Final answer:
The average rate of change of the function f(x) = -2x² + 3x + 4 from x = -3 to x = 2 is found by evaluating the function at both points and then computing the slope of the secant line. The result is an average rate of change of 5.
Step-by-step explanation:
The average rate of change of a function between two points is equivalent to the slope of the secant line that passes through these points on the graph of the function. To find the average rate of change of the function f(x) = -2x² + 3x + 4 from x = -3 to x = 2, we need to calculate the difference in the function values at these points and divide by the difference in x-values.
First, we find the function values:
- f(-3) = -2(-3)² + 3(-3) + 4 = -18 - 9 + 4 = -23
- f(2) = -2(2)² + 3(2) + 4 = -8 + 6 + 4 = 2
Next, we calculate the average rate of change using the formula:
\[ \text{Average Rate of Change} = \frac{f(2) - f(-3)}{2 - (-3)} \]
Substituting the values we found:
\[ \text{Average Rate of Change} = \frac{2 - (-23)}{2 - (-3)} = \frac{25}{5} = 5 \]
Therefore, the average rate of change of the function f(x) from x = -3 to x = 2 is 5 units per unit interval.