given the function ( f(x)=-2x - 5 ).\nfind the average rate of change of ( f ) from ( x=-2 ) to ( x = 3…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) from (x=a) to (x = b) is given by (\frac{f(b)-f(a)}{b - a}). Here, (a=-2) and (b = 3).

Step2: Calculate (f(-2))

Substitute (x=-2) into (f(x)=-2x^{2}-5). [ \begin{align*} f(-2)&=-2\times(-2)^{2}-5\ &=-2\times4 - 5\ &=-8-5\ &=-13 \end{align*} ]

Step3: Calculate (f(3))

Substitute (x = 3) into (f(x)=-2x^{2}-5). [ \begin{align*} f(3)&=-2\times3^{2}-5\ &=-2\times9-5\ &=-18 - 5\ &=-23 \end{align*} ]

Step4: Calculate the average rate of change

Substitute (f(-2)=-13), (f(3)=-23), (a=-2), and (b = 3) into the formula (\frac{f(b)-f(a)}{b - a}). [ \begin{align*} \frac{f(3)-f(-2)}{3-(-2)}&=\frac{-23-(-13)}{3 + 2}\ &=\frac{-23 + 13}{5}\ &=\frac{-10}{5}\ &=-2 \end{align*} ]

Answer:

(-2)