Question

the function f(x)=x² + 2x - 48 is graphed below. determine the slope of the secant line of f for each of the intervals indicated in the table. find the average rate of change of f on the interval -1,x₂ for each value of x₂ shown in the table. write your answers as decimals.

x₂	average roc

|0|
|-0.9|
|-0.99|
|-0.999|

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change (ROC) of a function $y = f(x)$ on the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$ and $b = x_2$, and $f(x)=x^{2}+2x - 48$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$:
$f(-1)=(-1)^{2}+2\times(-1)-48=1 - 2-48=-49$.

Step3: When $x_2 = 0$

First, find $f(0)$: $f(0)=0^{2}+2\times0 - 48=-48$.
Then, calculate the average ROC: $\frac{f(0)-f(-1)}{0-(-1)}=\frac{-48-(-49)}{1}=\frac{-48 + 49}{1}=1$.

Step4: When $x_2=-0.9$

Find $f(-0.9)=(-0.9)^{2}+2\times(-0.9)-48=0.81-1.8 - 48=-48.99$.
Calculate the average ROC: $\frac{f(-0.9)-f(-1)}{-0.9-(-1)}=\frac{-48.99-(-49)}{-0.9 + 1}=\frac{-48.99 + 49}{0.1}=0.01\div0.1 = 0.1$.

Step5: When $x_2=-0.99$

Find $f(-0.99)=(-0.99)^{2}+2\times(-0.99)-48=0.9801-1.98-48=-48.9999$.
Calculate the average ROC: $\frac{f(-0.99)-f(-1)}{-0.99-(-1)}=\frac{-48.9999-(-49)}{-0.99 + 1}=\frac{-48.9999 + 49}{0.01}=0.0001\div0.01 = 0.01$.

Step6: When $x_2=-0.999$

Find $f(-0.999)=(-0.999)^{2}+2\times(-0.999)-48=0.998001-1.998-48=-48.999999$.
Calculate the average ROC: $\frac{f(-0.999)-f(-1)}{-0.999-(-1)}=\frac{-48.999999-(-49)}{-0.999 + 1}=\frac{-48.999999 + 49}{0.001}=0.000001\div0.001 = 0.001$.

Answer:

$x_2$	Average ROC
-0.9	0.1
-0.99	0.01
-0.999	0.001