Find the average rate of change between $$ x=-1 $$ and $$ x=2 $$, given the function, $$ f(x)=2 x^{2}+4 x $$

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To find the average rate of change of the function $$ f(x) = 2x^2 + 4x $$ between $$ x = -1 $$ and $$ x = 2 $$, we use the formula for the average rate of change:

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

where $$ a = -1 $$ and $$ b = 2 $$.

First, we need to calculate $$ f(-1) $$ and $$ f(2) $$.

1. Calculate $$ f(-1) $$:
   $$
   f(-1) = 2(-1)^2 + 4(-1) = 2(1) - 4 = 2 - 4 = -2
   $$

2. Calculate $$ f(2) $$:
   $$
   f(2) = 2(2)^2 + 4(2) = 2(4) + 8 = 8 + 8 = 16
   $$

Now we can substitute these values into the average rate of change formula:

$$
\text{Average Rate of Change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{16 - (-2)}{2 + 1} = \frac{16 + 2}{3} = \frac{18}{3} = 6
$$

Thus, the average rate of change of the function $$ f(x) = 2x^2 + 4x $$ between $$ x = -1 $$ and $$ x = 2 $$ is

$$
\boxed{6}
$$
The average rate of change is 6.

Find the average rate of change between \( x=-1 \) and \( x=2 \), given the function, \( f(x)=2 x^{2}+4 x \)

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