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To find the average rate of change of \( f(x) = 2x^2 - 9 \) between the given pairs of points, we use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

### Part (a):
The points are \( (1, -7) \) and \( (3, 9) \), where these represent the coordinates \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \).

The values for \( x_1 = 1 \) and \( x_2 = 3 \), so:

\[
f(x_1) = f(1) = 2(1)^2 - 9 = 2 - 9 = -7
\]

\[
f(x_2) = f(3) = 2(3)^2 - 9 = 18 - 9 = 9
\]

Thus, the average rate of change is:

\[
\frac{f(3) - f(1)}{3 - 1} = \frac{9 - (-7)}{3 - 1} = \frac{9 + 7}{2} = \frac{16}{2} = 8
\]

### Part (b):
For this part, we are asked to find the average rate of change between \( (x, f(x)) \) and \( (x + h, f(x + h)) \).

The average rate of change formula is:

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

Substitute \( f(x) = 2x^2 - 9 \):

\[
f(x + h) = 2(x + h)^2 - 9 = 2(x^2 + 2xh + h^2) - 9 = 2x^2 + 4xh + 2h^2 - 9
\]

Now, the difference \( f(x + h) - f(x) \) is:

\[
f(x + h) - f(x) = (2x^2 + 4xh + 2h^2 - 9) - (2x^2 - 9) = 4xh + 2h^2
\]

So, the average rate of change is:

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

Thus, the average rate of change between \( (x, f(x)) \) and \( (x + h, f(x + h)) \) is \( 4x + 2h \).

---

Would you like further clarification or details on any part? Here are five additional related questions to explore:

1. How does the average rate of change relate to the derivative?
2. What is the significance of \( h \) in the second part, and what happens when \( h \to 0 \)?
3. How can we interpret the result \( 4x + 2h \) geometrically?
4. How would the average rate of change differ if the function was cubic instead of quadratic?
5. Can the average rate of change ever be negative, and under what conditions?

**Tip**: The average rate of change can be seen as the slope of the secant line connecting two points on the graph of the function. This concept generalizes to the derivative when the interval shrinks to zero.

Find the average rate of change of f(x) = 2x^2 - 9 between each of the pairs of points: (a) Between (1, -7) and (3, 9), and (b) Between (x, f(x)) and (x + h, f(x + h)).

Find the average rate of change for the quadratic function f(x) = 2x^2 - 9 between specific points. Learn how to apply the difference quotient and understand the relationship between the secant line and the function's slope. Suitable for high school calculus students.

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