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

Accepted 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
$$
\begin{align*}
\frac{(5t - 1)(t + 1)}{t + 1}&=5t-1\quad(t
eq - 1)
\end{align*}
$$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the average rate of change formula

Step2: Calculate \(f(-1)\)

Step3: Calculate \(f(t)\)

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

Step5: Factor the numerator

Step6: Simplify the expression

Answer: