Question

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

Explanation:

Step1: Find $f(2)$ and $f(t)$

Substitute $x = 2$ into $f(x)$: $f(2)=-2\times2^{2}+4=-8 + 4=-4$.
Substitute $x=t$ into $f(x)$: $f(t)=-2t^{2}+4$.

Step2: Use 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, $a = 2$, $b=t$, so the average rate of change is $\frac{f(t)-f(2)}{t - 2}$.
Substitute $f(2)=-4$ and $f(t)=-2t^{2}+4$ into the formula:
\[

$$\begin{align*} \frac{(-2t^{2}+4)-(-4)}{t - 2}&=\frac{-2t^{2}+4 + 4}{t - 2}\\ &=\frac{-2t^{2}+8}{t - 2}\\ &=\frac{-2(t^{2}-4)}{t - 2}\\ &=\frac{-2(t + 2)(t - 2)}{t - 2}\\ &=-2(t + 2) \end{align*}$$

\]

Answer:

$-2(t + 2)$