question\ngiven ( f(x)=-x^{2}+5x - 1 ), find the average rate of change of ( f(x) ) from ( x = 3 ) to ( x =…

question\ngiven ( f(x)=-x^{2}+5x - 1 ), 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)=-x^{2}+5x - 1 ), 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:

$-t + 2$

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 = 3) and (b=t), and (f(x)=-x^{2}+5x - 1).

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

  • Calculate (f(3)): Substitute (x = 3) into (f(x)): (f(3)=-(3)^{2}+5\times3-1=-9 + 15-1=5).
  • Calculate (f(t)): Substitute (x=t) into (f(x)): (f(t)=-t^{2}+5t - 1).

Step3: Compute the average rate of change

[ \begin{align*} \frac{f(t)-f(3)}{t - 3}&=\frac{(-t^{2}+5t - 1)-5}{t - 3}\ &=\frac{-t^{2}+5t-6}{t - 3}\ &=\frac{-(t^{2}-5t + 6)}{t - 3}\ &=\frac{-(t-2)(t - 3)}{t - 3}\ &=-t + 2 \end{align*} ]