Question

let g(x)=2x^2 - 7. (a) find the average rate of change from - 9 to 8. (b) find an equation of the secant line containing (-9,g(-9)) and (8,g(8)). (a) the average rate of change from - 9 to 8 is (square). (simplify your answer.)

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ from $x = a$ to $x = b$ is given by $\frac{g(b)-g(a)}{b - a}$. Here, $a=-9$, $b = 8$, and $g(x)=2x^{2}-7$.

Step2: Calculate $g(-9)$ and $g(8)$

$g(-9)=2(-9)^{2}-7=2\times81 - 7=162-7 = 155$.
$g(8)=2(8)^{2}-7=2\times64 - 7=128 - 7=121$.

Step3: Compute the average rate of change

$\frac{g(8)-g(-9)}{8-(-9)}=\frac{121 - 155}{8 + 9}=\frac{-34}{17}=-2$.

Step4: Recall the point - slope form for the secant line

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. We know $m=-2$, and we can use the point $(-9,g(-9))=(-9,155)$.

Step5: Write the equation of the secant line

$y - 155=-2(x + 9)$.
Expand to get $y-155=-2x-18$.
Then $y=-2x + 137$.

Answer:

(a) - 2
(b) $y=-2x + 137$