Answer :
To find the average rate of change of the function \( f(x) = 2x + 9 \) from \( x_1 = -4 \) to \( x_2 = -2 \), we follow these steps:
1. Evaluate the function at \( x_1 \):
[tex]\[ f(x_1) = f(-4) = 2(-4) + 9 = -8 + 9 = 1 \][/tex]
So, \( f(-4) = 1 \).
2. Evaluate the function at \( x_2 \):
[tex]\[ f(x_2) = f(-2) = 2(-2) + 9 = -4 + 9 = 5 \][/tex]
So, \( f(-2) = 5 \).
3. Calculate the average rate of change:
The formula for the average rate of change between two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) is:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Substituting in the values we found:
[tex]\[ \text{Average rate of change} = \frac{5 - 1}{-2 - (-4)} = \frac{5 - 1}{-2 + 4} = \frac{4}{2} = 2 \][/tex]
So, the average rate of change of [tex]\( f(x) \)[/tex] from [tex]\( x_1 = -4 \)[/tex] to [tex]\( x_2 = -2 \)[/tex] is [tex]\( \boxed{2} \)[/tex].
1. Evaluate the function at \( x_1 \):
[tex]\[ f(x_1) = f(-4) = 2(-4) + 9 = -8 + 9 = 1 \][/tex]
So, \( f(-4) = 1 \).
2. Evaluate the function at \( x_2 \):
[tex]\[ f(x_2) = f(-2) = 2(-2) + 9 = -4 + 9 = 5 \][/tex]
So, \( f(-2) = 5 \).
3. Calculate the average rate of change:
The formula for the average rate of change between two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) is:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Substituting in the values we found:
[tex]\[ \text{Average rate of change} = \frac{5 - 1}{-2 - (-4)} = \frac{5 - 1}{-2 + 4} = \frac{4}{2} = 2 \][/tex]
So, the average rate of change of [tex]\( f(x) \)[/tex] from [tex]\( x_1 = -4 \)[/tex] to [tex]\( x_2 = -2 \)[/tex] is [tex]\( \boxed{2} \)[/tex].
