describe and correct the error a student made...

describe and correct the error a student made in finding the average rate of change for f(x)=0.5x² over the interval -4≤x≤ -2. find the slope of the line that passes through (-4, -8) and (-2, -2). (-2 - (-8))/(-2 - (-4)) = 6/2 = 3 x a. 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.) 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 (-4,8),(-2,2). (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 denominator. the correct slope is. (type an integer or a simplified fraction.) correct the students error. the correct average rate of change is. (type an integer or a simplified fraction.)

Answer

# Explanation: ## Step1: Find function values at endpoints 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 \((-2,2)\). ## Step2: Calculate the average rate of change The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here \(a=-4\), \(b = - 2\), \(f(-4)=8\), \(f(-2)=2\). So the average rate of change is \(\frac{2 - 8}{-2-(-4)}=\frac{-6}{2}=-3\). # Answer: C. \((-4,8),(-2,2)\) The correct average rate of change is \(-3\)