let f(x)=x² + 3x - 4. what is the average rate of change for the quadratic function from x = -3 to x = 8?

Answer

Explanation:

Step1: Recall 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, $a=-3$, $b = 8$, and $f(x)=x^{2}+3x - 4$.

Step2: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)$: [ \begin{align*} f(-3)&=(-3)^{2}+3\times(-3)-4\ &=9-9 - 4\ &=-4 \end{align*} ]

Step3: Calculate $f(8)$

Substitute $x = 8$ into $f(x)$: [ \begin{align*} f(8)&=8^{2}+3\times8-4\ &=64 + 24-4\ &=84 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(8)-f(-3)}{8-(-3)}&=\frac{84-(-4)}{8 + 3}\ &=\frac{84 + 4}{11}\ &=\frac{88}{11}\ &=8 \end{align*} ]