question given ( f(x)=5 x^{2}+4 x - 4 ), find the average rate of change of ( f(x) ) from ( x=-1 ) to ( x =…

question given ( f(x)=5 x^{2}+4 x - 4 ), find the average rate of change of ( f(x) ) from ( x=-1 ) to ( x = t ). give your answer in terms of ( t ). simplify your answer. provide your answer below:

question given ( f(x)=5 x^{2}+4 x - 4 ), find the average rate of change of ( f(x) ) from ( x=-1 ) to ( x = t ). give your answer in terms of ( t ). simplify your answer. provide your answer below:

Answer

Answer:

$5t - 1$

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=-1) and (b = t), and (f(x)=5x^{2}+4x - 4).

Step2: Calculate (f(-1))

Substitute (x=-1) into (f(x)): [ \begin{align*} f(-1)&=5(-1)^{2}+4(-1)-4\ &=5\times1-4 - 4\ &=5-8\ &=-3 \end{align*} ]

Step3: Calculate (f(t))

Substitute (x = t) into (f(x)): (f(t)=5t^{2}+4t-4)

Step4: Substitute into the average - rate - of - change formula

[ \begin{align*} \frac{f(t)-f(-1)}{t-(-1)}&=\frac{(5t^{2}+4t - 4)-(-3)}{t + 1}\ &=\frac{5t^{2}+4t-4 + 3}{t + 1}\ &=\frac{5t^{2}+4t-1}{t + 1} \end{align*} ]

Step5: Factor the numerator

Factor (5t^{2}+4t-1=(5t - 1)(t + 1))

Step6: Simplify the expression

[ \begin{align*} \frac{(5t - 1)(t + 1)}{t + 1}&=5t-1\quad(t\neq - 1) \end{align*} ]