The graph of \( f(x)=-2 x-8 \) and \( g(x)=-2 x^{2}-8 x \) are represented in the diagrem, wherein \( f(x) \) and \( g(x) \) intersect at \( (-4 ; 0) \) and at \( (1 ;-10) \) belos:

To find the intersections of linear and quadratic functions like \( f(x) \) and \( g(x) \), we can set them equal to each other. By solving the equation \( -2x - 8 = -2x^2 - 8x \), we can simplify it to \( -2x^2 - 6x - 8 = 0 \). Dividing through by -2 gives the easier form \( x^2 + 3x + 4 = 0 \). From here, we can use the quadratic formula to solve for \( x \), allowing us to confirm or explore the intersection points graphically or numerically. In the context of real-world applications, the intersections of \( f(x) \) and \( g(x) \) might represent different rates of change or profit and cost scenarios in business. For instance, if \( f(x) \) represents a straight line of income from sales and \( g(x) \) represents the increasing costs associated with production, the points of intersection can help determine break-even points – essential information for making business decisions!