Bot GPT 4-o mini To find the average rate of change of the function \( f(x) = x^2 + 7x + 10 \) from \( x = -20 \) to \( x = -15 \), we can use the formula:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = -20 \) and \( b = -15 \).
First, we need to calculate \( f(-20) \) and \( f(-15) \).
Calculating \( f(-20) \): \[ f(-20) = (-20)^2 + 7(-20) + 10 \] \[ = 400 - 140 + 10 \] \[ = 400 - 140 + 10 = 270 \]
Calculating \( f(-15) \): \[ f(-15) = (-15)^2 + 7(-15) + 10 \] \[ = 225 - 105 + 10 \] \[ = 225 - 105 + 10 = 130 \]
Now we have:
- \( f(-20) = 270 \)
- \( f(-15) = 130 \)
Now, we can plug these values into the average rate of change formula:
\[ \text{Average rate of change} = \frac{f(-15) - f(-20)}{-15 - (-20)} \] \[ = \frac{130 - 270}{-15 + 20} \] \[ = \frac{130 - 270}{5} \] \[ = \frac{-140}{5} = -28 \]
Therefore, the average rate of change for \( f(x) \) from \( x = -20 \) to \( x = -15 \) is \( -28 \).
The correct response is:
A) -28
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