question
given ( f(x)=-x^{2}+5x - 1 ), find the average rate of change of ( f(x) ) from ( x = 3 ) to ( x = t ).
give your answer in terms of ( t ).simplify your answer.
provide your answer below:

Question

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

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

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

Step3: Compute the average rate of change

Answer: