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.

In this case, the total cost is x3−10x2+43xx3−10x2+43x size 12{x rSup { size 8{3} } - "10"x rSup { size 8{2} } +"43"x} {} and the number of items is xx size 12{x} {}. So the average cost per item is

What the question is asking, mathematically, is: what value of xx size 12{x} {} makes this function the lowest?

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

The graph opens up (since the x−52x−52 size 12{ left (x - 5 right ) rSup { size 8{2} } } {}term is positive), and has its vertex at 5,185,18 size 12{ left (5,"18" right )} {} (since it is moved 5 to the right and 18 up). So it would look something like this:

Figure 1

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 xx size 12{x} {} items, the graph shows what A(x)A(x) size 12{A \( x \) } {}, the average cost per item, will be.

The point 5,185,18 size 12{ left (5,"18" right )} {} is the lowest point on the graph. It is possible for xx size 12{x} {} to be higher or lower than 5, but it is never possible for AA size 12{A} {} to be lower than 18. So if its goal is to minimize average cost, their best strategy is to manufacture 5 items, which will bring their average cost to $18/item.

Content actions

Add module to:

My Favorites Login Required (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own Login Required (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Footer

This work is licensed by Kenny M. Felder under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Kenny M. Felder on Dec 30, 2008 4:49 pm -0600.