Inside Collection (Textbook): Advanced Algebra II: Conceptual Explanations

Summary: This module covers how to solve problems through the graphic of quadratic functions.

Surprisingly, there is a fairly substantial class of real world problems that can be solved by graphing quadratic functions.

These problems are commonly known as “optimization problems” because they involve the question: “When does this important function reach its *maximum*?” (Or sometimes, its *minimum*?) In real life, of course, there are many things we want to maximize—a company wants to maximize its revenue, a baseball player his batting average, a car designer the leg room in front of the driver. And there are many things we want to minimize—a company wants to minimize its costs, a baseball player his errors, a car designer the amount of gas used. Mathematically, this is done by writing a function for that quantity and finding where that function reaches its highest or lowest point.

If a company manufactures
*minimize* its *average cost per item*?

“Average cost per item” is the total cost, divided by the number of items. For instance, if it costs $600 to manufacture 50 items, then the average cost per item was $12. It is important for companies to minimize average cost because this enables them to sell at a low price.

In this case, the total cost is
*average cost per item* is

What the question is asking, mathematically, is: what value of
*lowest*?

Well, suppose we were to graph this function. We would complete the square by rewriting it as:

The graph opens up (since the

I have graphed only the first quadrant, because negative values are not relevant for this problem (why?).

The real question here is, what can we learn from that graph? Every point on that graph represents one possibility for our company: if they manufacture

The point
*lowest point on the graph*. It is possible for

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