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The problem is asking for two things:

1. **Average Rate of Change of \( y = f(x) = x^2 - 3x \) on the interval [2, 4]**.
   
   The formula for the average rate of change of a function \( f(x) \) over an interval \([a, b]\) is:

   \[
   \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
   \]

   Here, \( a = 2 \) and \( b = 4 \). Let's calculate:

   \[
   f(4) = 4^2 - 3(4) = 16 - 12 = 4
   \]
   \[
   f(2) = 2^2 - 3(2) = 4 - 6 = -2
   \]

   Now, applying the average rate of change formula:

   \[
   \frac{f(4) - f(2)}{4 - 2} = \frac{4 - (-2)}{2} = \frac{4 + 2}{2} = \frac{6}{2} = 3
   \]

   So, the average rate of change is \( 3 \).

2. **Instantaneous Rate of Change (Derivative) at \( x = 2 \)**.
   
   The instantaneous rate of change is the derivative of \( f(x) = x^2 - 3x \). Let's first find \( f'(x) \):

   \[
   f'(x) = 2x - 3
   \]

   Now, substitute \( x = 2 \) into \( f'(x) \):

   \[
   f'(2) = 2(2) - 3 = 4 - 3 = 1
   \]

   So, the instantaneous rate of change at \( x = 2 \) is \( 1 \).

### Summary:
- Average rate of change over the interval [2, 4] is \( 3 \).
- Instantaneous rate of change at \( x = 2 \) is \( 1 \).

Do you need any additional details or have any questions?

Here are 5 related questions to expand on this:
1. How does the average rate of change differ from the instantaneous rate of change in terms of graphical interpretation?
2. How would the average rate of change change if the interval were [1, 3] instead of [2, 4]?
3. Can the instantaneous rate of change be negative for this function? If so, at what values of \( x \)?
4. How does the derivative of the function relate to its graphical slope at any point?
5. What does the second derivative of this function tell us about its concavity?

**Tip:** The average rate of change is geometrically the slope of the secant line between two points on the curve, while the instantaneous rate of change is the slope of the tangent line at a single point.

Let y = f(x) = x^2 - 3x. Determine the average rate of change of y with respect to x across the interval x = 2 to x = 4 and the instantaneous rate of change of y at x = 2.

This problem covers finding the average rate of change of the function y = x^2 - 3x over the interval [2, 4] and the instantaneous rate of change at x = 2. It includes step-by-step solutions using derivatives and the difference quotient.

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