let g(x)=7x² - 8. (a) find the average rate of change from - 5 to 7. (b) find an equation of the secant line…

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 $\frac{g(b)-g(a)}{b - a}$. Here, $a=-5$, $b = 7$, and $g(x)=7x^{2}-8$.

Step2: Calculate $g(-5)$

Substitute $x=-5$ into $g(x)$: $g(-5)=7\times(-5)^{2}-8=7\times25 - 8=175-8 = 167$.

Step3: Calculate $g(7)$

Substitute $x = 7$ into $g(x)$: $g(7)=7\times7^{2}-8=7\times49-8=343 - 8=335$.

Step4: Calculate the average rate of change

$\frac{g(7)-g(-5)}{7-(-5)}=\frac{335 - 167}{7 + 5}=\frac{168}{12}=14$.

Answer:

14