the regions a, b, c, and d in the figure below are bounded by the graph of the function f and the x - axis…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a=-8$, $b = 8$, so $b - a=8-(-8)=16$.

Step2: Determine the definite - integral value

The definite integral $\int_{-8}^{8}f(x)dx$ is the net - signed area between the graph of $y = f(x)$ and the $x$ - axis. Areas above the $x$ - axis are positive and areas below the $x$ - axis are negative. So, $\int_{-8}^{8}f(x)dx=A - B + C - D$. Given $A = 12$, $B = 6$, $C = 7$, $D = 5$, then $\int_{-8}^{8}f(x)dx=12-6 + 7-5=8$.

Step3: Calculate the average value

Using the average - value formula $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$, substitute $b - a = 16$ and $\int_{-8}^{8}f(x)dx = 8$. We get $\bar{y}=\frac{8}{16}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$