Question

f(b)-f(a)/b-a to calculate the average rate of change over the interval [-2,-1] given the function table y =3^x+2

Answer 1

To calculate the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\), we can use the formula for the average rate of change:

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

Here, \( a = -2 \) and \( b = -1 \). We first need to find \( f(-2) \) and \( f(-1) \).

1. **Calculate \( f(-2) \)**:
\[
f(-2) = 3^{-2} + 2 = \frac{1}{3^2} + 2 = \frac{1}{9} + 2 = \frac{1}{9} + \frac{18}{9} = \frac{19}{9}
\]

2. **Calculate \( f(-1) \)**:
\[
f(-1) = 3^{-1} + 2 = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}
\]

3. **Now plug \( f(-2) \) and \( f(-1) \) into the average rate of change formula**:
\[
\text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{\frac{7}{3} - \frac{19}{9}}{-1 + 2}
\]

4. **Simplify the denominator**:
\[
-1 + 2 = 1
\]

5. **Now simplify the numerator** (\( \frac{7}{3} - \frac{19}{9} \)):
To subtract these fractions, we need a common denominator. The least common denominator between 3 and 9 is 9.
\[
\frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9}
\]
Thus,
\[
\frac{7}{3} - \frac{19}{9} = \frac{21}{9} - \frac{19}{9} = \frac{21 - 19}{9} = \frac{2}{9}
\]

6. **Now, substitute back**:
\[
\text{Average Rate of Change} = \frac{\frac{2}{9}}{1} = \frac{2}{9}
\]

Therefore, the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\) is \(\frac{2}{9}\).