Focal Points and Mapping Diagrams




The point of this project is to organize and present the material on Linear Maps, Focal Points, and Mapping Diagrams. Here are some definitions and ideas you may want to include.




Definition: A Mapping Diagram for the function $f$ consists of the following.

  1. Two parallel vertical lines, the left hand one called the input axis and the righthand one called the output axis. Coordinate systems are set up on these lines, each with an origin and with positive coordinates on top, negatives below. It is generally assumed that the scales on both axes are the same.

  2. One or more lines, each connecting some $x$ on the input axis, with $f(x)$ on the output axis. These are called mapping lines, and are denoted, for example, $x\rightarrow f(x)$, $u\rightarrow f(u)$, etc.




Definition: A mapping diagram is said to have a focal point if either of the following conditions hold.

  1. There is a point $P$ such that every possible mapping line from $x$ to $f(x)$ passes through $P$, OR

  2. All the mapping lines are parallel. If this is the case, we say that the focal point is at infinity.

Theorem A: If a function is linear (i.e. is of the form $f(x)=ax+b$), then its mapping diagram has a focal point (which may be at infinity).

Theorem B: If the mapping diagram of a function has a focal point, then the function is linear.




NOTES:

  1. Theorem B is the converse of Theorem A.

  2. To prove Theorem A, suppose that the line $x=0$ is the input axis, and the line $x=1$ is the output axis; let $f(t)=at+b$. Find the equations of the mapping lines: $u\rightarrow au+b$ (i.e. joining $(0,u)$ with $(1,au+b)$ and $v\rightarrow av+b$, and calculate the point where they intersect. Explain why this point must line on every mapping line.

  3. To prove Theorem B, set up the input and output axes as above, and suppose that $P=(p,q)$ is a focal point. Find the equation of the line passing through the input $t$, represented as $\left( 0,t\right) $, and $P$, and see where it hits the output axis; this must be the point $(1,f(t))$. Compute this point and show that $f(t)$ must be of the form $at+b$ (where $a$ and $b$ will be in terms of $p$ and $q$).




Finally, discuss, for linear functions $at+b$, the relation between the position of the focal point (between or outside the input/output axes and how close it is to them), the value of $a$ (positive, negative, and the size of MATH), and whether the function magnifies or contracts lengths. You can express this in sentences, or make a table.