question\ngiven ( f(x)=-2 x^{2}+4 x-3 ), find the average rate of change of ( f(x) ) from ( x=-3 ) to ( x=t…

question\ngiven ( f(x)=-2 x^{2}+4 x-3 ), find the average rate of change of ( f(x) ) from ( x=-3 ) to ( x=t ).\ngive your answer in terms of ( t ). simplify your answer.\nprovide your answer below:

question\ngiven ( f(x)=-2 x^{2}+4 x-3 ), find the average rate of change of ( f(x) ) from ( x=-3 ) to ( x=t ).\ngive your answer in terms of ( t ). simplify your answer.\nprovide your answer below:

Answer

Answer:

$-2t - 2$

Explanation:

Step1: Recall the average rate of change formula

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=-3) and (b = t), and (f(x)=-2x^{2}+4x - 3).

Step2: Find (f(t)) and (f(-3))

  • For (f(t)): Substitute (x=t) into (f(x)), we get (f(t)=-2t^{2}+4t - 3).
  • For (f(-3)): Substitute (x = - 3) into (f(x)): [ \begin{align*} f(-3)&=-2(-3)^{2}+4(-3)-3\ &=-2\times9-12 - 3\ &=-18-12-3\ &=-33 \end{align*} ]

Step3: Calculate (f(t)-f(-3))

[ \begin{align*} f(t)-f(-3)&=(-2t^{2}+4t - 3)-(-33)\ &=-2t^{2}+4t - 3 + 33\ &=-2t^{2}+4t+30 \end{align*} ]

Step4: Calculate the average rate of change (\frac{f(t)-f(-3)}{t-(-3)})

[ \begin{align*} \frac{f(t)-f(-3)}{t + 3}&=\frac{-2t^{2}+4t + 30}{t + 3}\ &=\frac{-2(t^{2}-2t - 15)}{t + 3}\ &=\frac{-2(t-5)(t + 3)}{t + 3}\ &=-2t-2 \end{align*} ]