Question

describe and correct the error a student made in finding the average rate of change for ( f(x) = 0.5x^2 ) over the interval ( -4 leq x leq -2 )

find the slope of the
line that passes through
( (-4, -8) ) and ( (-2, -2) ).
( \frac{-2 - (-8)}{-2 - (-4)} = \frac{6}{2} = 3 )

describe the student’s error. select the correct choice below and fill in the answer box to complete your choice.

a. the points were correct, but the student made an error in calculating the denominator. the correct slope is
(type an integer or a simplified fraction.)

b. the points were correct and the student correctly simplified the numerator and denominator, but they did not simplify the fraction correctly. the correctly simplified fraction is
(type an integer or a simplified fraction.)

c. the points used to find the slope were incorrect. the correct points are
(type an ordered pair. use a comma to separate answers as needed.)

d. the points were correct, but the student made an error in calculating the numerator. the correct slope is
(type an integer or a simplified fraction.)

Explanation:

Step 1: Find the correct function values

To find the average rate of change of \( f(x) = 0.5x^2 \) over \(-4 \leq x \leq -2\), we first calculate \( f(-4) \) and \( f(-2) \).

For \( x = -4 \):
\( f(-4)=0.5\times(-4)^2 = 0.5\times16 = 8 \)

For \( x = -2 \):
\( f(-2)=0.5\times(-2)^2 = 0.5\times4 = 2 \)

So the correct points are \((-4, 8)\) and \((-2, 2)\), not \((-4, -8)\) and \((-2, -2)\) as the student used. Thus, the student's error is in the points (Option C).

Step 2: Calculate the average rate of change (slope)

The formula for the average rate of change (slope \( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Using \((x_1, y_1)=(-4, 8)\) and \((x_2, y_2)=(-2, 2)\):
\( m=\frac{2 - 8}{-2 - (-4)}=\frac{-6}{2}=-3 \)

(for the multiple-choice part):
The student used incorrect \( y \)-values (function outputs) for \( x = -4 \) and \( x = -2 \). The correct \( f(-4)=8 \) (not -8) and \( f(-2)=2 \) (not -2), so the points were wrong (Option C). The correct points are \((-4, 8)\) and \((-2, 2)\).

Answer:

C. The points used to find the slope were incorrect. The correct points are \((-4, 8)\), \((-2, 2)\)
The correct slope (average rate of change) is \(-3\) (if calculating the slope, but for the option, we just confirm the correct points and the error type as per the question's multiple - choice context)