Question

Find the average rate of change for f(x) = x2
+ 7x + 10 from x = −20 to x = −15.
Responses
A −28−28
B −36−36
C 2828
D 36

Answers

Answered by GPT-4o 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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