question
given ( f(x)=-2 x^{2}+4 x-3 ), 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:
$-2t - 2$
# 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 $\frac{f(t)-f(-3)}{t-(-3)}$
$$
\begin{align*}
\frac{f(t)-f(-3)}{t + 3}&=\frac{-2t^{2}+4t + 30}{t + 3}\
&=\frac{-2(t^{2}-2t - 15)}{t + 3}\
&=\frac{-2(t-5)(t + 3)}{t + 3}\
&=-2t-2
\end{align*}
$$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the average rate of change formula

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

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

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

Answer: