on a curve gives the instantaneous rate of change of the function at that specific point.

Find the slope of the graph of y = f(x) at the designated point. f(x) = 3x^2 - 5x + 4; (2, 6).

Learn how to find the slope of the graph of y = f(x) at a specific point. This problem involves calculus concepts, such as derivatives and the slope of a curve, suitable for students in Grades 11-12.

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