given the function ( f(x)=-4 x^{2}+5 x + 3 ).\nfind the average rate of change of ( f ) from ( x=-5 ) to ( x…

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 (\frac{f(b)-f(a)}{b - a}). Here, (a=-5) and (b = 4).

Step2: Calculate (f(-5))

Substitute (x=-5) into (f(x)=-4x^{2}+5x + 3). [ \begin{align*} f(-5)&=-4\times(-5)^{2}+5\times(-5)+3\ &=-4\times25-25 + 3\ &=-100-25+3\ &=-122 \end{align*} ]

Step3: Calculate (f(4))

Substitute (x = 4) into (f(x)=-4x^{2}+5x + 3). [ \begin{align*} f(4)&=-4\times4^{2}+5\times4+3\ &=-4\times16 + 20+3\ &=-64+20 + 3\ &=-41 \end{align*} ]

Step4: Calculate the average rate of change

Substitute (f(-5)=-122), (f(4)=-41), (a=-5), and (b = 4) into (\frac{f(b)-f(a)}{b - a}). [ \begin{align*} \frac{f(4)-f(-5)}{4-(-5)}&=\frac{-41-(-122)}{4 + 5}\ &=\frac{-41 + 122}{9}\ &=\frac{81}{9}\ &=9 \end{align*} ]

Answer:

(9)